%------------------------------------------------------------------------------
% File     : ITP214_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Hoare_Triple 00428_012651
% 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    : 0025_Hoare_Triple_00428_012651 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.1.0
% Syntax   : Number of formulae    : 15849 (3984 unt;4511 typ;   0 def)
%            Number of atoms       : 27975 (8886 equ)
%            Maximal formula atoms :   38 (   2 avg)
%            Number of connectives : 18184 (2318   ~; 488   |;1542   &)
%                                         (2929 <=>;10907  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       : 2447 ( 785 fml;1662 var)
%            Number of X terms     :  655 (   0  []; 622 ite;  33 let)
%            Number of types       : 1004 (1003 usr)
%            Number of type conns  : 4588 (2904   >;1684   *;   0   +;   0  <<)
%            Number of predicates  :  218 ( 215 usr;   2 prp; 0-4 aty)
%            Number of functors    : 3303 (3303 usr; 614 con; 0-8 aty)
%            Number of variables   : 30210 (29658   !; 552   ?;30210   :)
% 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 15:41:23.176
%------------------------------------------------------------------------------
% Could-be-implicit typings (1003)
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tff(ty_n_t__fun_It__Int__Oint_Mt__Set__Oset_I_Eo_J_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Onatural_M_Eo_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Onatural_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(ty_n_t__Option__Ooption_It__Assertions__Oassn_J,type,
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tff(ty_n_t__Heap__Oref_It__Product____Type__Ounit_J,type,
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tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
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tff(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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tff(ty_n_t__Option__Ooption_It__Set__Oset_I_Eo_J_J,type,
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tff(ty_n_t__fun_It__Product____Type__Ounit_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Product____Type__Ounit_J,type,
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tff(ty_n_t__Heap____Time____Monad__OHeap_Itf__b_J,type,
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tff(ty_n_t__Heap____Time____Monad__OHeap_Itf__a_J,type,
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tff(ty_n_t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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tff(ty_n_t__fun_Itf__b_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_Itf__a_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
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tff(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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tff(ty_n_t__List__Olist_It__String__Ochar_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
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tff(ty_n_t__Heap__Oarray_It__Int__Oint_J,type,
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tff(ty_n_t__List__Olist_It__Rat__Orat_J,type,
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tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
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tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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tff(ty_n_t__Code____Numeral__Onatural,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__Product____Type__Ounit,type,
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tff(ty_n_t__Option__Ooption_I_Eo_J,type,
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tff(ty_n_t__List__Olist_Itf__b_J,type,
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tff(ty_n_t__List__Olist_Itf__a_J,type,
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tff(ty_n_t__Heap__Oarray_I_Eo_J,type,
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tff(ty_n_t__List__Olist_I_Eo_J,type,
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tff(ty_n_t__fun_Itf__b_M_Eo_J,type,
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tff(ty_n_t__fun_Itf__a_M_Eo_J,type,
    fun_a_o: $tType ).

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

tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
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tff(ty_n_t__String__Ochar,type,
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tff(ty_n_t__Rat__Orat,type,
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tff(ty_n_t__Num__Onum,type,
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tff(ty_n_t__Nat__Onat,type,
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tff(ty_n_t__Int__Oint,type,
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tff(ty_n_tf__b,type,
    b: $tType ).

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

% Explicit typings (3508)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: set_o > fun_fu2881495813246281945un_o_o ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: set_o > fun_fu8188496484073410517nt_int ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: set_int > fun_fu354264921998911491_int_o ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: set_int > fun_fu8759299743271765508er_int ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: fun_Co3009391273526075439nt_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: set_Code_integer > fun_fu7527167023631164035eger_o ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: set_Code_integer > fun_fu1493779525457017043nt_int ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: set_o > fun_fu865690600344432797un_o_o ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: set_o > fun_fu4640364191220304065at_nat ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_nat_fun_o_nat > fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: set_nat > fun_fu2885402556529144793_nat_o ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: set_nat > fun_fu4618758991028193283_o_nat ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: set_Code_integer > fun_fu7601192970096278275eger_o ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: set_Code_integer > fun_fu8336550301028287791at_nat ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_na5492297256294505719er_nat > fun_Co3312552843764780407at_nat ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: set_nat > fun_fu165634461008800387_nat_o ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: set_nat > fun_fu792289584686185660er_nat ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_se6260736226359567993nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: set_nat > fun_fu4073184632390761289at_nat ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: set_nat > fun_fu5572043292636673283_o_nat ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: set_o > fun_fu4422690180167752641at_nat ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: set_nat > fun_fu4176442768802536380er_nat ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: set_Code_integer > fun_fu1095011443884336175at_nat ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: set_nat > fun_fu731591423980087811_o_int ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_o_fun_nat_int > fun_nat_fun_o_int ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: set_o > fun_fu1432006813369012345at_int ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: set_nat > fun_fu202463024092729020er_int ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_Co822469698833053651at_int > fun_na3002214111362778963er_int ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: set_Code_integer > fun_fu4854240044958656303at_int ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: set_int > fun_fu5188028542856013059_o_int ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: set_o > fun_fu8188496484073410517nt_int ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: set_int > fun_fu8759299743271765508er_int ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: set_Code_integer > fun_fu1493779525457017043nt_int ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: set_o > fun_fu4640364191220304065at_nat ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: set_nat > fun_fu4618758991028193283_o_nat ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: set_Code_integer > fun_fu8336550301028287791at_nat ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: set_nat > fun_fu792289584686185660er_nat ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: set_o > fun_fu8716660765737269145un_o_o ).

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

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: set_Code_integer > fun_fu855086506144988854eger_o ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: set_Code_integer > fun_fu4083202260958266008eger_o ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: set_o > fun_fu6337961365400742745un_o_o ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: set_o > fun_fu2641566564518291801un_o_o ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: set_Code_integer > fun_fu4523568072766896123eger_o ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: set_Code_integer > fun_fu7141499545569947298eger_o ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: set_o > fun_fu5915781709770235737un_o_o ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: set_o > fun_fu3819205551299168222un_o_o ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: set_o > fun_fu8716660765737269145un_o_o ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: set_nat > fun_fu5838931486547983824_nat_o ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: set_nat > fun_fu8532066432991188095_nat_o ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: set_int > fun_fu5271299750659287320_int_o ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: set_int > fun_fu63825462011493995_int_o ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: set_Code_integer > fun_fu855086506144988854eger_o ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: set_Code_integer > fun_fu4083202260958266008eger_o ).

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

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: set_o > fun_fu3143923321121214633un_o_o ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: set_o > fun_fu5861359551626625881un_o_o ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: set_nat > fun_fu1116345184836134341_nat_o ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: set_nat > fun_fu3109912920687205353_nat_o ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: set_int > fun_fu2934462529731303693_int_o ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: set_int > fun_fu607692679349591509_int_o ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: set_Code_integer > fun_fu4844139638962010283eger_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: set_Code_integer > fun_fu4735791719682850818eger_o ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: set_o > fun_fu6337961365400742745un_o_o ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: set_o > fun_fu2641566564518291801un_o_o ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_o_Code_integer > fun_fu7181763470137098718nteger ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: set_Code_integer > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_Co8499238673658658103nteger > fun_fu2729256655946442782nteger ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_o_rat > fun_fu830079035465870425_o_rat ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_Code_integer_rat > fun_fu991479512444375705er_rat ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_o_nat > fun_fu7210065372943547985_o_nat ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: fun_list_nat_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_list_nat_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_se4101287982975402108nt_nat > fun_se7249947736802449410_int_o ).

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

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: fun_se4101287982975402108nt_nat > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_set_nat_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_set_nat_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_Pr2602117396471556196at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: fun_Pr2602117396471556196at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: fun_o_Code_integer > fun_fu7181763470137098718nteger ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_nat_Code_integer > fun_fu7898940055406642654nteger ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_int_Code_integer > fun_fu6361890901329191134nteger ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_Co8499238673658658103nteger > fun_fu2729256655946442782nteger ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_o_assn > fun_fu7714003655882793385o_assn ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: fun_nat_assn > fun_fu2792963925203051945t_assn ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: fun_int_assn > fun_fu2048648494261043369t_assn ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: fun_Co4490034186337852034r_assn > fun_fu1898041513128675305r_assn ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: fun_o_rat > fun_fu830079035465870425_o_rat ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: rat > fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_list_nat_nat > fun_li1025265955262268170_nat_o ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: fun_se4101287982975402108nt_nat > fun_se2092758488442489134_int_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: fun_set_nat_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_nat_o ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: fun_Pr2602117396471556196at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: fun_o_Code_integer > fun_se6637447080540490323nteger ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: fun_int_Code_integer > fun_se6579715966475752165nteger ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: fun_Co8499238673658658103nteger > fun_se5087743179055239239nteger ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_o_rat > fun_set_o_fun_o_rat ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: fun_Code_integer_rat > fun_se8983711159220854466er_rat ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: fun_o_nat > fun_set_o_fun_o_nat ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: fun_int_nat > fun_se7317719546745285464nt_nat ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: fun_Code_integer_nat > fun_se6140325459843756218er_nat ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: fun_o_int > fun_set_o_fun_o_int ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: fun_Co8499238673658658103nteger > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__afu____,type,
    aTP_Lamm_afu: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__afv____,type,
    aTP_Lamm_afv: fun_nat_int > fun_nat_o ).

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: fun_Code_integer_int > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: fun_list_nat_nat > fun_fu4472246792138249213_nat_o ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: fun_se4101287982975402108nt_nat > fun_fu6634461749290962705_int_o ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: fun_set_nat_nat > fun_fu1237770272497958833_nat_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_nat_o ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: fun_Pr2602117396471556196at_nat > fun_fu7237799034916548881_nat_o ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: fun_o_Code_integer > fun_se6637447080540490323nteger ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: fun_int_Code_integer > fun_se6579715966475752165nteger ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: fun_Co8499238673658658103nteger > fun_se5087743179055239239nteger ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: fun_o_assn > fun_set_o_fun_o_assn ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: fun_int_assn > fun_se6298406198028402864t_assn ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: fun_Co4490034186337852034r_assn > fun_se7799640332748030482r_assn ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: fun_o_rat > fun_set_o_fun_o_rat ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: fun_int_rat > fun_se937733209267607904nt_rat ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: fun_Code_integer_rat > fun_se8983711159220854466er_rat ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: fun_o_nat > fun_set_o_fun_o_nat ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: fun_nat_Code_integer > fun_nat_o ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: fun_int_Code_integer > fun_int_o ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: fun_Co8499238673658658103nteger > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: fun_nat_assn > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: fun_int_assn > fun_int_o ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: fun_Co4490034186337852034r_assn > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: fun_int_o > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: heap_e7401611519738050253t_unit > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aha____,type,
    aTP_Lamm_aha: fun_Code_integer_o > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: fun_int_o > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: fun_Code_integer_o > fun_fu2498091903632033272er_nat ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: fun_Code_integer_o > fun_fu1007793439646807308er_int ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: fun_nat_o > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: fun_nat_o > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: fun_nat_o > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: fun_int_o > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: fun_list_nat_o > fun_fu1597678101255499801at_rat ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: fun_int_o > fun_fu5832215286117076717t_assn ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: fun_Code_integer_o > fun_fu4755565544374692864r_assn ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: fun_nat_o > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: code_integer > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: code_integer > fun_fu2498091903632033272er_nat ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: code_integer > fun_fu1007793439646807308er_int ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: $o > fun_fu9039538216874201369_o_rat ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: $o > fun_fu3581538110357139729_o_nat ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: $o > fun_fu9026017786105829997_o_int ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: int > fun_fu1617438229336688381nt_rat ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: int > fun_fu973771059875708133nt_nat ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: fun_int_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: fun_int_nat > fun_nat_fun_int_o ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: fun_Code_integer_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: set_o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: fun_Code_integer_nat > fun_na6318815019539281885eger_o ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: fun_Code_integer_nat > fun_na6318815019539281885eger_o ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: fun_Code_integer_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: fun_Code_integer_int > fun_in4891666191172872833eger_o ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: fun_Code_integer_int > fun_in4891666191172872833eger_o ).

tff(sy_c_ATP_058Lamm__aif____,type,
    aTP_Lamm_aif: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aig____,type,
    aTP_Lamm_aig: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: fun_nat_int > fun_int_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: fun_int_Code_natural > fun_Co6749200540738471926atural ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: fun_int_Code_natural > fun_Co7174943067365291894_int_o ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: fun_int_Code_natural > fun_Co7174943067365291894_int_o ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: fun_Co3734316457315071362atural > fun_Co8514937155073724839atural ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: fun_Co3734316457315071362atural > fun_Co2157892728433036165eger_o ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: fun_Co3734316457315071362atural > fun_Co2157892728433036165eger_o ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: fun_nat_Code_natural > fun_Co740390889742897050atural ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: fun_nat_Code_natural > fun_Co196715998180962002_nat_o ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: fun_nat_Code_natural > fun_Co196715998180962002_nat_o ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: fun_list_nat_nat > fun_na7671319193681171066at_nat ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: fun_list_nat_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: fun_list_nat_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__aja____,type,
    aTP_Lamm_aja: code_integer > fun_fu4755565544374692864r_assn ).

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: $o > fun_fu556125837173103209o_assn ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: int > fun_fu5832215286117076717t_assn ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: list_nat > fun_he986703168080474088it_nat ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: list_o > fun_he2323575226933966612it_nat ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: list_int > fun_he8488442066948866628it_nat ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ajr____,type,
    aTP_Lamm_ajr: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: set_o > fun_fu5257111247350578329un_o_o ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__aju____,type,
    aTP_Lamm_aju: set_int > fun_fu7864316720324808067_int_o ).

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: set_Code_integer > fun_fu1992263728252513447eger_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: set_se6260736226359567993nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: set_o > fun_fu7756609662939579309un_o_o ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: set_o > fun_fu6781074197733401937un_o_o ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: set_int > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: set_Code_integer > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: set_o > fun_fu7756609662939579309un_o_o ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: set_nat > fun_fu8207122332086543297_nat_o ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: set_Code_integer > fun_fu1992263728252513447eger_o ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: set_o > fun_fu7722316861827675049un_o_o ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: set_nat > fun_fu2798942646354680517_nat_o ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: set_int > fun_fu2231310910465984013_int_o ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: set_Code_integer > fun_fu4213695556130342827eger_o ).

tff(sy_c_ATP_058Lamm__akm____,type,
    aTP_Lamm_akm: nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: nat > fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: int > fun_list_nat_int ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: int > fun_set_nat_int ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: rat > fun_int_rat ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: code_integer > fun_int_Code_integer ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: rat > fun_list_nat_rat ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: rat > fun_set_nat_rat ).

tff(sy_c_ATP_058Lamm__akz____,type,
    aTP_Lamm_akz: fun_int_o > fun_int_nat ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: fun_Code_integer_o > fun_Code_integer_nat ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: fun_Code_integer_o > fun_Code_integer_int ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: fun_nat_o > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: fun_int_o > fun_int_Code_integer ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: fun_Code_integer_o > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: fun_nat_o > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__alg____,type,
    aTP_Lamm_alg: fun_nat_o > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__alh____,type,
    aTP_Lamm_alh: fun_int_o > fun_int_int ).

tff(sy_c_ATP_058Lamm__ali____,type,
    aTP_Lamm_ali: fun_list_nat_o > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__alj____,type,
    aTP_Lamm_alj: fun_num_num ).

tff(sy_c_ATP_058Lamm__alk____,type,
    aTP_Lamm_alk: set_Pr958786334691620121nt_int > fun_na6120508140265747663_int_o ).

tff(sy_c_ATP_058Lamm__all____,type,
    aTP_Lamm_all: set_list_nat > fun_na1108963418173300452_nat_o ).

tff(sy_c_ATP_058Lamm__alm____,type,
    aTP_Lamm_alm: set_set_nat > fun_na3711450577076729278_nat_o ).

tff(sy_c_ATP_058Lamm__aln____,type,
    aTP_Lamm_aln: set_nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__alo____,type,
    aTP_Lamm_alo: set_Code_integer > fun_na7849980549906154535eger_o ).

tff(sy_c_ATP_058Lamm__alp____,type,
    aTP_Lamm_alp: set_Pr1261947904930325089at_nat > fun_na4518284863240432007_nat_o ).

tff(sy_c_ATP_058Lamm__alq____,type,
    aTP_Lamm_alq: set_int > fun_na8635581178960477080_int_o ).

tff(sy_c_ATP_058Lamm__alr____,type,
    aTP_Lamm_alr: fun_o_Code_integer > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__als____,type,
    aTP_Lamm_als: fun_nat_Code_integer > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__alt____,type,
    aTP_Lamm_alt: fun_int_Code_integer > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__alu____,type,
    aTP_Lamm_alu: fun_o_rat > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__alv____,type,
    aTP_Lamm_alv: fun_nat_rat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__alw____,type,
    aTP_Lamm_alw: fun_int_rat > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__alx____,type,
    aTP_Lamm_alx: fun_o_nat > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__aly____,type,
    aTP_Lamm_aly: fun_nat_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__alz____,type,
    aTP_Lamm_alz: fun_int_nat > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ama____,type,
    aTP_Lamm_ama: fun_o_int > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__amb____,type,
    aTP_Lamm_amb: set_o > fun_fu4605530428521801667un_o_o ).

tff(sy_c_ATP_058Lamm__amc____,type,
    aTP_Lamm_amc: set_o > fun_fu4605530428521801667un_o_o ).

tff(sy_c_ATP_058Lamm__amd____,type,
    aTP_Lamm_amd: set_o > fun_fu4625457934454704336_o_nat ).

tff(sy_c_ATP_058Lamm__ame____,type,
    aTP_Lamm_ame: set_int > fun_fu2630858903579709668_o_nat ).

tff(sy_c_ATP_058Lamm__amf____,type,
    aTP_Lamm_amf: set_Code_integer > fun_fu5820097178109300859eger_o ).

tff(sy_c_ATP_058Lamm__amg____,type,
    aTP_Lamm_amg: set_Code_integer > fun_fu9082890457842154773_o_nat ).

tff(sy_c_ATP_058Lamm__amh____,type,
    aTP_Lamm_amh: set_int > fun_fu7616189625787487449_int_o ).

tff(sy_c_ATP_058Lamm__ami____,type,
    aTP_Lamm_ami: set_o > fun_fu6500062642585991509un_o_o ).

tff(sy_c_ATP_058Lamm__amj____,type,
    aTP_Lamm_amj: set_o > fun_fu2395600473883792738nt_nat ).

tff(sy_c_ATP_058Lamm__amk____,type,
    aTP_Lamm_amk: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__aml____,type,
    aTP_Lamm_aml: set_int > fun_fu2566930710778725758nt_nat ).

tff(sy_c_ATP_058Lamm__amm____,type,
    aTP_Lamm_amm: set_Code_integer > fun_fu5813846785102284803eger_o ).

tff(sy_c_ATP_058Lamm__amn____,type,
    aTP_Lamm_amn: set_Code_integer > fun_fu716243638958221197nt_nat ).

tff(sy_c_ATP_058Lamm__amo____,type,
    aTP_Lamm_amo: set_Code_integer > fun_fu5848294987682527897eger_o ).

tff(sy_c_ATP_058Lamm__amp____,type,
    aTP_Lamm_amp: set_o > fun_fu5285309056923805367un_o_o ).

tff(sy_c_ATP_058Lamm__amq____,type,
    aTP_Lamm_amq: set_o > fun_fu2439999276549649860er_nat ).

tff(sy_c_ATP_058Lamm__amr____,type,
    aTP_Lamm_amr: set_int > fun_fu3779984648686517950er_nat ).

tff(sy_c_ATP_058Lamm__ams____,type,
    aTP_Lamm_ams: set_Code_integer > fun_fu8958601512183576259eger_o ).

tff(sy_c_ATP_058Lamm__amt____,type,
    aTP_Lamm_amt: set_Code_integer > fun_fu8958601512183576259eger_o ).

tff(sy_c_ATP_058Lamm__amu____,type,
    aTP_Lamm_amu: set_Code_integer > fun_fu6385016345819965645er_nat ).

tff(sy_c_ATP_058Lamm__amv____,type,
    aTP_Lamm_amv: set_nat > fun_fu6443148758731022984_o_nat ).

tff(sy_c_ATP_058Lamm__amw____,type,
    aTP_Lamm_amw: fun_int_nat ).

tff(sy_c_ATP_058Lamm__amx____,type,
    aTP_Lamm_amx: fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__amy____,type,
    aTP_Lamm_amy: fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__amz____,type,
    aTP_Lamm_amz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ana____,type,
    aTP_Lamm_ana: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anb____,type,
    aTP_Lamm_anb: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__anc____,type,
    aTP_Lamm_anc: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__and____,type,
    aTP_Lamm_and: fun_Code_integer_rat > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__ane____,type,
    aTP_Lamm_ane: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__anf____,type,
    aTP_Lamm_anf: fun_Code_integer_nat > fun_fu7724909810345500241er_nat ).

tff(sy_c_ATP_058Lamm__ang____,type,
    aTP_Lamm_ang: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__anh____,type,
    aTP_Lamm_anh: fun_Code_integer_int > fun_fu403917627589892361er_int ).

tff(sy_c_ATP_058Lamm__ani____,type,
    aTP_Lamm_ani: fun_Pr8317260874932935276at_rat > fun_fu3997796857297124035at_rat ).

tff(sy_c_ATP_058Lamm__anj____,type,
    aTP_Lamm_anj: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ank____,type,
    aTP_Lamm_ank: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__anl____,type,
    aTP_Lamm_anl: fun_int_nat > fun_int_nat ).

tff(sy_c_ATP_058Lamm__anm____,type,
    aTP_Lamm_anm: fun_list_nat_nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__ann____,type,
    aTP_Lamm_ann: fun_set_nat_nat > fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__ano____,type,
    aTP_Lamm_ano: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__anp____,type,
    aTP_Lamm_anp: set_nat > fun_fu2757636626010989786nt_nat ).

tff(sy_c_ATP_058Lamm__anq____,type,
    aTP_Lamm_anq: set_nat > fun_fu513762604956806682er_nat ).

tff(sy_c_ATP_058Lamm__anr____,type,
    aTP_Lamm_anr: code_natural > fun_Co9168753396221475450atural ).

tff(sy_c_ATP_058Lamm__ans____,type,
    aTP_Lamm_ans: produc7822875418678951345atural > fun_Co1890133979025130057atural ).

tff(sy_c_ATP_058Lamm__ant____,type,
    aTP_Lamm_ant: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__anu____,type,
    aTP_Lamm_anu: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__anv____,type,
    aTP_Lamm_anv: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__anw____,type,
    aTP_Lamm_anw: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__anx____,type,
    aTP_Lamm_anx: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__any____,type,
    aTP_Lamm_any: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__anz____,type,
    aTP_Lamm_anz: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: heap_e7401611519738050253t_unit > fun_he4135464443191253079_nat_o ).

tff(sy_c_ATP_058Lamm__aoa____,type,
    aTP_Lamm_aoa: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__aob____,type,
    aTP_Lamm_aob: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__aoc____,type,
    aTP_Lamm_aoc: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__aod____,type,
    aTP_Lamm_aod: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__aoe____,type,
    aTP_Lamm_aoe: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aof____,type,
    aTP_Lamm_aof: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aog____,type,
    aTP_Lamm_aog: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aoh____,type,
    aTP_Lamm_aoh: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aoi____,type,
    aTP_Lamm_aoi: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aoj____,type,
    aTP_Lamm_aoj: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aok____,type,
    aTP_Lamm_aok: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aol____,type,
    aTP_Lamm_aol: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aom____,type,
    aTP_Lamm_aom: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aon____,type,
    aTP_Lamm_aon: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aoo____,type,
    aTP_Lamm_aoo: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aop____,type,
    aTP_Lamm_aop: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aoq____,type,
    aTP_Lamm_aoq: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aor____,type,
    aTP_Lamm_aor: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aos____,type,
    aTP_Lamm_aos: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aot____,type,
    aTP_Lamm_aot: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aou____,type,
    aTP_Lamm_aou: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aov____,type,
    aTP_Lamm_aov: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aow____,type,
    aTP_Lamm_aow: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aox____,type,
    aTP_Lamm_aox: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aoy____,type,
    aTP_Lamm_aoy: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aoz____,type,
    aTP_Lamm_aoz: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__apa____,type,
    aTP_Lamm_apa: fun_Co1981900195452921617nteger ).

tff(sy_c_ATP_058Lamm__apb____,type,
    aTP_Lamm_apb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apc____,type,
    aTP_Lamm_apc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apd____,type,
    aTP_Lamm_apd: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__ape____,type,
    aTP_Lamm_ape: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__apf____,type,
    aTP_Lamm_apf: fun_int_int ).

tff(sy_c_ATP_058Lamm__apg____,type,
    aTP_Lamm_apg: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__aph____,type,
    aTP_Lamm_aph: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__api____,type,
    aTP_Lamm_api: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__apj____,type,
    aTP_Lamm_apj: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__apk____,type,
    aTP_Lamm_apk: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__apl____,type,
    aTP_Lamm_apl: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__apm____,type,
    aTP_Lamm_apm: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__apn____,type,
    aTP_Lamm_apn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apo____,type,
    aTP_Lamm_apo: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__app____,type,
    aTP_Lamm_app: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apq____,type,
    aTP_Lamm_apq: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__apr____,type,
    aTP_Lamm_apr: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__aps____,type,
    aTP_Lamm_aps: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apt____,type,
    aTP_Lamm_apt: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__apu____,type,
    aTP_Lamm_apu: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__apv____,type,
    aTP_Lamm_apv: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__apw____,type,
    aTP_Lamm_apw: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__apx____,type,
    aTP_Lamm_apx: fun_int_nat ).

tff(sy_c_ATP_058Lamm__apy____,type,
    aTP_Lamm_apy: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__apz____,type,
    aTP_Lamm_apz: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__aqa____,type,
    aTP_Lamm_aqa: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aqb____,type,
    aTP_Lamm_aqb: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__aqc____,type,
    aTP_Lamm_aqc: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aqd____,type,
    aTP_Lamm_aqd: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aqe____,type,
    aTP_Lamm_aqe: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aqf____,type,
    aTP_Lamm_aqf: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__aqg____,type,
    aTP_Lamm_aqg: fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__aqh____,type,
    aTP_Lamm_aqh: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aqi____,type,
    aTP_Lamm_aqi: fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_Pr1099658227000185396_nat_o > fun_fu714798943701216869_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_Pr8551490117392284871at_nat > fun_se3981709065971081742_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_Pr4329608150637261639at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_Pr8551490117392284871at_nat > fun_Pr1099658227000185396_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_Pr4329608150637261639at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_se5750412846314922030_nat_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_se5750412846314922030_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_assn_fun_assn_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_assn_fun_assn_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: set_Pr8551490117392284871at_nat > fun_se2186744213282142921at_nat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: option2498585697089621389at_nat > fun_op3311566623873074537at_nat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_Pr4329608150637261639at_nat > fun_se7901873761543946185at_nat ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: option1583680563626158861at_nat > fun_op291006984297869545at_nat ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_nat > fun_se7316093745529328985et_nat ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: option_set_nat > fun_op386870758422531628et_nat ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: option_nat > fun_op7680796716376270858on_nat ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: int > fun_int_option_int ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: option_int > fun_op7417050415941327134on_int ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_Pr1312406793078187350et_nat > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_Pr4040394263813967220et_nat > fun_fu2611741329380808165_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: set_Pr7600907837789447088it_nat > fun_b_540864168722188329_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_Pr7098220151150636591it_nat > fun_a_2467521563176467050_nat_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_Pr8551490117392284871at_nat > fun_mu7838234450475261998_nat_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_Pr4329608150637261639at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: produc7388388658123137530it_nat > fun_Pr856651894144386551_nat_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: produc3260487557148687353it_nat > fun_Pr5746006151868627512_nat_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: produc8664842809031399944it_nat > fun_Pr8032424337255632745_nat_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_num_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_int_int > fun_fu7117095973476296707um_int ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_int_option_num > fun_fu7964671872712412353on_num ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_int_num > fun_fu5322398707847370225um_num ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_option_num_int > fun_fu1947565687807238081um_int ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_op8989173174992673105on_num > fun_fu2987389051837124479on_num ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_option_num_num > fun_fu152868422178311599um_num ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_num_int > fun_fu726849872193282289um_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_num_option_num > fun_fu4719417816141531311on_num ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_num_num > fun_fu8155524643419131615um_num ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_he8413527710729349838unit_o > fun_fu2568544640974905491it_nat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_he8413527710729349838unit_o > fun_fu9056938625493858259it_nat ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_he8413527710729349838unit_o > fun_fu2492676192234506515it_nat ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_num_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: option_set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: option_assn > fun_assn_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_Pr774283325640823912it_nat > fun_Pr6873217934956908984it_nat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_Pr774283325640823912it_nat > fun_op7235540884721919880it_nat ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_Pr7750636411365408485it_nat > fun_Pr7916406164169562037it_nat ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_Pr7750636411365408485it_nat > fun_op5883308099655130629it_nat ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_Pr1003294667014919570it_nat > fun_Pr3631287881653221602it_nat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_Pr1003294667014919570it_nat > fun_op4112804339236761266it_nat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_num_fun_num_num > fun_nu1049378303802735384on_num ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_num_fun_num_num > fun_op3323321805596408808on_num ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: heap_Time_Heap_a > fun_Pr5215307506330550056Heap_a ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: heap_Time_Heap_b > fun_Pr5286341546376896041Heap_b ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: heap_T5738788834812785303t_unit > fun_Pr292408699144966967t_unit ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_o_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_int_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_he6342020312717187381unit_b > fun_he8897818899970768042it_nat ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_he6342020308413958580unit_a > fun_he1921015881043195817it_nat ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_he5123621388056232515t_unit > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: b > fun_he168995381318928090it_nat ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: a > fun_he8889679731534893401it_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: product_unit > fun_he4944555133437423336it_nat ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: ref_Product_unit > fun_Pr2477578491906815678t_assn ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_Pr5876161306168176259nteger > fun_Co1561183069634336733nteger ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_Pr5876161306168176259nteger > fun_fu4800041728187187127nteger ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: product_unit > fun_he4944555133437423336it_nat ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: b > fun_he168995381318928090it_nat ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: a > fun_he8889679731534893401it_nat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_int_int ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_int_int ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_a_assn > fun_assn_fun_a_assn ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_b_assn > fun_assn_fun_b_assn ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_Pr6626665806141890376t_assn > fun_as4019953064511417593t_assn ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: assn > fun_a_fun_a_assn ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: assn > fun_b_fun_b_assn ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: assn > fun_Pr2477578491906815678t_assn ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_Pr4796943543331071514_nat_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: assn > fun_assn_option_assn ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: option_assn > fun_assn_option_assn ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_Pr1261947904930325089at_nat > fun_se5244420045238757833at_nat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: option8963830502488799655at_nat > fun_se5244420045238757833at_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_nat > fun_se7316093745529328985et_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: option_set_nat > fun_se7316093745529328985et_nat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: option_nat > fun_nat_option_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_Pr3238374608701176162_int_o > fun_fu5399310930634420241_int_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: option2108551362046372146_int_o > fun_fu5399310930634420241_int_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_Pr1261947904930325089at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: nat > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_int_int ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: rat > fun_nat_fun_rat_rat ).

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

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: array_nat > fun_li4928566814967809234t_assn ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: array_o > fun_li3807767012653957926t_assn ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: array_int > fun_li1703715468755832110t_assn ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_se7249947736802449410_int_o > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_Pr1099658227000185396_nat_o > fun_fu714798943701216869_nat_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_Pr3059359313727237556_nat_o > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: assn > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_list_nat_o > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_se7249947736802449410_int_o > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: set_se6260736226359567993nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_set_nat > fun_set_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: set_o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_se6260736226359567993nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: nat > fun_li6911922195568332558t_assn ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: nat > fun_li4611866707255273568o_assn ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: nat > fun_li8672845860709154922t_assn ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: nat > fun_li124330075790197867t_assn ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_na1668100129766719005at_nat > fun_Pr557098546508347797at_nat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_na5914473681949881652_nat_o > fun_Pr3455331679158536364_nat_o ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_int_o > fun_fu8090991196261985865_int_o ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_int_o > fun_fu6703568167625668877_int_o ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_o > fun_fu2677729659939697685_nat_o ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_nat_o > fun_fu1617828088232643505_nat_o ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: set_Pr958786334691620121nt_int > fun_se2817990011555733942_int_o ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: set_Pr958786334691620121nt_int > fun_se4456751357192657760_int_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: set_Pr1261947904930325089at_nat > fun_se3165749655728557022_nat_o ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: set_Pr1261947904930325089at_nat > fun_se218054151627931304_nat_o ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_b_8426044978055006235Heap_b > fun_b_7001994053037931259it_nat ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_b_8426044978055006235Heap_b > fun_b_7697183073795684005it_nat ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_a_6164325696594227994Heap_b > fun_a_9184149311047732028it_nat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_a_6164325696594227994Heap_b > fun_a_7024012846491921188it_nat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_b_8355010938008660250Heap_a > fun_b_989364880692008698it_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_b_8355010938008660250Heap_a > fun_b_4099525269183078948it_nat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_a_6093291656547882009Heap_a > fun_a_3171520138701809467it_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_a_6093291656547882009Heap_a > fun_a_3426355041879316131it_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_int_option_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: heap_Time_Heap_b > fun_fu400886066185544210it_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: heap_Time_Heap_a > fun_fu2612457627315080531it_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: heap_Time_Heap_b > fun_fu2643727258468172498it_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: heap_Time_Heap_a > fun_fu4855298819597708819it_nat ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_b_6677255003012645673t_unit > fun_b_4161654760626720265it_nat ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_b_6677255003012645673t_unit > fun_b_3682875105831643059it_nat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: heap_Time_Heap_b > fun_fu8161990777412754it_nat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_a_2990225006182978344t_unit > fun_a_6373226321756256586it_nat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_a_2990225006182978344t_unit > fun_a_4942204877778174258it_nat ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: heap_Time_Heap_a > fun_fu6488529940522614227it_nat ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_fu2352890940381120596et_nat > fun_fu7582738562687708775_nat_o ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_fu1905361174868762994et_nat > fun_fu4927615759029954091_nat_o ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_b_7832390750199427630it_nat > fun_fu6044858605386232627_nat_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_a_1313742400928293293it_nat > fun_fu4836394952439539509_nat_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_int_option_int > fun_fu7409062071265970153_int_o ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_int_nat > fun_se5614109888078709212_int_o ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: produc3843707927480180839at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: produc3843707927480180839at_nat > fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: product_prod_nat_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: $o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: set_Pr958786334691620121nt_int > fun_se3135557758678015630_int_o ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: produc3843707927480180839at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: set_Pr958786334691620121nt_int > fun_fu5088349107139067777_int_o ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: product_prod_nat_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: produc4166570645942440679at_nat > fun_Pr1099658227000185396_nat_o ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: multis2468970476368604999at_nat > fun_mu1615448358963118404_nat_o ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: set_Pr1261947904930325089at_nat > fun_se4507453553354261854_nat_o ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: nat > fun_na683342808802396953_nat_o ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: int > fun_in7694643900704578053_int_o ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_Co1105491363547889651atural > fun_fu8850208375822146947atural ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_op8989173174992673105on_num > fun_fu1574286537503582443on_num ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_option_num_nat > fun_fu1502755112258691281at_nat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_option_num_o > fun_fu2897328509465618819_nat_o ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_nat_option_num > fun_fu5130958774822480081on_num ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_o_option_num > fun_fu2797749360310127211on_num ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_o_nat > fun_fu318872218122388049at_nat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_o_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_o_Code_integer > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_o_rat > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_o_int > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_se7586849273513198688nteger > fun_fu5510336631606277565nteger ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_se4101287982975402108nt_nat > fun_fu8621421401018972867nt_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_Co3583066429914935172er_nat ).

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

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_na2338080690716861136at_nat > fun_na2338080690716861136at_nat ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_na2338080690716861136at_nat > fun_na1945018436707187408at_nat ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: nat > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_Code_integer > fun_nat_Code_integer ).

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

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

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_na2338080690716861136at_nat > fun_na2338080690716861136at_nat ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: nat > fun_fu695563712076452518at_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: nat > fun_fu695563712076452518at_nat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: nat > fun_ra8505079968215247908at_rat ).

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

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: nat > fun_ra8505079968215247908at_rat ).

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

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_fun_nat_int ).

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

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

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

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: code_natural > fun_Co740390889742897050atural ).

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

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: code_integer > fun_Co6226381647958292740nteger ).

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

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: nat > fun_nat_Code_integer ).

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

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_nat_int ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_int_int ).

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

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

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

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_fun_nat_int > fun_se7912642235559603452at_int ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_fun_int_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_fun_int_int > fun_se876191773397849432nt_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_int_fun_nat_int > fun_se876191773397849432nt_int ).

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

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_int_fun_nat_int > fun_se2640714827120536920at_int ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_int_int > fun_nat_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_fun_nat_nat > fun_nat_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_int > fun_na7654559027617327379at_int ).

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

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

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: nat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: code_natural > fun_nat_Code_natural ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_assn > fun_nat_assn ).

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

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

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_Code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_assn > fun_na2454419004871237530n_assn ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: code_integer > fun_na8135612474337863287nteger ).

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

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

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_nat_assn > fun_nat_assn ).

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

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

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

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_rat > fun_ra7317100687414072407at_rat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_o_rat > fun_ra154387494674143703at_rat ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_int_rat > fun_ra1107035566227760819at_rat ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_int > fun_in2640216123419753727at_int ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_int_int > fun_in5653523039088217947at_int ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_nat > fun_na6244580715232974919at_nat ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_o_nat > fun_na2977190036508382167at_nat ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_int_nat > fun_na34515594046663331at_nat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_o_int > fun_in3146844979458463959at_int ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_fu6327154928193479288nteger ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: nat > fun_fu3959123160307459545t_assn ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: code_integer > fun_Co3544761417380347684eger_o ).

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

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: set_Pr958786334691620121nt_int > fun_se7249947736802449410_int_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_nat_int > fun_nat_fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: list_nat > fun_ar6592047341168336478it_nat ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: list_o > fun_ar1474576826808783682it_nat ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: list_int > fun_ar554763049074313054it_nat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: code_integer > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: $o > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: code_integer > fun_fu722387130092441185er_rat ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: $o > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: nat > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: int > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: code_integer > fun_fu1763652423599918571nteger ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: $o > fun_fu704708528062463619o_assn ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: nat > fun_fu2995346897264222311t_assn ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: int > fun_fu6568868615275847455t_assn ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: code_integer > fun_fu5082522747898945281r_assn ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: $o > fun_fu1989461620956114563_o_rat ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: $o > fun_fu2786480252100984515nteger ).

tff(sy_c_ATP_058Lamm__xx____,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: fun_se1909708154429037836et_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Num__Onum_J,type,
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    sup_su5525570899277871387at_nat: fun_se177437144654672861at_nat ).

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tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
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tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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tff(sy_c_Lattices__Big_Osemilattice__order__set_001t__Assertions__Oassn,type,
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tff(sy_c_List_Oappend_001t__Int__Oint,type,
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tff(sy_c_List_Oappend_001t__Nat__Onat,type,
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tff(sy_c_List_Obutlast_001t__Nat__Onat,type,
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tff(sy_c_List_Odrop_001t__Nat__Onat,type,
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tff(sy_c_List_Ofind_001t__Nat__Onat,type,
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    cons_P6512896166579812791at_nat: ( product_prod_nat_nat * list_P6011104703257516679at_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Olist_OCons_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,
    cons_P4136552807766418642it_nat: ( produc8664842809031399944it_nat * list_P626663023886443800it_nat ) > list_P626663023886443800it_nat ).

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    cons_P8169920869477312234et_nat: ( produc8799533983703829792et_nat * list_P8836981409019564976et_nat ) > list_P8836981409019564976et_nat ).

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    cons_P5637129515692979848et_nat: ( produc3112576180687729086et_nat * list_P4783563340545138382et_nat ) > list_P4783563340545138382et_nat ).

tff(sy_c_List_Olist_OCons_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,
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tff(sy_c_List_Olist_OCons_001t__Rat__Orat,type,
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tff(sy_c_List_Olist_OCons_001tf__a,type,
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tff(sy_c_List_Olist_OCons_001tf__b,type,
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tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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tff(sy_c_List_Onth_001_Eo,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
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tff(sy_c_List_Onth_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Num__Onum,type,
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tff(sy_c_List_Onth_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Rat__Orat,type,
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tff(sy_c_List_Onth_001tf__b,type,
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tff(sy_c_List_Oproduct_001_Eo_001_Eo,type,
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tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_List_Orotate1_001_Eo,type,
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tff(sy_c_List_Orotate1_001t__Int__Oint,type,
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tff(sy_c_List_Orotate1_001t__Nat__Onat,type,
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tff(sy_c_List_Orotate_001_Eo,type,
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tff(sy_c_List_Orotate_001t__Int__Oint,type,
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tff(sy_c_List_Orotate_001t__Nat__Onat,type,
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tff(sy_c_List_Osorted__wrt_001_Eo,type,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
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tff(sy_c_List_Osorted__wrt_001t__Rat__Orat,type,
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tff(sy_c_List_Otake_001t__Nat__Onat,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_List_Ozip_001_Eo_001_Eo,type,
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tff(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
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tff(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
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tff(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
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tff(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_OEps__Opt_001t__Int__Oint,type,
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tff(sy_c_Misc_OEps__Opt_001t__Num__Onum,type,
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tff(sy_c_Misc_Obijective_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Orel__of_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oset__to__map_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Oset__to__map_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oslice_001_Eo,type,
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tff(sy_c_Misc_Oslice_001t__Int__Oint,type,
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tff(sy_c_Misc_Oslice_001t__Nat__Onat,type,
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tff(sy_c_Misc_Othe__default_001t__Num__Onum,type,
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tff(sy_c_Misc_Othe__default_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Misc_Othe__default_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Misc_Othe__default_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Multiset_Oms__strict,type,
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tff(sy_c_Multiset_Oms__weak,type,
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tff(sy_c_Multiset_Opw__leq,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Int__Oint,type,
    case_nat_int: ( int * fun_nat_int ) > fun_nat_int ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Olist__decode,type,
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tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod_nat_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod_nat_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: nat > set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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tff(sy_c_Num_Onum_OBit0,type,
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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 ).

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    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: fun_num_rat ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Num_Osqr,type,
    sqr: fun_num_num ).

tff(sy_c_Option_Ocombine__options_001t__Num__Onum,type,
    combine_options_num: ( fun_num_fun_num_num * option_num ) > fun_op8989173174992673105on_num ).

tff(sy_c_Option_Ocombine__options_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,
    combin4318129983670048329it_nat: ( fun_Pr1003294667014919570it_nat * option8956607266484857688it_nat * option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

tff(sy_c_Option_Ocombine__options_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,
    combin5819755802923621690it_nat: ( fun_Pr7750636411365408485it_nat * option3562590408128118217it_nat * option3562590408128118217it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin724284867043296059it_nat: ( fun_Pr774283325640823912it_nat * option4065278094766928714it_nat * option4065278094766928714it_nat ) > option4065278094766928714it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Assertions__Oassn,type,
    none_assn: option_assn ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_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__Int__Oint_Mt__Int__Oint_J,type,
    none_P2377608414092835994nt_int: option4624381673175914239nt_int ).

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_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P7891021755743879167et_nat: option4955102875562834416et_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    none_P2758821087663475997et_nat: option5899113203645146382et_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_ONone_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P6779099274072355161it_nat: option4065278094766928714it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_I_Eo_J,type,
    none_set_o: option_set_o ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
    none_set_int: option_set_int ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    none_s625347054029921090at_nat: option8963830502488799655at_nat ).

tff(sy_c_Option_Ooption_ONone_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    none_f6270875889056561217_int_o: option2108551362046372146_int_o ).

tff(sy_c_Option_Ooption_OSome_001_Eo,type,
    some_o: $o > option_o ).

tff(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
    some_assn: assn > option_assn ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: fun_int_option_int ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: fun_nat_option_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__Heap__Oarray_I_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P3509045262911171395it_nat: produc8032495849646651230it_nat > option1013472963438093988it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oarray_It__Int__Oint_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P8650484732927216191it_nat: produc7196121022310644452it_nat > option2555668343544256180it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oarray_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P8206036975937309155it_nat: produc6751673265320737416it_nat > option4277301481530639448it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    some_P1054239925786823975it_nat: produc6653097349344004940it_nat > option233860712434008220it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    some_P4184893108420464158nt_int: product_prod_int_int > option4624381673175914239nt_int ).

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_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    some_P994283844982041979et_nat: produc8799533983703829792et_nat > option4955102875562834416et_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    some_P3006888012174695577et_nat: produc3112576180687729086et_nat > option5899113203645146382et_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_OSome_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P2818173045054083285it_nat: fun_Pr4959652356039452225it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_I_Eo_J,type,
    some_set_o: set_o > option_set_o ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
    some_set_int: set_int > option_set_int ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    some_s287724117700012716at_nat: set_Pr8551490117392284871at_nat > option2498585697089621389at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    some_s147305329494351046at_nat: set_Pr1261947904930325089at_nat > option8963830502488799655at_nat ).

tff(sy_c_Option_Ooption_OSome_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,
    some_s5890477192898017836at_nat: set_Pr4329608150637261639at_nat > option1583680563626158861at_nat ).

tff(sy_c_Option_Ooption_OSome_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    some_f3270274345126892989_int_o: fun_Pr3238374608701176162_int_o > option2108551362046372146_int_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Assertions__Oassn,type,
    case_option_o_assn: ( $o * fun_assn_o * option_assn ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Int__Oint,type,
    case_option_o_int: ( $o * fun_int_o * option_int ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Nat__Onat,type,
    case_option_o_nat: ( $o * fun_nat_o * option_nat ) > $o ).

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_001_Eo_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o7395119307266030407it_nat: ( $o * fun_Pr856651894144386551_nat_o ) > fun_op3313016124244812839_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Rat__Orat,type,
    case_option_o_rat: ( $o * fun_rat_o * option_rat ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    case_o223999843215110191et_int: ( $o * fun_set_int_o * option_set_int ) > $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__Int__Oint,type,
    case_option_nat_int: ( nat * fun_int_nat * option_int ) > nat ).

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__Assertions__Oassn_J_001t__Assertions__Oassn,type,
    case_o4484465799723439917n_assn: ( option_assn * fun_assn_option_assn * option_assn ) > option_assn ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Int__Oint_J_001t__Int__Oint,type,
    case_o390784466056649525nt_int: ( option_int * fun_int_option_int * option_int ) > option_int ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
    case_o7429725398727453821at_nat: ( option_nat * fun_nat_option_nat * option_nat ) > option_nat ).

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_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_o2963978774867076333it_nat: ( option8956607266484857688it_nat * fun_Pr7644814098969791837it_nat * option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

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_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__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o2632329678403290591it_nat: ( option8956607266484857688it_nat * fun_Pr2992940411974564687it_nat * option4065278094766928714it_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 ).

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    case_o965627003169710416it_nat: ( option3562590408128118217it_nat * fun_Pr7206221373966655808it_nat * option4065278094766928714it_nat ) > option3562590408128118217it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_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_o1773283271472524752it_nat: ( option4065278094766928714it_nat * fun_Pr157300717155555520it_nat * option3562590408128118217it_nat ) > option4065278094766928714it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_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__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o5901184372446974929it_nat: ( option4065278094766928714it_nat * fun_Pr4959652356039452225it_nat * option4065278094766928714it_nat ) > option4065278094766928714it_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
    case_o4054078431260844265et_nat: ( option_set_nat * fun_se7316093745529328985et_nat * option_set_nat ) > option_set_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    case_o8757249484648823045at_nat: ( option2498585697089621389at_nat * fun_se2186744213282142921at_nat * option2498585697089621389at_nat ) > option2498585697089621389at_nat ).

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tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member5617269971687963298it_nat: ( produc8961450480463052793it_nat * set_Pr4389693562480114009it_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member6099555550032318734et_nat: ( produc8111630337999740517et_nat * set_Pr719794911490849221et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member8566619992076573584nt_int: ( produc1219242969750017639nt_int * set_Pr2560585780119916871nt_int ) > $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__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member2216856746449346506et_nat: ( produc8893060842726229537et_nat * set_Pr9066344408807579265et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    member2136425100634443654et_nat: ( produc505541996239113949et_nat * set_Pr7057080801110549565et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_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__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,
    member5335690527091456380it_nat: ( produc9011797661310329043it_nat * set_Pr2819221443900773171it_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    member6820296301096096254it_nat: ( produc1273031398460193109it_nat * set_Pr5508209795250834101it_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    member4307123515891402160nteger: ( produc6491284506569428743nteger * set_Pr7577011563204128103nteger ) > $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_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 > fun_se7788542033685991124_nat_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member4513763875858039863et_nat: ( produc8799533983703829792et_nat * set_Pr1312406793078187350et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member9174161442226468181et_nat: ( produc3112576180687729086et_nat * set_Pr4040394263813967220et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    member3012145478770660624_nat_o: ( produc6671579610312156647_nat_o * set_Pr2890355626087285447_nat_o ) > $o ).

tff(sy_c_member_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,
    member4554412811331277712it_nat: ( produc3260487557148687353it_nat * set_Pr7098220151150636591it_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: ( product_prod_a_a * set_Product_prod_a_a ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member8682313912305727889it_nat: ( produc7388388658123137530it_nat * set_Pr7600907837789447088it_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: ( product_prod_b_b * set_Product_prod_b_b ) > $o ).

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

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

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > fun_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 > fun_se9106460625052465826_int_o ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: fun_b_assn ).

tff(sy_v_R,type,
    r: fun_a_assn ).

tff(sy_v_as____,type,
    as: set_nat ).

tff(sy_v_f,type,
    f: heap_Time_Heap_a ).

tff(sy_v_g,type,
    g: fun_a_6164325696594227994Heap_b ).

tff(sy_v_h_H_H____,type,
    h: heap_e7401611519738050253t_unit ).

tff(sy_v_h_H____,type,
    h2: heap_e7401611519738050253t_unit ).

tff(sy_v_h____,type,
    h3: heap_e7401611519738050253t_unit ).

tff(sy_v_rf____,type,
    rf: a ).

tff(sy_v_rg____,type,
    rg: b ).

tff(sy_v_t_H_H____,type,
    t: nat ).

tff(sy_v_t_H____,type,
    t2: nat ).

% Relevant facts (11309)
tff(fact_0__092_060open_062new__addrs_Ah_H_A_Inew__addrs_Ah_Aas_Ah_H_J_Ah_H_H_A_061_Anew__addrs_Ah_Aas_Ah_H_H_092_060close_062,axiom,
    hoare_new_addrs(h2,hoare_new_addrs(h3,as,h2),h) = hoare_new_addrs(h3,as,h) ).

% \<open>new_addrs h' (new_addrs h as h') h'' = new_addrs h as h''\<close>
tff(fact_1__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_092_060close_062,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(p),produc7507926704131184380et_nat(h3,as)) ).

% \<open>(h, as) \<Turnstile> P\<close>
tff(fact_2_new__addr__refl,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] : hoare_new_addrs(Ha,Asa,Ha) = Asa ).

% new_addr_refl
tff(fact_3_POST__G,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(q,rg)),produc7507926704131184380et_nat(h,hoare_new_addrs(h2,hoare_new_addrs(h3,as,h2),h))) ).

% POST_G
tff(fact_4_POST__F,axiom,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(r,rf)),produc7507926704131184380et_nat(h2,hoare_new_addrs(h3,as,h2))) ).

% POST_F
tff(fact_5_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( rep_assn(X) = rep_assn(Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_6_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_7_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X2: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X1),X2) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_8_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X2: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X1),X2) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_9_prod_Oinject,axiom,
    ! [X1: b,X2: produc6653097349344004940it_nat,Y1: b,Y2: produc6653097349344004940it_nat] :
      ( ( aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X1),X2) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_10_prod_Oinject,axiom,
    ! [X1: a,X2: produc6653097349344004940it_nat,Y1: a,Y2: produc6653097349344004940it_nat] :
      ( ( aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X1),X2) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_11_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_12_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_13_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_14_old_Oprod_Oinject,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,A2: b,B2: produc6653097349344004940it_nat] :
      ( ( aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_15_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat] :
      ( ( aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_16_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] : X != produc7507926704131184380et_nat(H,As) ).

% one_assn_raw.cases
tff(fact_17_times__assn__raw_Ocases,axiom,
    ! [X: produc3112576180687729086et_nat] :
      ~ ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As: set_nat] : X != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,P),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Q),produc7507926704131184380et_nat(H,As))) ).

% times_assn_raw.cases
tff(fact_18_prod__induct4,axiom,
    ! [Pa: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: heap_e7401611519738050253t_unit,D: set_nat] : aa_Pro2671238023309943196_nat_o(Pa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B3),produc7507926704131184380et_nat(C,D))))
     => aa_Pro2671238023309943196_nat_o(Pa,X) ) ).

% prod_induct4
tff(fact_19_prod__induct3,axiom,
    ! [Pa: fun_Pr2620521676128474833_nat_o,X: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: heap_e7401611519738050253t_unit,C: set_nat] : aa_Pro5643009357358181178_nat_o(Pa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),produc7507926704131184380et_nat(B3,C)))
     => aa_Pro5643009357358181178_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_20_prod__induct3,axiom,
    ! [Pa: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(Pa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B3),C)))
     => aa_Pro2671238023309943196_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_21_prod__induct3,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,X: produc7388388658123137530it_nat] :
      ( ! [A3: b,B3: heap_e7401611519738050253t_unit,C: nat] : aa_Pro1537530206668979808_nat_o(Pa,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),produc584006145561248582it_nat(B3,C)))
     => aa_Pro1537530206668979808_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_22_prod__induct3,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat] :
      ( ! [A3: a,B3: heap_e7401611519738050253t_unit,C: nat] : aa_Pro6426884464393220769_nat_o(Pa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),produc584006145561248582it_nat(B3,C)))
     => aa_Pro6426884464393220769_nat_o(Pa,X) ) ).

% prod_induct3
tff(fact_23_prod__cases4,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: heap_e7401611519738050253t_unit,D: set_nat] : Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B3),produc7507926704131184380et_nat(C,D))) ).

% prod_cases4
tff(fact_24_prod__cases3,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: heap_e7401611519738050253t_unit,C: set_nat] : Y != aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),produc7507926704131184380et_nat(B3,C)) ).

% prod_cases3
tff(fact_25_prod__cases3,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat] : Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,B3),C)) ).

% prod_cases3
tff(fact_26_prod__cases3,axiom,
    ! [Y: produc7388388658123137530it_nat] :
      ~ ! [A3: b,B3: heap_e7401611519738050253t_unit,C: nat] : Y != aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),produc584006145561248582it_nat(B3,C)) ).

% prod_cases3
tff(fact_27_prod__cases3,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A3: a,B3: heap_e7401611519738050253t_unit,C: nat] : Y != aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),produc584006145561248582it_nat(B3,C)) ).

% prod_cases3
tff(fact_28_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_29_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_30_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_31_Pair__inject,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,A2: b,B2: produc6653097349344004940it_nat] :
      ( ( aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_32_Pair__inject,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat] :
      ( ( aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_33_prod__cases,axiom,
    ! [Pa: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A3: int,B3: int] : aa_Pro4592068669816069595_int_o(Pa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3))
     => aa_Pro4592068669816069595_int_o(Pa,P2) ) ).

% prod_cases
tff(fact_34_prod__cases,axiom,
    ! [Pa: fun_Pr2620521676128474833_nat_o,P2: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(Pa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3))
     => aa_Pro5643009357358181178_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_35_prod__cases,axiom,
    ! [Pa: fun_Pr1227177680593602867_nat_o,P2: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(Pa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3))
     => aa_Pro2671238023309943196_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_36_prod__cases,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,P2: produc7388388658123137530it_nat] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] : aa_Pro1537530206668979808_nat_o(Pa,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3))
     => aa_Pro1537530206668979808_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_37_prod__cases,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,P2: produc3260487557148687353it_nat] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] : aa_Pro6426884464393220769_nat_o(Pa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3))
     => aa_Pro6426884464393220769_nat_o(Pa,P2) ) ).

% prod_cases
tff(fact_38_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y3: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) ).

% surj_pair
tff(fact_39_surj__pair,axiom,
    ! [P2: produc8799533983703829792et_nat] :
    ? [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] : P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3) ).

% surj_pair
tff(fact_40_surj__pair,axiom,
    ! [P2: produc3112576180687729086et_nat] :
    ? [X3: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] : P2 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3) ).

% surj_pair
tff(fact_41_surj__pair,axiom,
    ! [P2: produc7388388658123137530it_nat] :
    ? [X3: b,Y3: produc6653097349344004940it_nat] : P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3) ).

% surj_pair
tff(fact_42_surj__pair,axiom,
    ! [P2: produc3260487557148687353it_nat] :
    ? [X3: a,Y3: produc6653097349344004940it_nat] : P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) ).

% surj_pair
tff(fact_43_bex2I,axiom,
    ! [A: int,B: int,S: set_Pr958786334691620121nt_int,Pa: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),S)
     => ( ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),S)
         => aa_int_o(aa_int_fun_int_o(Pa,A),B) )
       => ? [A3: int,B3: int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3)),S)
            & aa_int_o(aa_int_fun_int_o(Pa,A3),B3) ) ) ) ).

% bex2I
tff(fact_44_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat,Pa: fun_fu8419763957888848259_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B),S)
     => ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B),S)
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Pa,A),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3),S)
            & aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Pa,A3),B3) ) ) ) ).

% bex2I
tff(fact_45_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat,Pa: fun_fu2611741329380808165_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B),S)
     => ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B),S)
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Pa,A),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3),S)
            & aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Pa,A3),B3) ) ) ) ).

% bex2I
tff(fact_46_bex2I,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,S: set_Pr7600907837789447088it_nat,Pa: fun_b_540864168722188329_nat_o] :
      ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B),S)
     => ( ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B),S)
         => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Pa,A),B) )
       => ? [A3: b,B3: produc6653097349344004940it_nat] :
            ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3),S)
            & aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Pa,A3),B3) ) ) ) ).

% bex2I
tff(fact_47_bex2I,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,S: set_Pr7098220151150636591it_nat,Pa: fun_a_2467521563176467050_nat_o] :
      ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),S)
     => ( ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),S)
         => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,A),B) )
       => ? [A3: a,B3: produc6653097349344004940it_nat] :
            ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3),S)
            & aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,A3),B3) ) ) ) ).

% bex2I
tff(fact_48_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) ).

% old.prod.exhaust
tff(fact_49_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] : Y != aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_50_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] : Y != aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_51_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7388388658123137530it_nat] :
      ~ ! [A3: b,B3: produc6653097349344004940it_nat] : Y != aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A3: a,B3: produc6653097349344004940it_nat] : Y != aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_53_T1,axiom,
    hoare_hoare_triple_a(p,f,r) ).

% T1
tff(fact_54_T2,axiom,
    ! [X: a] : hoare_hoare_triple_b(aa_a_assn(r,X),aa_a_H8829438778223708363Heap_b(g,X),q) ).

% T2
tff(fact_55_EX__G,axiom,
    aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_a_H8829438778223708363Heap_b(g,rf)),h2) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,rg),produc584006145561248582it_nat(h,t))) ).

% EX_G
tff(fact_56_LIM__G,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(h2)),lim_Product_unit(h)) ).

% LIM_G
tff(fact_57_LIM__F,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(h3)),lim_Product_unit(h2)) ).

% LIM_F
tff(fact_58_EX__F,axiom,
    aa_hea4724376117875161946it_nat(heap_Time_execute_a(f),h3) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,rf),produc584006145561248582it_nat(h2,t2))) ).

% EX_F
tff(fact_59_mem__Collect__eq,axiom,
    ! [A: $o,Pa: fun_o_o] :
      ( aa_set_o_o(member_o((A)),collect_o(Pa))
    <=> aa_o_o(Pa,(A)) ) ).

% mem_Collect_eq
tff(fact_60_mem__Collect__eq,axiom,
    ! [A: list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(Pa))
    <=> aa_list_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_61_mem__Collect__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),collec5210948495886036740nt_int(Pa))
    <=> aa_set6170482117830564987_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_62_mem__Collect__eq,axiom,
    ! [A: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(Pa))
    <=> aa_set_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_63_mem__Collect__eq,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(Pa))
    <=> aa_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_64_mem__Collect__eq,axiom,
    ! [A: int,Pa: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(Pa))
    <=> aa_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_65_Collect__mem__eq,axiom,
    ! [A4: set_o] : collect_o(aTP_Lamm_a(A4)) = A4 ).

% Collect_mem_eq
tff(fact_66_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : collect_list_nat(aTP_Lamm_aa(A4)) = A4 ).

% Collect_mem_eq
tff(fact_67_Collect__mem__eq,axiom,
    ! [A4: set_se6260736226359567993nt_int] : collec5210948495886036740nt_int(aTP_Lamm_ab(A4)) = A4 ).

% Collect_mem_eq
tff(fact_68_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A4)) = A4 ).

% Collect_mem_eq
tff(fact_69_Collect__mem__eq,axiom,
    ! [A4: set_nat] : collect_nat(aTP_Lamm_ad(A4)) = A4 ).

% Collect_mem_eq
tff(fact_70_Collect__mem__eq,axiom,
    ! [A4: set_int] : collect_int(aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_71_Collect__cong,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(Pa,X3)
        <=> aa_list_nat_o(Qa,X3) )
     => ( collect_list_nat(Pa) = collect_list_nat(Qa) ) ) ).

% Collect_cong
tff(fact_72_Collect__cong,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( aa_set6170482117830564987_int_o(Pa,X3)
        <=> aa_set6170482117830564987_int_o(Qa,X3) )
     => ( collec5210948495886036740nt_int(Pa) = collec5210948495886036740nt_int(Qa) ) ) ).

% Collect_cong
tff(fact_73_Collect__cong,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(Pa,X3)
        <=> aa_set_nat_o(Qa,X3) )
     => ( collect_set_nat(Pa) = collect_set_nat(Qa) ) ) ).

% Collect_cong
tff(fact_74_Collect__cong,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
        <=> aa_nat_o(Qa,X3) )
     => ( collect_nat(Pa) = collect_nat(Qa) ) ) ).

% Collect_cong
tff(fact_75_Collect__cong,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
        <=> aa_int_o(Qa,X3) )
     => ( collect_int(Pa) = collect_int(Qa) ) ) ).

% Collect_cong
tff(fact_76_bijective__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(bijective_int_int,Ra)
    <=> ( ! [X4: int,Y4: int,Z: int] :
            ( ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4)),Ra)
              & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Z)),Ra) )
           => ( Y4 = Z ) )
        & ! [X4: int,Y4: int,Z: int] :
            ( ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Z)),Ra)
              & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y4),Z)),Ra) )
           => ( X4 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_77_bijective__def,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat] :
      ( biject6574446902887564708et_nat(Ra)
    <=> ( ! [X4: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat,Z: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Y4),Ra)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Z),Ra) )
           => ( Y4 = Z ) )
        & ! [X4: fun_Pr6460944699146269935_nat_o,Y4: fun_Pr6460944699146269935_nat_o,Z: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Z),Ra)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Y4),Z),Ra) )
           => ( X4 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_78_bijective__def,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat] :
      ( biject4066639966817638850et_nat(Ra)
    <=> ( ! [X4: fun_Pr6460944699146269935_nat_o,Y4: produc8799533983703829792et_nat,Z: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Y4),Ra)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Z),Ra) )
           => ( Y4 = Z ) )
        & ! [X4: fun_Pr6460944699146269935_nat_o,Y4: fun_Pr6460944699146269935_nat_o,Z: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Z),Ra)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Y4),Z),Ra) )
           => ( X4 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_79_bijective__def,axiom,
    ! [Ra: set_Pr7600907837789447088it_nat] :
      ( biject3496835315794560510it_nat(Ra)
    <=> ( ! [X4: b,Y4: produc6653097349344004940it_nat,Z: produc6653097349344004940it_nat] :
            ( ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X4),Y4),Ra)
              & member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X4),Z),Ra) )
           => ( Y4 = Z ) )
        & ! [X4: b,Y4: b,Z: produc6653097349344004940it_nat] :
            ( ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X4),Z),Ra)
              & member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y4),Z),Ra) )
           => ( X4 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_80_bijective__def,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat] :
      ( biject8592306251674886141it_nat(Ra)
    <=> ( ! [X4: a,Y4: produc6653097349344004940it_nat,Z: produc6653097349344004940it_nat] :
            ( ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Y4),Ra)
              & member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Z),Ra) )
           => ( Y4 = Z ) )
        & ! [X4: a,Y4: a,Z: produc6653097349344004940it_nat] :
            ( ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Z),Ra)
              & member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y4),Z),Ra) )
           => ( X4 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_81_ssubst__Pair__rhs,axiom,
    ! [R: int,S2: int,Ra: set_Pr958786334691620121nt_int,S3: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R),S2)),Ra)
     => ( ( S3 = S2 )
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R),S3)),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_82_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr6460944699146269935_nat_o,S2: produc3658429121746597890et_nat,Ra: set_Pr1312406793078187350et_nat,S3: produc3658429121746597890et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,R),S2),Ra)
     => ( ( S3 = S2 )
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,R),S3),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_83_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr6460944699146269935_nat_o,S2: produc8799533983703829792et_nat,Ra: set_Pr4040394263813967220et_nat,S3: produc8799533983703829792et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,R),S2),Ra)
     => ( ( S3 = S2 )
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,R),S3),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_84_ssubst__Pair__rhs,axiom,
    ! [R: b,S2: produc6653097349344004940it_nat,Ra: set_Pr7600907837789447088it_nat,S3: produc6653097349344004940it_nat] :
      ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R),S2),Ra)
     => ( ( S3 = S2 )
       => member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R),S3),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_85_ssubst__Pair__rhs,axiom,
    ! [R: a,S2: produc6653097349344004940it_nat,Ra: set_Pr7098220151150636591it_nat,S3: produc6653097349344004940it_nat] :
      ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),S2),Ra)
     => ( ( S3 = S2 )
       => member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),S3),Ra) ) ) ).

% ssubst_Pair_rhs
tff(fact_86_curryI,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(Fa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(Fa),A),B) ) ).

% curryI
tff(fact_87_curryI,axiom,
    ! [Fa: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro5643009357358181178_nat_o(Fa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(Fa),A),B) ) ).

% curryI
tff(fact_88_curryI,axiom,
    ! [Fa: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro2671238023309943196_nat_o(Fa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(Fa),A),B) ) ).

% curryI
tff(fact_89_curryI,axiom,
    ! [Fa: fun_Pr856651894144386551_nat_o,A: b,B: produc6653097349344004940it_nat] :
      ( aa_Pro1537530206668979808_nat_o(Fa,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B))
     => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(produc8043051624866120150_nat_o(Fa),A),B) ) ).

% curryI
tff(fact_90_curryI,axiom,
    ! [Fa: fun_Pr5746006151868627512_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro6426884464393220769_nat_o(Fa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))
     => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(produc3709033845735585303_nat_o(Fa),A),B) ) ).

% curryI
tff(fact_91_le__some__optE,axiom,
    ! [M: set_int,X: option_set_int] :
      ( ord_le353528952715127954et_int(some_set_int(M),X)
     => ~ ! [M2: set_int] :
            ( ( X = some_set_int(M2) )
           => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,M),M2) ) ) ).

% le_some_optE
tff(fact_92_le__some__optE,axiom,
    ! [M: rat,X: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),X)
     => ~ ! [M2: rat] :
            ( ( X = some_rat(M2) )
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),M2) ) ) ).

% le_some_optE
tff(fact_93_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,M)),X)
     => ~ ! [M2: num] :
            ( ( X = aa_num_option_num(some_num,M2) )
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),M2) ) ) ).

% le_some_optE
tff(fact_94_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ord_le5914376470875661696on_nat(aa_nat_option_nat(some_nat,M),X)
     => ~ ! [M2: nat] :
            ( ( X = aa_nat_option_nat(some_nat,M2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),M2) ) ) ).

% le_some_optE
tff(fact_95_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,M),X)
     => ~ ! [M2: int] :
            ( ( X = aa_int_option_int(some_int,M2) )
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),M2) ) ) ).

% le_some_optE
tff(fact_96_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),C2)
       => ( ( C2 = D2 )
         => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_97_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ( C2 = D2 )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_98_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ( C2 = D2 )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_99_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ( C2 = D2 )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_100_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ( C2 = D2 )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_101_hoare__triple__preI,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H)
         => hoare_hoare_triple_a(Pa,C2,Qa) )
     => hoare_hoare_triple_a(Pa,C2,Qa) ) ).

% hoare_triple_preI
tff(fact_102_hoare__triple__preI,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H)
         => hoare_hoare_triple_b(Pa,C2,Qa) )
     => hoare_hoare_triple_b(Pa,C2,Qa) ) ).

% hoare_triple_preI
tff(fact_103_hoare__triple__preI,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H)
         => hoare_8945653483474564448t_unit(Pa,C2,Qa) )
     => hoare_8945653483474564448t_unit(Pa,C2,Qa) ) ).

% hoare_triple_preI
tff(fact_104_curryD,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(Fa),A),B)
     => aa_Pro4592068669816069595_int_o(Fa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% curryD
tff(fact_105_curryD,axiom,
    ! [Fa: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(Fa),A),B)
     => aa_Pro5643009357358181178_nat_o(Fa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B)) ) ).

% curryD
tff(fact_106_curryD,axiom,
    ! [Fa: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(Fa),A),B)
     => aa_Pro2671238023309943196_nat_o(Fa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B)) ) ).

% curryD
tff(fact_107_curryD,axiom,
    ! [Fa: fun_Pr856651894144386551_nat_o,A: b,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(produc8043051624866120150_nat_o(Fa),A),B)
     => aa_Pro1537530206668979808_nat_o(Fa,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B)) ) ).

% curryD
tff(fact_108_curryD,axiom,
    ! [Fa: fun_Pr5746006151868627512_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(produc3709033845735585303_nat_o(Fa),A),B)
     => aa_Pro6426884464393220769_nat_o(Fa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B)) ) ).

% curryD
tff(fact_109_curryE,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(Fa),A),B)
     => aa_Pro4592068669816069595_int_o(Fa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% curryE
tff(fact_110_curryE,axiom,
    ! [Fa: fun_Pr2620521676128474833_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(Fa),A),B)
     => aa_Pro5643009357358181178_nat_o(Fa,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B)) ) ).

% curryE
tff(fact_111_curryE,axiom,
    ! [Fa: fun_Pr1227177680593602867_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(Fa),A),B)
     => aa_Pro2671238023309943196_nat_o(Fa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B)) ) ).

% curryE
tff(fact_112_curryE,axiom,
    ! [Fa: fun_Pr856651894144386551_nat_o,A: b,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(produc8043051624866120150_nat_o(Fa),A),B)
     => aa_Pro1537530206668979808_nat_o(Fa,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B)) ) ).

% curryE
tff(fact_113_curryE,axiom,
    ! [Fa: fun_Pr5746006151868627512_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(produc3709033845735585303_nat_o(Fa),A),B)
     => aa_Pro6426884464393220769_nat_o(Fa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B)) ) ).

% curryE
tff(fact_114_less__eq__option__Some,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le353528952715127954et_int(some_set_int(X),some_set_int(Y))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% less_eq_option_Some
tff(fact_115_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(X),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% less_eq_option_Some
tff(fact_116_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,X)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% less_eq_option_Some
tff(fact_117_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ord_le5914376470875661696on_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% less_eq_option_Some
tff(fact_118_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,X),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% less_eq_option_Some
tff(fact_119_option_Oinject,axiom,
    ! [X2: produc7388388658123137530it_nat,Y2: produc7388388658123137530it_nat] :
      ( ( aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_120_option_Oinject,axiom,
    ! [X2: produc3260487557148687353it_nat,Y2: produc3260487557148687353it_nat] :
      ( ( aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_121_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_122_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_123_dual__order_Orefl,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A) ).

% dual_order.refl
tff(fact_124_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_125_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_126_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_127_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_128_order__refl,axiom,
    ! [X: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),X) ).

% order_refl
tff(fact_129_order__refl,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X) ).

% order_refl
tff(fact_130_order__refl,axiom,
    ! [X: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),X) ).

% order_refl
tff(fact_131_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_132_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_133__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rg_Ah_H_H_At_H_H_O_A_092_060lbrakk_062execute_A_Ig_Arf_J_Ah_H_A_061_ASome_A_Irg_M_Ah_H_H_M_At_H_H_J_059_A_Ih_H_H_M_Anew__addrs_Ah_H_A_Inew__addrs_Ah_Aas_Ah_H_J_Ah_H_H_J_A_092_060Turnstile_062_AQ_Arg_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_H_A_092_060and_062_Aa_A_092_060notin_062_Anew__addrs_Ah_Aas_Ah_H_125_Ah_H_Ah_H_H_059_Alim_Ah_H_A_092_060le_062_Alim_Ah_H_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Rg: b,H2: heap_e7401611519738050253t_unit] :
        ( ? [T: nat] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_a_H8829438778223708363Heap_b(g,rf)),h2) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Rg),produc584006145561248582it_nat(H2,T)))
       => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(q,Rg)),produc7507926704131184380et_nat(H2,hoare_new_addrs(h2,hoare_new_addrs(h3,as,h2),H2)))
         => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_af),h2),H2)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(h2)),lim_Product_unit(H2)) ) ) ) ).

% \<open>\<And>thesis. (\<And>rg h'' t''. \<lbrakk>execute (g rf) h' = Some (rg, h'', t''); (h'', new_addrs h' (new_addrs h as h') h'') \<Turnstile> Q rg; relH {a. a < lim h' \<and> a \<notin> new_addrs h as h'} h' h''; lim h' \<le> lim h''\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_134__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rf_Ah_H_At_H_O_A_092_060lbrakk_062execute_Af_Ah_A_061_ASome_A_Irf_M_Ah_H_M_At_H_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas_Ah_H_J_A_092_060Turnstile_062_AR_Arf_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_Ah_Ah_H_059_Alim_Ah_A_092_060le_062_Alim_Ah_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Rf: a,H3: heap_e7401611519738050253t_unit] :
        ( ? [T2: nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(f),h3) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Rf),produc584006145561248582it_nat(H3,T2)))
       => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(r,Rf)),produc7507926704131184380et_nat(H3,hoare_new_addrs(h3,as,H3)))
         => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_ag),h3),H3)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(h3)),lim_Product_unit(H3)) ) ) ) ).

% \<open>\<And>thesis. (\<And>rf h' t'. \<lbrakk>execute f h = Some (rf, h', t'); (h', new_addrs h as h') \<Turnstile> R rf; relH {a. a < lim h \<and> a \<notin> as} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_135_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc7388388658123137530it_nat,X: produc7388388658123137530it_nat] : the_de4670738602788725795it_nat(Uu,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X)) = X ).

% the_default.simps(1)
tff(fact_136_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc3260487557148687353it_nat,X: produc3260487557148687353it_nat] : the_de542837501814275618it_nat(Uu,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X)) = X ).

% the_default.simps(1)
tff(fact_137_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc8664842809031399944it_nat,X: produc8664842809031399944it_nat] : the_de2487931475039285041it_nat(Uu,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X)) = X ).

% the_default.simps(1)
tff(fact_138_the__default_Osimps_I1_J,axiom,
    ! [Uu: num,X: num] : aa_option_num_num(the_default_num(Uu),aa_num_option_num(some_num,X)) = X ).

% the_default.simps(1)
tff(fact_139_hoare__triple__effect,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_a(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ? [H3: heap_e7401611519738050253t_unit,R2: a,T3: nat] :
            ( aa_nat_o(heap_Time_effect_a(C2,Ha,H3,R2),T3)
            & aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R2)),produc7507926704131184380et_nat(H3,hoare_new_addrs(Ha,Asa,H3))) ) ) ) ).

% hoare_triple_effect
tff(fact_140_hoare__triple__effect,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_b(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ? [H3: heap_e7401611519738050253t_unit,R2: b,T3: nat] :
            ( aa_nat_o(heap_Time_effect_b(C2,Ha,H3,R2),T3)
            & aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(Qa,R2)),produc7507926704131184380et_nat(H3,hoare_new_addrs(Ha,Asa,H3))) ) ) ) ).

% hoare_triple_effect
tff(fact_141_hoare__triple__effect,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_8945653483474564448t_unit(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ? [H3: heap_e7401611519738050253t_unit,R2: product_unit,T3: nat] :
            ( aa_nat_o(heap_T6553295506729943825t_unit(C2,Ha,H3,R2),T3)
            & aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R2)),produc7507926704131184380et_nat(H3,hoare_new_addrs(Ha,Asa,H3))) ) ) ) ).

% hoare_triple_effect
tff(fact_142_hoare__triple__success,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_a(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => aa_hea7994211679605476167unit_o(heap_Time_success_a(C2),Ha) ) ) ).

% hoare_triple_success
tff(fact_143_hoare__triple__success,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_b(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => aa_hea7994211679605476167unit_o(heap_Time_success_b(C2),Ha) ) ) ).

% hoare_triple_success
tff(fact_144_hoare__triple__success,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_8945653483474564448t_unit(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => aa_hea7994211679605476167unit_o(heap_T584514906347983379t_unit(C2),Ha) ) ) ).

% hoare_triple_success
tff(fact_145_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As2: fun_int_set_int] :
      ( bNF_Ca583493526879471924et_int(R,As2)
    <=> ! [I: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J)),R)
         => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(As2,I)),aa_int_set_int(As2,J)) ) ) ).

% relChain_def
tff(fact_146_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As2: fun_int_rat] :
      ( bNF_Ca1332973979827979050nt_rat(R,As2)
    <=> ! [I: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J)),R)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(As2,I)),aa_int_rat(As2,J)) ) ) ).

% relChain_def
tff(fact_147_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As2: fun_int_num] :
      ( bNF_Ca7748807862925029228nt_num(R,As2)
    <=> ! [I: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J)),R)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(As2,I)),aa_int_num(As2,J)) ) ) ).

% relChain_def
tff(fact_148_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As2: fun_int_nat] :
      ( bNF_Ca1968104039914474786nt_nat(R,As2)
    <=> ! [I: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J)),R)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(As2,I)),aa_int_nat(As2,J)) ) ) ).

% relChain_def
tff(fact_149_relChain__def,axiom,
    ! [R: set_Pr958786334691620121nt_int,As2: fun_int_int] :
      ( bNF_Ca1965613569405424510nt_int(R,As2)
    <=> ! [I: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J)),R)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(As2,I)),aa_int_int(As2,J)) ) ) ).

% relChain_def
tff(fact_150_Heap__eqI,axiom,
    ! [Fa: heap_Time_Heap_b,Ga: heap_Time_Heap_b] :
      ( ! [H: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),H) = aa_hea5227063804513972443it_nat(heap_Time_execute_b(Ga),H)
     => ( Fa = Ga ) ) ).

% Heap_eqI
tff(fact_151_Heap__eqI,axiom,
    ! [Fa: heap_Time_Heap_a,Ga: heap_Time_Heap_a] :
      ( ! [H: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),H) = aa_hea4724376117875161946it_nat(heap_Time_execute_a(Ga),H)
     => ( Fa = Ga ) ) ).

% Heap_eqI
tff(fact_152_RH__G,axiom,
    aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_af),h2),h) ).

% RH_G
tff(fact_153_RH__F,axiom,
    aa_hea7994211679605476167unit_o(relH(collect_nat(aTP_Lamm_ag),h3),h2) ).

% RH_F
tff(fact_154_less__option__Some,axiom,
    ! [X: assn,Y: assn] :
      ( ord_less_option_assn(some_assn(X),some_assn(Y))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y) ) ).

% less_option_Some
tff(fact_155_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(X),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% less_option_Some
tff(fact_156_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,aa_num_option_num(some_num,X)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% less_option_Some
tff(fact_157_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ord_less_option_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% less_option_Some
tff(fact_158_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ord_less_option_int(aa_int_option_int(some_int,X),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% less_option_Some
tff(fact_159_relH__subset,axiom,
    ! [Bs: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_hea7994211679605476167unit_o(relH(Bs,Ha),H_a)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Asa),Bs)
       => aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a) ) ) ).

% relH_subset
tff(fact_160_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X) ).

% lt_ex
tff(fact_161_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),X) ).

% lt_ex
tff(fact_162_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_1) ).

% gt_ex
tff(fact_163_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X_1) ).

% gt_ex
tff(fact_164_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X),X_1) ).

% gt_ex
tff(fact_165_dense,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Y) ) ) ).

% dense
tff(fact_166_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_167_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_168_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_169_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_170_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_171_order_Oasym,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A) ) ).

% order.asym
tff(fact_172_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_173_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_174_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_175_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_176_ord__eq__less__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( A = B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_177_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_178_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_179_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_180_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_181_ord__less__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( B = C2 )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_182_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_183_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_184_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_185_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_186_less__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y5),X3)
             => aa_nat_o(Pa,Y5) )
         => aa_nat_o(Pa,X3) )
     => aa_nat_o(Pa,A) ) ).

% less_induct
tff(fact_187_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_188_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_189_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_190_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_191_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ( X != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_cases
tff(fact_192_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( ( X != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_cases
tff(fact_193_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( ( X != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_cases
tff(fact_194_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( ( X != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_cases
tff(fact_195_dual__order_Oasym,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% dual_order.asym
tff(fact_196_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_197_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_198_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_199_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_200_dual__order_Oirrefl,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),A) ).

% dual_order.irrefl
tff(fact_201_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_202_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_203_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_204_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_205_exists__least__iff,axiom,
    ! [Pa: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ? [N: nat] :
          ( aa_nat_o(Pa,N)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
             => ~ aa_nat_o(Pa,M3) ) ) ) ).

% exists_least_iff
tff(fact_206_linorder__less__wlog,axiom,
    ! [Pa: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(Pa,A3),B3) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(Pa,A3),A3)
       => ( ! [A3: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(Pa,B3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(Pa,A3),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_207_linorder__less__wlog,axiom,
    ! [Pa: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(Pa,A3),B3) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(Pa,A3),A3)
       => ( ! [A3: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(Pa,B3),A3)
             => aa_num_o(aa_num_fun_num_o(Pa,A3),B3) )
         => aa_num_o(aa_num_fun_num_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_208_linorder__less__wlog,axiom,
    ! [Pa: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,A3),A3)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,B3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_209_linorder__less__wlog,axiom,
    ! [Pa: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(Pa,A3),A3)
       => ( ! [A3: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(Pa,B3),A3)
             => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(Pa,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_210_order_Ostrict__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% order.strict_trans
tff(fact_211_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans
tff(fact_212_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans
tff(fact_213_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans
tff(fact_214_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans
tff(fact_215_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_216_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_217_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_218_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_219_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_220_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: 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,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_221_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_222_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_223_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: 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,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_224_order_Ostrict__implies__not__eq,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_225_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_226_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_227_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_228_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_229_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_230_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_231_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_232_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_233_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_234_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neqE
tff(fact_235_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_neqE
tff(fact_236_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE
tff(fact_237_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE
tff(fact_238_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_asym
tff(fact_239_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_asym
tff(fact_240_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_asym
tff(fact_241_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_asym
tff(fact_242_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_asym
tff(fact_243_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_244_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_245_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_246_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_247_order__less__asym_H,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A) ) ).

% order_less_asym'
tff(fact_248_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_249_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_250_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_251_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_252_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z3)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z3) ) ) ).

% order_less_trans
tff(fact_253_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3) ) ) ).

% order_less_trans
tff(fact_254_order__less__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3) ) ) ).

% order_less_trans
tff(fact_255_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3) ) ) ).

% order_less_trans
tff(fact_256_order__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3) ) ) ).

% order_less_trans
tff(fact_257_ord__eq__less__subst,axiom,
    ! [A: assn,Fa: fun_assn_assn,B: assn,C2: assn] :
      ( ( A = aa_assn_assn(Fa,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,X3)),aa_assn_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_258_ord__eq__less__subst,axiom,
    ! [A: rat,Fa: fun_assn_rat,B: assn,C2: assn] :
      ( ( A = aa_assn_rat(Fa,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,X3)),aa_assn_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_259_ord__eq__less__subst,axiom,
    ! [A: num,Fa: fun_assn_num,B: assn,C2: assn] :
      ( ( A = aa_assn_num(Fa,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(Fa,X3)),aa_assn_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_assn_num(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_260_ord__eq__less__subst,axiom,
    ! [A: nat,Fa: fun_assn_nat,B: assn,C2: assn] :
      ( ( A = aa_assn_nat(Fa,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(Fa,X3)),aa_assn_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_assn_nat(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_261_ord__eq__less__subst,axiom,
    ! [A: int,Fa: fun_assn_int,B: assn,C2: assn] :
      ( ( A = aa_assn_int(Fa,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(Fa,X3)),aa_assn_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_assn_int(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_262_ord__eq__less__subst,axiom,
    ! [A: assn,Fa: fun_rat_assn,B: rat,C2: rat] :
      ( ( A = aa_rat_assn(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_263_ord__eq__less__subst,axiom,
    ! [A: rat,Fa: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_264_ord__eq__less__subst,axiom,
    ! [A: num,Fa: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_265_ord__eq__less__subst,axiom,
    ! [A: nat,Fa: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_266_ord__eq__less__subst,axiom,
    ! [A: int,Fa: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(Fa,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_267_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_assn(Fa,B) = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,X3)),aa_assn_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_268_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_rat,C2: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_rat(Fa,B) = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,X3)),aa_assn_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_269_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_num,C2: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_num(Fa,B) = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(Fa,X3)),aa_assn_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_270_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_nat(Fa,B) = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(Fa,X3)),aa_assn_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_271_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( ( aa_assn_int(Fa,B) = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(Fa,X3)),aa_assn_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_272_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_assn,C2: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_assn(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_273_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_274_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_275_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_276_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_277_order__less__irrefl,axiom,
    ! [X: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),X) ).

% order_less_irrefl
tff(fact_278_order__less__irrefl,axiom,
    ! [X: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X) ).

% order_less_irrefl
tff(fact_279_order__less__irrefl,axiom,
    ! [X: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),X) ).

% order_less_irrefl
tff(fact_280_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_281_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_282_order__less__subst1,axiom,
    ! [A: assn,Fa: fun_assn_assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(Fa,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,X3)),aa_assn_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_283_order__less__subst1,axiom,
    ! [A: assn,Fa: fun_rat_assn,B: rat,C2: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_284_order__less__subst1,axiom,
    ! [A: assn,Fa: fun_num_assn,B: num,C2: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(Fa,X3)),aa_num_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_285_order__less__subst1,axiom,
    ! [A: assn,Fa: fun_nat_assn,B: nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(Fa,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,X3)),aa_nat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_286_order__less__subst1,axiom,
    ! [A: assn,Fa: fun_int_assn,B: int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(Fa,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(Fa,X3)),aa_int_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_287_order__less__subst1,axiom,
    ! [A: rat,Fa: fun_assn_rat,B: assn,C2: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(Fa,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,X3)),aa_assn_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_288_order__less__subst1,axiom,
    ! [A: rat,Fa: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_289_order__less__subst1,axiom,
    ! [A: rat,Fa: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_290_order__less__subst1,axiom,
    ! [A: rat,Fa: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(Fa,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_291_order__less__subst1,axiom,
    ! [A: rat,Fa: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(Fa,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,X3)),aa_int_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(Fa,C2)) ) ) ) ).

% order_less_subst1
tff(fact_292_order__less__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,X3)),aa_assn_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_293_order__less__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_rat,C2: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,X3)),aa_assn_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_294_order__less__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_num,C2: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(Fa,X3)),aa_assn_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_assn_num(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_295_order__less__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(Fa,X3)),aa_assn_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_296_order__less__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(Fa,X3)),aa_assn_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_297_order__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_assn,C2: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_298_order__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_rat,C2: 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(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_299_order__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_300_order__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_301_order__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_302_order__less__not__sym,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_not_sym
tff(fact_303_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_not_sym
tff(fact_304_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_not_sym
tff(fact_305_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_not_sym
tff(fact_306_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_not_sym
tff(fact_307_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,Pa: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_308_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_309_order__less__imp__triv,axiom,
    ! [X: num,Y: num,Pa: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_310_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_311_order__less__imp__triv,axiom,
    ! [X: int,Y: int,Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X)
       => (Pa) ) ) ).

% order_less_imp_triv
tff(fact_312_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      | ( X = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_less_linear
tff(fact_313_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      | ( X = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_less_linear
tff(fact_314_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      | ( X = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_less_linear
tff(fact_315_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      | ( X = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_less_linear
tff(fact_316_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_317_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_318_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_319_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_320_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_321_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_322_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_323_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_324_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_325_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_326_order__less__imp__not__less,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% order_less_imp_not_less
tff(fact_327_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_328_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% order_less_imp_not_less
tff(fact_329_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% order_less_imp_not_less
tff(fact_330_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% order_less_imp_not_less
tff(fact_331_relH__trans,axiom,
    ! [Asa: set_nat,H1: heap_e7401611519738050253t_unit,H22: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,H1),H22)
     => ( aa_hea7994211679605476167unit_o(relH(Asa,H22),H32)
       => aa_hea7994211679605476167unit_o(relH(Asa,H1),H32) ) ) ).

% relH_trans
tff(fact_332_relH__sym,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => aa_hea7994211679605476167unit_o(relH(Asa,H_a),Ha) ) ).

% relH_sym
tff(fact_333_leD,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y) ) ).

% leD
tff(fact_334_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y) ) ).

% leD
tff(fact_335_leD,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% leD
tff(fact_336_leD,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% leD
tff(fact_337_leD,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% leD
tff(fact_338_leD,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% leD
tff(fact_339_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% leI
tff(fact_340_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% leI
tff(fact_341_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% leI
tff(fact_342_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% leI
tff(fact_343_nless__le,axiom,
    ! [A: assn,B: assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_344_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
    <=> ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_345_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_346_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_347_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_348_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_349_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_350_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_351_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_352_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_353_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_354_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_355_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_356_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_357_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_358_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_359_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_360_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_361_dense__ge,axiom,
    ! [Z3: rat,Y: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3) ) ).

% dense_ge
tff(fact_362_dense__le,axiom,
    ! [Y: rat,Z3: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3) ) ).

% dense_le
tff(fact_363_less__le__not__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ) ).

% less_le_not_le
tff(fact_364_less__le__not__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X) ) ) ).

% less_le_not_le
tff(fact_365_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ) ).

% less_le_not_le
tff(fact_366_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ) ).

% less_le_not_le
tff(fact_367_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_368_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% less_le_not_le
tff(fact_369_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% not_le_imp_less
tff(fact_370_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y) ) ).

% not_le_imp_less
tff(fact_371_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% not_le_imp_less
tff(fact_372_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ).

% not_le_imp_less
tff(fact_373_order_Oorder__iff__strict,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_374_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_375_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_376_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_377_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_378_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_379_order_Ostrict__iff__order,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_380_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_381_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_382_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_383_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_384_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_385_order_Ostrict__trans1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% order.strict_trans1
tff(fact_386_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),C2) ) ) ).

% order.strict_trans1
tff(fact_387_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_388_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans1
tff(fact_389_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_390_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans1
tff(fact_391_order_Ostrict__trans2,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% order.strict_trans2
tff(fact_392_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),C2) ) ) ).

% order.strict_trans2
tff(fact_393_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_394_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans2
tff(fact_395_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_396_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans2
tff(fact_397_order_Ostrict__iff__not,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ) ).

% order.strict_iff_not
tff(fact_398_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_399_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_400_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_401_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_402_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_403_dense__ge__bounded,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),X)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3) ) ) ).

% dense_ge_bounded
tff(fact_404_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z3) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3) ) ) ).

% dense_le_bounded
tff(fact_405_dual__order_Oorder__iff__strict,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_406_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_407_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_408_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_409_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_410_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_411_dual__order_Ostrict__iff__order,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_412_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_413_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_414_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_415_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_416_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_417_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_418_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,C2),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_419_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_420_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_421_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_422_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_423_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_424_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_425_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_426_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_427_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_428_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_429_dual__order_Ostrict__iff__not,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_430_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_431_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_432_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_433_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_434_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_435_order_Ostrict__implies__order,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).

% order.strict_implies_order
tff(fact_436_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% order.strict_implies_order
tff(fact_437_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_438_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_439_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_440_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_441_dual__order_Ostrict__implies__order,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_442_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_443_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_444_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_445_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_446_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_447_order__le__less,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_448_order__le__less,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_449_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_450_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_451_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_452_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_453_order__less__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_454_order__less__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_455_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_456_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_457_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_458_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_459_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_not_le
tff(fact_460_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_not_le
tff(fact_461_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_not_le
tff(fact_462_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_not_le
tff(fact_463_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_not_less
tff(fact_464_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_not_less
tff(fact_465_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_not_less
tff(fact_466_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_not_less
tff(fact_467_order__less__imp__le,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% order_less_imp_le
tff(fact_468_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% order_less_imp_le
tff(fact_469_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% order_less_imp_le
tff(fact_470_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% order_less_imp_le
tff(fact_471_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_472_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_less_imp_le
tff(fact_473_order__le__neq__trans,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( ( A != B )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).

% order_le_neq_trans
tff(fact_474_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( ( A != B )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_475_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_476_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_477_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_478_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_479_order__neq__le__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( A != B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ) ).

% order_neq_le_trans
tff(fact_480_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_481_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_482_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_483_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_484_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_485_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),Z3)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z3) ) ) ).

% order_le_less_trans
tff(fact_486_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Y),Z3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Z3) ) ) ).

% order_le_less_trans
tff(fact_487_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3) ) ) ).

% order_le_less_trans
tff(fact_488_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3) ) ) ).

% order_le_less_trans
tff(fact_489_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3) ) ) ).

% order_le_less_trans
tff(fact_490_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3) ) ) ).

% order_le_less_trans
tff(fact_491_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z3)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z3) ) ) ).

% order_less_le_trans
tff(fact_492_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),Z3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Z3) ) ) ).

% order_less_le_trans
tff(fact_493_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3) ) ) ).

% order_less_le_trans
tff(fact_494_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3) ) ) ).

% order_less_le_trans
tff(fact_495_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3) ) ) ).

% order_less_le_trans
tff(fact_496_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3) ) ) ).

% order_less_le_trans
tff(fact_497_order__le__less__subst1,axiom,
    ! [A: assn,Fa: fun_assn_assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(Fa,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,X3)),aa_assn_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_498_order__le__less__subst1,axiom,
    ! [A: assn,Fa: fun_rat_assn,B: rat,C2: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_rat_assn(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_499_order__le__less__subst1,axiom,
    ! [A: assn,Fa: fun_num_assn,B: num,C2: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_num_assn(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(Fa,X3)),aa_num_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_500_order__le__less__subst1,axiom,
    ! [A: assn,Fa: fun_nat_assn,B: nat,C2: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_nat_assn(Fa,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,X3)),aa_nat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_nat_assn(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_501_order__le__less__subst1,axiom,
    ! [A: assn,Fa: fun_int_assn,B: int,C2: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_int_assn(Fa,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(Fa,X3)),aa_int_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_int_assn(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_502_order__le__less__subst1,axiom,
    ! [A: rat,Fa: fun_assn_rat,B: assn,C2: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_assn_rat(Fa,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,X3)),aa_assn_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_assn_rat(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_503_order__le__less__subst1,axiom,
    ! [A: rat,Fa: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_504_order__le__less__subst1,axiom,
    ! [A: rat,Fa: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_505_order__le__less__subst1,axiom,
    ! [A: rat,Fa: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(Fa,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_506_order__le__less__subst1,axiom,
    ! [A: rat,Fa: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(Fa,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,X3)),aa_int_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(Fa,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_507_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_assn,C2: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_508_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_509_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_510_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_511_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_512_order__le__less__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_assn,C2: assn] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_num_assn(Fa,X3)),aa_num_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_513_order__le__less__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_514_order__le__less__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,X3)),aa_num_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_515_order__le__less__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,X3)),aa_num_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_516_order__le__less__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,X3)),aa_num_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(Fa,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_517_order__less__le__subst1,axiom,
    ! [A: assn,Fa: fun_rat_assn,B: rat,C2: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_rat_assn(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_518_order__less__le__subst1,axiom,
    ! [A: rat,Fa: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_519_order__less__le__subst1,axiom,
    ! [A: num,Fa: fun_rat_num,B: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_520_order__less__le__subst1,axiom,
    ! [A: nat,Fa: fun_rat_nat,B: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_521_order__less__le__subst1,axiom,
    ! [A: int,Fa: fun_rat_int,B: rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_522_order__less__le__subst1,axiom,
    ! [A: assn,Fa: fun_num_assn,B: num,C2: num] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_num_assn(Fa,X3)),aa_num_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_num_assn(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_523_order__less__le__subst1,axiom,
    ! [A: rat,Fa: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_524_order__less__le__subst1,axiom,
    ! [A: num,Fa: fun_num_num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,X3)),aa_num_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_525_order__less__le__subst1,axiom,
    ! [A: nat,Fa: fun_num_nat,B: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,X3)),aa_num_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_526_order__less__le__subst1,axiom,
    ! [A: int,Fa: fun_num_int,B: num,C2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,X3)),aa_num_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(Fa,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_527_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,X3)),aa_assn_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_528_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_assn,C2: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,X3)),aa_rat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_529_order__less__le__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_assn,C2: assn] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_num_assn(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(Fa,X3)),aa_num_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_num_assn(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_530_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,Fa: fun_nat_assn,C2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(Fa,B)),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,X3)),aa_nat_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_531_order__less__le__subst2,axiom,
    ! [A: int,B: int,Fa: fun_int_assn,C2: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(Fa,B)),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(Fa,X3)),aa_int_assn(Fa,Y3)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_532_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,Fa: fun_assn_rat,C2: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_assn_rat(Fa,B)),C2)
       => ( ! [X3: assn,Y3: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,X3)),aa_assn_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_533_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_534_order__less__le__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_535_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,Fa: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,B)),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_536_order__less__le__subst2,axiom,
    ! [A: int,B: int,Fa: fun_int_rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,B)),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,X3)),aa_int_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_537_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_538_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),X) ) ).

% linorder_le_less_linear
tff(fact_539_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ).

% linorder_le_less_linear
tff(fact_540_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ).

% linorder_le_less_linear
tff(fact_541_order__le__imp__less__or__eq,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_542_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_543_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_544_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_545_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_546_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_547_exists__leI,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),N2)
           => ~ aa_nat_o(Pa,N3) )
       => aa_nat_o(Pa,N2) )
     => ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),N2)
          & aa_nat_o(Pa,N4) ) ) ).

% exists_leI
tff(fact_548_mod__relH,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Pa: assn] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
      <=> aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H_a,Asa)) ) ) ).

% mod_relH
tff(fact_549_hoare__triple__def,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( hoare_hoare_triple_a(Pa,C2,Qa)
    <=> ! [H4: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H4,As3))
         => ? [R3: a,H5: heap_e7401611519738050253t_unit] :
              ( ? [T4: nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(C2),H4) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R3),produc584006145561248582it_nat(H5,T4)))
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R3)),produc7507926704131184380et_nat(H5,hoare_new_addrs(H4,As3,H5)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(H4),As3)),H4),H5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H4)),lim_Product_unit(H5)) ) ) ) ).

% hoare_triple_def
tff(fact_550_hoare__triple__def,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( hoare_hoare_triple_b(Pa,C2,Qa)
    <=> ! [H4: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H4,As3))
         => ? [R3: b,H5: heap_e7401611519738050253t_unit] :
              ( ? [T4: nat] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(C2),H4) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R3),produc584006145561248582it_nat(H5,T4)))
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(Qa,R3)),produc7507926704131184380et_nat(H5,hoare_new_addrs(H4,As3,H5)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(H4),As3)),H4),H5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H4)),lim_Product_unit(H5)) ) ) ) ).

% hoare_triple_def
tff(fact_551_hoare__triple__def,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(Pa,C2,Qa)
    <=> ! [H4: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H4,As3))
         => ? [R3: product_unit,H5: heap_e7401611519738050253t_unit] :
              ( ? [T4: nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C2),H4) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R3,produc584006145561248582it_nat(H5,T4)))
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R3)),produc7507926704131184380et_nat(H5,hoare_new_addrs(H4,As3,H5)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(H4),As3)),H4),H5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H4)),lim_Product_unit(H5)) ) ) ) ).

% hoare_triple_def
tff(fact_552_hoare__tripleI,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H,As))
         => ? [R4: a,H6: heap_e7401611519738050253t_unit,T5: nat] :
              ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(C2),H) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R4),produc584006145561248582it_nat(H6,T5))) )
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R4)),produc7507926704131184380et_nat(H6,hoare_new_addrs(H,As,H6)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(H),As)),H),H6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H)),lim_Product_unit(H6)) ) )
     => hoare_hoare_triple_a(Pa,C2,Qa) ) ).

% hoare_tripleI
tff(fact_553_hoare__tripleI,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H,As))
         => ? [R4: b,H6: heap_e7401611519738050253t_unit,T5: nat] :
              ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(C2),H) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R4),produc584006145561248582it_nat(H6,T5))) )
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(Qa,R4)),produc7507926704131184380et_nat(H6,hoare_new_addrs(H,As,H6)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(H),As)),H),H6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H)),lim_Product_unit(H6)) ) )
     => hoare_hoare_triple_b(Pa,C2,Qa) ) ).

% hoare_tripleI
tff(fact_554_hoare__tripleI,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H,As))
         => ? [R4: product_unit,H6: heap_e7401611519738050253t_unit,T5: nat] :
              ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C2),H) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R4,produc584006145561248582it_nat(H6,T5))) )
              & aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R4)),produc7507926704131184380et_nat(H6,hoare_new_addrs(H,As,H6)))
              & aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(H),As)),H),H6)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(H)),lim_Product_unit(H6)) ) )
     => hoare_8945653483474564448t_unit(Pa,C2,Qa) ) ).

% hoare_tripleI
tff(fact_555_hoare__tripleE,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_a(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ~ ! [R2: a,H3: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(C2),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H3,T3)))
             => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_a_assn(Qa,R2)),produc7507926704131184380et_nat(H3,hoare_new_addrs(Ha,Asa,H3)))
               => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(Ha),Asa)),Ha),H3)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Ha)),lim_Product_unit(H3)) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_556_hoare__tripleE,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_hoare_triple_b(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ~ ! [R2: b,H3: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(C2),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H3,T3)))
             => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(Qa,R2)),produc7507926704131184380et_nat(H3,hoare_new_addrs(Ha,Asa,H3)))
               => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(Ha),Asa)),Ha),H3)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Ha)),lim_Product_unit(H3)) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_557_hoare__tripleE,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( hoare_8945653483474564448t_unit(Pa,C2,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
       => ~ ! [R2: product_unit,H3: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C2),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H3,T3)))
             => ( aa_Pro653082453696054616_nat_o(rep_assn(aa_Product_unit_assn(Qa,R2)),produc7507926704131184380et_nat(H3,hoare_new_addrs(Ha,Asa,H3)))
               => ( aa_hea7994211679605476167unit_o(relH(collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ah(Ha),Asa)),Ha),H3)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Ha)),lim_Product_unit(H3)) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_558_successE,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_T584514906347983379t_unit(Fa),Ha)
     => ~ ! [R2: product_unit,H3: produc6653097349344004940it_nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,H3)) ) ).

% successE
tff(fact_559_successE,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_b(Fa),Ha)
     => ~ ! [R2: b,H3: produc6653097349344004940it_nat] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),H3)) ) ).

% successE
tff(fact_560_successE,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(Fa),Ha)
     => ~ ! [R2: a,H3: produc6653097349344004940it_nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),H3)) ) ).

% successE
tff(fact_561_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_562_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_563_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_564_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_565_le__cases3,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),X) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),X)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_566_le__cases3,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z3) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z3) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z3)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z3)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),X) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),X)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_567_le__cases3,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),X) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),X)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_568_le__cases3,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),X) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),X)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_569_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_570_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_571_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_572_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_573_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_574_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_575_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_576_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_577_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_578_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_579_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( ( B = C2 )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_580_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_581_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_582_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_583_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_584_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_585_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_586_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_587_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_588_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_589_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2) ) ) ).

% order.trans
tff(fact_590_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% order.trans
tff(fact_591_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% order.trans
tff(fact_592_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% order.trans
tff(fact_593_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% order.trans
tff(fact_594_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),Z3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z3) ) ) ).

% order_trans
tff(fact_595_order__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3) ) ) ).

% order_trans
tff(fact_596_order__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z3) ) ) ).

% order_trans
tff(fact_597_order__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3) ) ) ).

% order_trans
tff(fact_598_order__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3) ) ) ).

% order_trans
tff(fact_599_linorder__wlog,axiom,
    ! [Pa: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(Pa,A3),B3) )
     => ( ! [A3: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(Pa,B3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(Pa,A3),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_600_linorder__wlog,axiom,
    ! [Pa: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(Pa,A3),B3) )
     => ( ! [A3: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(Pa,B3),A3)
           => aa_num_o(aa_num_fun_num_o(Pa,A3),B3) )
       => aa_num_o(aa_num_fun_num_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_601_linorder__wlog,axiom,
    ! [Pa: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
     => ( ! [A3: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(Pa,B3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_602_linorder__wlog,axiom,
    ! [Pa: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
     => ( ! [A3: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(Pa,B3),A3)
           => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
       => aa_int_o(aa_int_fun_int_o(Pa,A),B) ) ) ).

% linorder_wlog
tff(fact_603_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_604_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_605_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_606_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_607_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_608_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_609_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_610_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_611_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_612_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_613_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A) ) ) ).

% dual_order.trans
tff(fact_614_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: 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,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% dual_order.trans
tff(fact_615_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% dual_order.trans
tff(fact_616_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% dual_order.trans
tff(fact_617_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: 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,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% dual_order.trans
tff(fact_618_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_619_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_620_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_621_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_622_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_623_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_624_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_625_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_626_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_627_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_628_order__subst1,axiom,
    ! [A: rat,Fa: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_629_order__subst1,axiom,
    ! [A: rat,Fa: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_630_order__subst1,axiom,
    ! [A: rat,Fa: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(Fa,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_631_order__subst1,axiom,
    ! [A: rat,Fa: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(Fa,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,X3)),aa_int_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_632_order__subst1,axiom,
    ! [A: num,Fa: fun_rat_num,B: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_633_order__subst1,axiom,
    ! [A: num,Fa: fun_num_num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,X3)),aa_num_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_634_order__subst1,axiom,
    ! [A: num,Fa: fun_nat_num,B: nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(Fa,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,X3)),aa_nat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_635_order__subst1,axiom,
    ! [A: num,Fa: fun_int_num,B: int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(Fa,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(Fa,X3)),aa_int_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_636_order__subst1,axiom,
    ! [A: nat,Fa: fun_rat_nat,B: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(Fa,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_637_order__subst1,axiom,
    ! [A: nat,Fa: fun_num_nat,B: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(Fa,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,X3)),aa_num_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(Fa,C2)) ) ) ) ).

% order_subst1
tff(fact_638_order__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_rat,C2: 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(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_639_order__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_640_order__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_641_order__subst2,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,B)),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_642_order__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_643_order__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,X3)),aa_num_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_644_order__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,X3)),aa_num_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_645_order__subst2,axiom,
    ! [A: num,B: num,Fa: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,B)),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,X3)),aa_num_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_646_order__subst2,axiom,
    ! [A: nat,B: nat,Fa: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,B)),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_647_order__subst2,axiom,
    ! [A: nat,B: nat,Fa: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,B)),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,X3)),aa_nat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,A)),C2) ) ) ) ).

% order_subst2
tff(fact_648_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% order_eq_refl
tff(fact_649_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% order_eq_refl
tff(fact_650_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% order_eq_refl
tff(fact_651_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% order_eq_refl
tff(fact_652_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% order_eq_refl
tff(fact_653_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_linear
tff(fact_654_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_linear
tff(fact_655_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_linear
tff(fact_656_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_linear
tff(fact_657_ord__eq__le__subst,axiom,
    ! [A: rat,Fa: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_658_ord__eq__le__subst,axiom,
    ! [A: num,Fa: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_659_ord__eq__le__subst,axiom,
    ! [A: nat,Fa: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_660_ord__eq__le__subst,axiom,
    ! [A: int,Fa: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(Fa,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_661_ord__eq__le__subst,axiom,
    ! [A: rat,Fa: fun_num_rat,B: num,C2: num] :
      ( ( A = aa_num_rat(Fa,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_662_ord__eq__le__subst,axiom,
    ! [A: num,Fa: fun_num_num,B: num,C2: num] :
      ( ( A = aa_num_num(Fa,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,X3)),aa_num_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_663_ord__eq__le__subst,axiom,
    ! [A: nat,Fa: fun_num_nat,B: num,C2: num] :
      ( ( A = aa_num_nat(Fa,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,X3)),aa_num_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_664_ord__eq__le__subst,axiom,
    ! [A: int,Fa: fun_num_int,B: num,C2: num] :
      ( ( A = aa_num_int(Fa,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,X3)),aa_num_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_665_ord__eq__le__subst,axiom,
    ! [A: rat,Fa: fun_nat_rat,B: nat,C2: nat] :
      ( ( A = aa_nat_rat(Fa,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_666_ord__eq__le__subst,axiom,
    ! [A: num,Fa: fun_nat_num,B: nat,C2: nat] :
      ( ( A = aa_nat_num(Fa,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,X3)),aa_nat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(Fa,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_667_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,X3)),aa_rat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_668_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,X3)),aa_rat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_669_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,X3)),aa_rat_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_670_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,Fa: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(Fa,B) = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,X3)),aa_rat_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_671_ord__le__eq__subst,axiom,
    ! [A: num,B: num,Fa: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(Fa,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,X3)),aa_num_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_672_ord__le__eq__subst,axiom,
    ! [A: num,B: num,Fa: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(Fa,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,X3)),aa_num_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_673_ord__le__eq__subst,axiom,
    ! [A: num,B: num,Fa: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(Fa,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,X3)),aa_num_nat(Fa,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_674_ord__le__eq__subst,axiom,
    ! [A: num,B: num,Fa: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(Fa,B) = C2 )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,X3)),aa_num_int(Fa,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_675_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,Fa: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(Fa,B) = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Fa,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_676_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,Fa: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(Fa,B) = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,X3)),aa_nat_num(Fa,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_677_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) ) ).

% linorder_le_cases
tff(fact_678_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X) ) ).

% linorder_le_cases
tff(fact_679_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ).

% linorder_le_cases
tff(fact_680_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) ) ).

% linorder_le_cases
tff(fact_681_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_682_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_683_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_684_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_685_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_686_effect__def,axiom,
    ! [C2: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,R: product_unit,N2: nat] :
      ( aa_nat_o(heap_T6553295506729943825t_unit(C2,Ha,H_a,R),N2)
    <=> ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C2),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H_a,N2))) ) ) ).

% effect_def
tff(fact_687_effect__def,axiom,
    ! [C2: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,R: b,N2: nat] :
      ( aa_nat_o(heap_Time_effect_b(C2,Ha,H_a,R),N2)
    <=> ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(C2),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R),produc584006145561248582it_nat(H_a,N2))) ) ) ).

% effect_def
tff(fact_688_effect__def,axiom,
    ! [C2: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,R: a,N2: nat] :
      ( aa_nat_o(heap_Time_effect_a(C2,Ha,H_a,R),N2)
    <=> ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(C2),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H_a,N2))) ) ) ).

% effect_def
tff(fact_689_effectI,axiom,
    ! [C2: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,R: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(C2),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(H_a,N2))) )
     => aa_nat_o(heap_T6553295506729943825t_unit(C2,Ha,H_a,R),N2) ) ).

% effectI
tff(fact_690_effectI,axiom,
    ! [C2: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,R: b,H_a: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(C2),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R),produc584006145561248582it_nat(H_a,N2))) )
     => aa_nat_o(heap_Time_effect_b(C2,Ha,H_a,R),N2) ) ).

% effectI
tff(fact_691_effectI,axiom,
    ! [C2: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,R: a,H_a: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(C2),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(H_a,N2))) )
     => aa_nat_o(heap_Time_effect_a(C2,Ha,H_a,R),N2) ) ).

% effectI
tff(fact_692_nat__less__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        & ( M != N2 ) ) ) ).

% nat_less_le
tff(fact_693_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% less_imp_le_nat
tff(fact_694_le__eq__less__or__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( M = N2 ) ) ) ).

% le_eq_less_or_eq
tff(fact_695_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( M = N2 ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% less_or_eq_imp_le
tff(fact_696_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( ( M != N2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% le_neq_implies_less
tff(fact_697_less__mono__imp__le__mono,axiom,
    ! [Fa: fun_nat_nat,I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,I3)),aa_nat_nat(Fa,J3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,I2)),aa_nat_nat(Fa,J2)) ) ) ).

% less_mono_imp_le_mono
tff(fact_698_nat__descend__induct,axiom,
    ! [N2: nat,Pa: fun_nat_o,M: nat] :
      ( ! [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K)
         => aa_nat_o(Pa,K) )
     => ( ! [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),I4)
                 => aa_nat_o(Pa,I4) )
             => aa_nat_o(Pa,K) ) )
       => aa_nat_o(Pa,M) ) ) ).

% nat_descend_induct
tff(fact_699_minf_I8_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T6),X5) ) ).

% minf(8)
tff(fact_700_minf_I8_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T6),X5) ) ).

% minf(8)
tff(fact_701_minf_I8_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T6),X5) ) ).

% minf(8)
tff(fact_702_minf_I8_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T6),X5) ) ).

% minf(8)
tff(fact_703_minf_I6_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),T6) ) ).

% minf(6)
tff(fact_704_minf_I6_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),T6) ) ).

% minf(6)
tff(fact_705_minf_I6_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),T6) ) ).

% minf(6)
tff(fact_706_minf_I6_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T6) ) ).

% minf(6)
tff(fact_707_pinf_I8_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T6),X5) ) ).

% pinf(8)
tff(fact_708_pinf_I8_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T6),X5) ) ).

% pinf(8)
tff(fact_709_pinf_I8_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T6),X5) ) ).

% pinf(8)
tff(fact_710_pinf_I8_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T6),X5) ) ).

% pinf(8)
tff(fact_711_subset__Collect__conv,axiom,
    ! [S: set_list_nat,Pa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(S,collect_list_nat(Pa))
    <=> ! [X4: list_nat] :
          ( member_list_nat(X4,S)
         => aa_list_nat_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_712_subset__Collect__conv,axiom,
    ! [S: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(S),collec5210948495886036740nt_int(Pa))
    <=> ! [X4: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),S)
         => aa_set6170482117830564987_int_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_713_subset__Collect__conv,axiom,
    ! [S: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(S),collect_set_nat(Pa))
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),S)
         => aa_set_nat_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_714_subset__Collect__conv,axiom,
    ! [S: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),collect_nat(Pa))
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),S)
         => aa_nat_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_715_subset__Collect__conv,axiom,
    ! [S: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),collect_int(Pa))
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),S)
         => aa_int_o(Pa,X4) ) ) ).

% subset_Collect_conv
tff(fact_716_Collect__restrict,axiom,
    ! [X6: set_o,Pa: fun_o_o] : aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_717_Collect__restrict,axiom,
    ! [X6: set_list_nat,Pa: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(X6),Pa)),X6) ).

% Collect_restrict
tff(fact_718_Collect__restrict,axiom,
    ! [X6: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] : aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_719_Collect__restrict,axiom,
    ! [X6: set_set_nat,Pa: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_al(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_720_Collect__restrict,axiom,
    ! [X6: set_nat,Pa: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_721_Collect__restrict,axiom,
    ! [X6: set_int,Pa: fun_int_o] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(X6),Pa))),X6) ).

% Collect_restrict
tff(fact_722_prop__restrict,axiom,
    ! [X: $o,Z4: set_o,X6: set_o,Pa: fun_o_o] :
      ( aa_set_o_o(member_o((X)),Z4)
     => ( aa_set_o_o(ord_less_eq_set_o(Z4),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(X6),Pa)))
       => aa_o_o(Pa,(X)) ) ) ).

% prop_restrict
tff(fact_723_prop__restrict,axiom,
    ! [X: list_nat,Z4: set_list_nat,X6: set_list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(X,Z4)
     => ( ord_le6045566169113846134st_nat(Z4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(X6),Pa)))
       => aa_list_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_724_prop__restrict,axiom,
    ! [X: set_Pr958786334691620121nt_int,Z4: set_se6260736226359567993nt_int,X6: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),Z4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(Z4),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(X6),Pa)))
       => aa_set6170482117830564987_int_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_725_prop__restrict,axiom,
    ! [X: set_nat,Z4: set_set_nat,X6: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),Z4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Z4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_al(X6),Pa)))
       => aa_set_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_726_prop__restrict,axiom,
    ! [X: nat,Z4: set_nat,X6: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),Z4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(X6),Pa)))
       => aa_nat_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_727_prop__restrict,axiom,
    ! [X: int,Z4: set_int,X6: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(member_int(X),Z4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(X6),Pa)))
       => aa_int_o(Pa,X) ) ) ).

% prop_restrict
tff(fact_728_pinf_I1_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
           => ( ( aa_rat_o(Pa,X5)
                & aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                & aa_rat_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_729_pinf_I1_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
           => ( ( aa_num_o(Pa,X5)
                & aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                & aa_num_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_730_pinf_I1_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
           => ( ( aa_nat_o(Pa,X5)
                & aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                & aa_nat_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_731_pinf_I1_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
           => ( ( aa_int_o(Pa,X5)
                & aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                & aa_int_o(Q2,X5) ) ) ) ) ) ).

% pinf(1)
tff(fact_732_pinf_I2_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z5),X3)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
           => ( ( aa_rat_o(Pa,X5)
                | aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                | aa_rat_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_733_pinf_I2_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z5),X3)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
           => ( ( aa_num_o(Pa,X5)
                | aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                | aa_num_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_734_pinf_I2_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z5),X3)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
           => ( ( aa_nat_o(Pa,X5)
                | aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                | aa_nat_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_735_pinf_I2_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
           => ( ( aa_int_o(Pa,X5)
                | aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                | aa_int_o(Q2,X5) ) ) ) ) ) ).

% pinf(2)
tff(fact_736_pinf_I3_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(3)
tff(fact_737_pinf_I3_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(3)
tff(fact_738_pinf_I3_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(3)
tff(fact_739_pinf_I3_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(3)
tff(fact_740_pinf_I4_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(4)
tff(fact_741_pinf_I4_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(4)
tff(fact_742_pinf_I4_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(4)
tff(fact_743_pinf_I4_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( X5 != T6 ) ) ).

% pinf(4)
tff(fact_744_pinf_I5_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),T6) ) ).

% pinf(5)
tff(fact_745_pinf_I5_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X5),T6) ) ).

% pinf(5)
tff(fact_746_pinf_I5_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),T6) ) ).

% pinf(5)
tff(fact_747_pinf_I5_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T6) ) ).

% pinf(5)
tff(fact_748_pinf_I7_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T6),X5) ) ).

% pinf(7)
tff(fact_749_pinf_I7_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,T6),X5) ) ).

% pinf(7)
tff(fact_750_pinf_I7_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T6),X5) ) ).

% pinf(7)
tff(fact_751_pinf_I7_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,T6),X5) ) ).

% pinf(7)
tff(fact_752_minf_I1_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
           => ( ( aa_rat_o(Pa,X5)
                & aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                & aa_rat_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_753_minf_I1_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
           => ( ( aa_num_o(Pa,X5)
                & aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                & aa_num_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_754_minf_I1_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
           => ( ( aa_nat_o(Pa,X5)
                & aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                & aa_nat_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_755_minf_I1_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
           => ( ( aa_int_o(Pa,X5)
                & aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                & aa_int_o(Q2,X5) ) ) ) ) ) ).

% minf(1)
tff(fact_756_minf_I2_J,axiom,
    ! [Pa: fun_rat_o,P3: fun_rat_o,Qa: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
         => ( aa_rat_o(Pa,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z5)
           => ( aa_rat_o(Qa,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
           => ( ( aa_rat_o(Pa,X5)
                | aa_rat_o(Qa,X5) )
            <=> ( aa_rat_o(P3,X5)
                | aa_rat_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_757_minf_I2_J,axiom,
    ! [Pa: fun_num_o,P3: fun_num_o,Qa: fun_num_o,Q2: fun_num_o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
         => ( aa_num_o(Pa,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z5)
           => ( aa_num_o(Qa,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
           => ( ( aa_num_o(Pa,X5)
                | aa_num_o(Qa,X5) )
            <=> ( aa_num_o(P3,X5)
                | aa_num_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_758_minf_I2_J,axiom,
    ! [Pa: fun_nat_o,P3: fun_nat_o,Qa: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
         => ( aa_nat_o(Pa,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z5)
           => ( aa_nat_o(Qa,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
           => ( ( aa_nat_o(Pa,X5)
                | aa_nat_o(Qa,X5) )
            <=> ( aa_nat_o(P3,X5)
                | aa_nat_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_759_minf_I2_J,axiom,
    ! [Pa: fun_int_o,P3: fun_int_o,Qa: fun_int_o,Q2: fun_int_o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
         => ( aa_int_o(Pa,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(Qa,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
           => ( ( aa_int_o(Pa,X5)
                | aa_int_o(Qa,X5) )
            <=> ( aa_int_o(P3,X5)
                | aa_int_o(Q2,X5) ) ) ) ) ) ).

% minf(2)
tff(fact_760_minf_I3_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(3)
tff(fact_761_minf_I3_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(3)
tff(fact_762_minf_I3_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(3)
tff(fact_763_minf_I3_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(3)
tff(fact_764_minf_I4_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(4)
tff(fact_765_minf_I4_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(4)
tff(fact_766_minf_I4_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(4)
tff(fact_767_minf_I4_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( X5 != T6 ) ) ).

% minf(4)
tff(fact_768_minf_I5_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),T6) ) ).

% minf(5)
tff(fact_769_minf_I5_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X5),T6) ) ).

% minf(5)
tff(fact_770_minf_I5_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),T6) ) ).

% minf(5)
tff(fact_771_minf_I5_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T6) ) ).

% minf(5)
tff(fact_772_minf_I7_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T6),X5) ) ).

% minf(7)
tff(fact_773_minf_I7_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,T6),X5) ) ).

% minf(7)
tff(fact_774_minf_I7_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T6),X5) ) ).

% minf(7)
tff(fact_775_minf_I7_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,T6),X5) ) ).

% minf(7)
tff(fact_776_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_777_infinite__descent,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( ! [N5: nat] :
          ( ~ aa_nat_o(Pa,N5)
         => ? [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N5)
              & ~ aa_nat_o(Pa,M4) ) )
     => aa_nat_o(Pa,N2) ) ).

% infinite_descent
tff(fact_778_nat__less__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N5)
             => aa_nat_o(Pa,M4) )
         => aa_nat_o(Pa,N5) )
     => aa_nat_o(Pa,N2) ) ).

% nat_less_induct
tff(fact_779_less__irrefl__nat,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N2) ).

% less_irrefl_nat
tff(fact_780_less__not__refl3,axiom,
    ! [S2: nat,T6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),T6)
     => ( S2 != T6 ) ) ).

% less_not_refl3
tff(fact_781_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( M != N2 ) ) ).

% less_not_refl2
tff(fact_782_less__not__refl,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N2) ).

% less_not_refl
tff(fact_783_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% nat_neq_iff
tff(fact_784_Nat_Oex__has__greatest__nat,axiom,
    ! [Pa: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(Pa,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X3: nat] :
            ( aa_nat_o(Pa,X3)
            & ! [Y5: nat] :
                ( aa_nat_o(Pa,Y5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y5),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_785_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ).

% nat_le_linear
tff(fact_786_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( M = N2 ) ) ) ).

% le_antisym
tff(fact_787_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% eq_imp_le
tff(fact_788_le__trans,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2) ) ) ).

% le_trans
tff(fact_789_le__refl,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N2) ).

% le_refl
tff(fact_790_pinf_I6_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),T6) ) ).

% pinf(6)
tff(fact_791_pinf_I6_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X5)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),T6) ) ).

% pinf(6)
tff(fact_792_pinf_I6_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),T6) ) ).

% pinf(6)
tff(fact_793_pinf_I6_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T6) ) ).

% pinf(6)
tff(fact_794_execute__bind_I1_J,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_Pr5286341546376896041Heap_b] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_T757603679106148409unit_b(Fa,Ga)),Ha) = heap_T7616092557645711336rame_b(N2,aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_Pro7886978639879396314Heap_b(Ga,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_795_execute__bind_I1_J,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_Pr5215307506330550056Heap_a] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_T757603679106148408unit_a(Fa,Ga)),Ha) = heap_T7616092557645711335rame_a(N2,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_Pro7886978635576167513Heap_a(Ga,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_796_execute__bind_I1_J,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,X: b,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_b_8426044978055006235Heap_b] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_b_b(Fa,Ga)),Ha) = heap_T7616092557645711336rame_b(N2,aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(Ga,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_797_execute__bind_I1_J,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,X: b,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_b_8355010938008660250Heap_a] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_b_a(Fa,Ga)),Ha) = heap_T7616092557645711335rame_a(N2,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_b_H1867786018526481995Heap_a(Ga,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_798_execute__bind_I1_J,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_a_6164325696594227994Heap_b] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_a_b(Fa,Ga)),Ha) = heap_T7616092557645711336rame_b(N2,aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_a_H8829438778223708363Heap_b(Ga,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_799_execute__bind_I1_J,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(Fa,Ga)),Ha) = heap_T7616092557645711335rame_a(N2,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(Ga,X)),H_a)) ) ) ).

% execute_bind(1)
tff(fact_800_subset__antisym,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
       => ( A4 = B4 ) ) ) ).

% subset_antisym
tff(fact_801_psubsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( ( A4 != B4 )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4) ) ) ).

% psubsetI
tff(fact_802_subsetI,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_set_o_o(member_o((X3)),B4) )
     => aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ).

% subsetI
tff(fact_803_subsetI,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
         => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),B4) )
     => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4) ) ).

% subsetI
tff(fact_804_subsetI,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A4)
         => aa_set_set_nat_o(member_set_nat(X3),B4) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4) ) ).

% subsetI
tff(fact_805_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_set_nat_o(member_nat(X3),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ).

% subsetI
tff(fact_806_subsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_set_int_o(member_int(X3),B4) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4) ) ).

% subsetI
tff(fact_807_complete__interval,axiom,
    ! [A: nat,B: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(Pa,A)
       => ( ~ aa_nat_o(Pa,B)
         => ? [C: 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,C),B)
              & ! [X5: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X5)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),C) )
                 => aa_nat_o(Pa,X5) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D3) )
                     => aa_nat_o(Pa,X3) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_808_complete__interval,axiom,
    ! [A: int,B: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(Pa,A)
       => ( ~ aa_int_o(Pa,B)
         => ? [C: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
              & ! [X5: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X5)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X5),C) )
                 => aa_int_o(Pa,X5) )
              & ! [D3: int] :
                  ( ! [X3: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D3) )
                     => aa_int_o(Pa,X3) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_809_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_810_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_811_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_812_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_813_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_Pr292408699144966967t_unit,Y: product_unit,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(Ga,X)),H_a) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T2633723481742716231t_unit(Fa,Ga)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_814_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_Pr5286341546376896041Heap_b,Y: b,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_Pro7886978639879396314Heap_b(Ga,X)),H_a) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y),produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_T757603679106148409unit_b(Fa,Ga)),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y),produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_815_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit,X: product_unit,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_Pr5215307506330550056Heap_a,Y: a,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_Pro7886978635576167513Heap_a(Ga,X)),H_a) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_T757603679106148408unit_a(Fa,Ga)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_816_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,X: b,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_b_6677255003012645673t_unit,Y: product_unit,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_b_H8549598898354664410t_unit(Ga,X)),H_a) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T3413130826733729493t_unit(Fa,Ga)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_817_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,X: b,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_b_8426044978055006235Heap_b,Y: b,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(Ga,X)),H_a) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y),produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_b_b(Fa,Ga)),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y),produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_818_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,X: b,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_b_8355010938008660250Heap_a,Y: a,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_b_H1867786018526481995Heap_a(Ga,X)),H_a) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_b_a(Fa,Ga)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_819_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_a_2990225006182978344t_unit,Y: product_unit,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_a_H4862568901524997081t_unit(Ga,X)),H_a) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T8685611227969916822t_unit(Fa,Ga)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(Y,produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_820_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_a_6164325696594227994Heap_b,Y: b,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_a_H8829438778223708363Heap_b(Ga,X)),H_a) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y),produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_a_b(Fa,Ga)),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Y),produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_821_execute__bind__eq__SomeI,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,X: a,H_a: heap_e7401611519738050253t_unit,N2: nat,Ga: fun_a_6093291656547882009Heap_a,Y: a,H_a2: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(H_a,N2))) )
     => ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(Ga,X)),H_a) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H_a2,N6))) )
       => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(Fa,Ga)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Y),produc584006145561248582it_nat(H_a2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_822_new__addrs__def,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat,H_a: heap_e7401611519738050253t_unit] : hoare_new_addrs(Ha,Asa,H_a) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),collect_nat(aa_hea9023553427033036112_nat_o(aTP_Lamm_ao(Ha),H_a))) ).

% new_addrs_def
tff(fact_823_subset__Collect__iff,axiom,
    ! [B4: set_o,A4: set_o,Pa: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B4),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(B4),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa)))
      <=> ! [X4: $o] :
            ( aa_set_o_o(member_o((X4)),B4)
           => aa_o_o(Pa,(X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_824_subset__Collect__iff,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,Pa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A4)
     => ( ord_le6045566169113846134st_nat(B4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(A4),Pa)))
      <=> ! [X4: list_nat] :
            ( member_list_nat(X4,B4)
           => aa_list_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_825_subset__Collect__iff,axiom,
    ! [B4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(B4),A4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(B4),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(A4),Pa)))
      <=> ! [X4: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),B4)
           => aa_set6170482117830564987_int_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_826_subset__Collect__iff,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),A4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_al(A4),Pa)))
      <=> ! [X4: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X4),B4)
           => aa_set_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_827_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
           => aa_nat_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_828_subset__Collect__iff,axiom,
    ! [B4: set_int,A4: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa)))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),B4)
           => aa_int_o(Pa,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_829_subset__CollectI,axiom,
    ! [B4: set_o,A4: set_o,Qa: fun_o_o,Pa: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B4),A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),B4)
           => ( aa_o_o(Qa,(X3))
             => aa_o_o(Pa,(X3)) ) )
       => aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(B4),Qa))),collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_830_subset__CollectI,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,Qa: fun_list_nat_o,Pa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A4)
     => ( ! [X3: list_nat] :
            ( member_list_nat(X3,B4)
           => ( aa_list_nat_o(Qa,X3)
             => aa_list_nat_o(Pa,X3) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(B4),Qa)),collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_831_subset__CollectI,axiom,
    ! [B4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,Qa: fun_se7249947736802449410_int_o,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(B4),A4)
     => ( ! [X3: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),B4)
           => ( aa_set6170482117830564987_int_o(Qa,X3)
             => aa_set6170482117830564987_int_o(Pa,X3) ) )
       => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(B4),Qa))),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_832_subset__CollectI,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Qa: fun_set_nat_o,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),A4)
     => ( ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),B4)
           => ( aa_set_nat_o(Qa,X3)
             => aa_set_nat_o(Pa,X3) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_al(B4),Qa))),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_al(A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_833_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Qa: fun_nat_o,Pa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),B4)
           => ( aa_nat_o(Qa,X3)
             => aa_nat_o(Pa,X3) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(B4),Qa))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_834_subset__CollectI,axiom,
    ! [B4: set_int,A4: set_int,Qa: fun_int_o,Pa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),B4)
           => ( aa_int_o(Qa,X3)
             => aa_int_o(Pa,X3) ) )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(B4),Qa))),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_835_conj__subset__def,axiom,
    ! [A4: set_list_nat,Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ap(Pa),Qa)))
    <=> ( ord_le6045566169113846134st_nat(A4,collect_list_nat(Pa))
        & ord_le6045566169113846134st_nat(A4,collect_list_nat(Qa)) ) ) ).

% conj_subset_def
tff(fact_836_conj__subset__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_aq(Pa),Qa)))
    <=> ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),collec5210948495886036740nt_int(Pa))
        & aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),collec5210948495886036740nt_int(Qa)) ) ) ).

% conj_subset_def
tff(fact_837_conj__subset__def,axiom,
    ! [A4: set_set_nat,Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ar(Pa),Qa)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),collect_set_nat(Pa))
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),collect_set_nat(Qa)) ) ) ).

% conj_subset_def
tff(fact_838_conj__subset__def,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_as(Pa),Qa)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),collect_nat(Pa))
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),collect_nat(Qa)) ) ) ).

% conj_subset_def
tff(fact_839_conj__subset__def,axiom,
    ! [A4: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_at(Pa),Qa)))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),collect_int(Pa))
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),collect_int(Qa)) ) ) ).

% conj_subset_def
tff(fact_840_UnCI,axiom,
    ! [C2: $o,B4: set_o,A4: set_o] :
      ( ( ~ aa_set_o_o(member_o((C2)),B4)
       => aa_set_o_o(member_o((C2)),A4) )
     => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) ) ).

% UnCI
tff(fact_841_UnCI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4) )
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),sup_su2047564715030645325nt_int(A4,B4)) ) ).

% UnCI
tff(fact_842_UnCI,axiom,
    ! [C2: set_nat,B4: set_set_nat,A4: set_set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(C2),B4)
       => aa_set_set_nat_o(member_set_nat(C2),A4) )
     => aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A4,B4)) ) ).

% UnCI
tff(fact_843_UnCI,axiom,
    ! [C2: int,B4: set_int,A4: set_int] :
      ( ( ~ aa_set_int_o(member_int(C2),B4)
       => aa_set_int_o(member_int(C2),A4) )
     => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) ) ).

% UnCI
tff(fact_844_UnCI,axiom,
    ! [C2: produc4166570645942440679at_nat,B4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( ~ member6689249552917799696at_nat(C2,B4)
       => member6689249552917799696at_nat(C2,A4) )
     => member6689249552917799696at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) ) ).

% UnCI
tff(fact_845_UnCI,axiom,
    ! [C2: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),B4)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),A4) )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ) ).

% UnCI
tff(fact_846_UnCI,axiom,
    ! [C2: nat,B4: set_nat,A4: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(C2),B4)
       => aa_set_nat_o(member_nat(C2),A4) )
     => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ) ).

% UnCI
tff(fact_847_Un__iff,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4))
    <=> ( aa_set_o_o(member_o((C2)),A4)
        | aa_set_o_o(member_o((C2)),B4) ) ) ).

% Un_iff
tff(fact_848_Un__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),sup_su2047564715030645325nt_int(A4,B4))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% Un_iff
tff(fact_849_Un__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A4,B4))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A4)
        | aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% Un_iff
tff(fact_850_Un__iff,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))
    <=> ( aa_set_int_o(member_int(C2),A4)
        | aa_set_int_o(member_int(C2),B4) ) ) ).

% Un_iff
tff(fact_851_Un__iff,axiom,
    ! [C2: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4))
    <=> ( member6689249552917799696at_nat(C2,A4)
        | member6689249552917799696at_nat(C2,B4) ) ) ).

% Un_iff
tff(fact_852_Un__iff,axiom,
    ! [C2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),A4)
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),B4) ) ) ).

% Un_iff
tff(fact_853_Un__iff,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
    <=> ( aa_set_nat_o(member_nat(C2),A4)
        | aa_set_nat_o(member_nat(C2),B4) ) ) ).

% Un_iff
tff(fact_854_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% nat_add_left_cancel_less
tff(fact_855_Un__subset__iff,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4),C3)
    <=> ( ord_le8081472938463900775at_nat(A4,C3)
        & ord_le8081472938463900775at_nat(B4,C3) ) ) ).

% Un_subset_iff
tff(fact_856_Un__subset__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C3)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),C3)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),C3) ) ) ).

% Un_subset_iff
tff(fact_857_Un__subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C3)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3) ) ) ).

% Un_subset_iff
tff(fact_858_Un__subset__iff,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)),C3)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),C3)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C3) ) ) ).

% Un_subset_iff
tff(fact_859_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% nat_add_left_cancel_le
tff(fact_860_sup__Some,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : sup_su5873096292857799009at_nat(some_s287724117700012716at_nat(X),some_s287724117700012716at_nat(Y)) = some_s287724117700012716at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% sup_Some
tff(fact_861_sup__Some,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : sup_su2273273666271716065at_nat(some_s5890477192898017836at_nat(X),some_s5890477192898017836at_nat(Y)) = some_s5890477192898017836at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_Some
tff(fact_862_sup__Some,axiom,
    ! [X: set_nat,Y: set_nat] : sup_su3598758113090618626et_nat(some_set_nat(X),some_set_nat(Y)) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_Some
tff(fact_863_sup__Some,axiom,
    ! [X: nat,Y: nat] : sup_sup_option_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_Some
tff(fact_864_sup__Some,axiom,
    ! [X: int,Y: int] : sup_sup_option_int(aa_int_option_int(some_int,X),aa_int_option_int(some_int,Y)) = aa_int_option_int(some_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_Some
tff(fact_865_relH__dist__union,axiom,
    ! [Asa: set_nat,As4: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As4),Ha),H_a)
    <=> ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
        & aa_hea7994211679605476167unit_o(relH(As4,Ha),H_a) ) ) ).

% relH_dist_union
tff(fact_866_psubsetD,axiom,
    ! [A4: set_o,B4: set_o,C2: $o] :
      ( aa_set_o_o(ord_less_set_o(A4),B4)
     => ( aa_set_o_o(member_o((C2)),A4)
       => aa_set_o_o(member_o((C2)),B4) ) ) ).

% psubsetD
tff(fact_867_psubsetD,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int,C2: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A4),B4)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% psubsetD
tff(fact_868_psubsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
     => ( aa_set_set_nat_o(member_set_nat(C2),A4)
       => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% psubsetD
tff(fact_869_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(C2),A4)
       => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% psubsetD
tff(fact_870_psubsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
     => ( aa_set_int_o(member_int(C2),A4)
       => aa_set_int_o(member_int(C2),B4) ) ) ).

% psubsetD
tff(fact_871_less__set__def,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),B4)
    <=> ord_less_fun_o_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_set_def
tff(fact_872_less__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A4),B4)
    <=> ord_le8970291702602496086_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4)) ) ).

% less_set_def
tff(fact_873_less__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4)) ) ).

% less_set_def
tff(fact_874_less__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_set_def
tff(fact_875_less__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4)) ) ).

% less_set_def
tff(fact_876_UnE,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4))
     => ( ~ aa_set_o_o(member_o((C2)),A4)
       => aa_set_o_o(member_o((C2)),B4) ) ) ).

% UnE
tff(fact_877_UnE,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),sup_su2047564715030645325nt_int(A4,B4))
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% UnE
tff(fact_878_UnE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A4,B4))
     => ( ~ aa_set_set_nat_o(member_set_nat(C2),A4)
       => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% UnE
tff(fact_879_UnE,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))
     => ( ~ aa_set_int_o(member_int(C2),A4)
       => aa_set_int_o(member_int(C2),B4) ) ) ).

% UnE
tff(fact_880_UnE,axiom,
    ! [C2: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4))
     => ( ~ member6689249552917799696at_nat(C2,A4)
       => member6689249552917799696at_nat(C2,B4) ) ) ).

% UnE
tff(fact_881_UnE,axiom,
    ! [C2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),A4)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),B4) ) ) ).

% UnE
tff(fact_882_UnE,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( ~ aa_set_nat_o(member_nat(C2),A4)
       => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% UnE
tff(fact_883_UnI1,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),A4)
     => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) ) ).

% UnI1
tff(fact_884_UnI1,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),sup_su2047564715030645325nt_int(A4,B4)) ) ).

% UnI1
tff(fact_885_UnI1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A4)
     => aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A4,B4)) ) ).

% UnI1
tff(fact_886_UnI1,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),A4)
     => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) ) ).

% UnI1
tff(fact_887_UnI1,axiom,
    ! [C2: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C2,A4)
     => member6689249552917799696at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) ) ).

% UnI1
tff(fact_888_UnI1,axiom,
    ! [C2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),A4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ) ).

% UnI1
tff(fact_889_UnI1,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A4)
     => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ) ).

% UnI1
tff(fact_890_UnI2,axiom,
    ! [C2: $o,B4: set_o,A4: set_o] :
      ( aa_set_o_o(member_o((C2)),B4)
     => aa_set_o_o(member_o((C2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) ) ).

% UnI2
tff(fact_891_UnI2,axiom,
    ! [C2: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),sup_su2047564715030645325nt_int(A4,B4)) ) ).

% UnI2
tff(fact_892_UnI2,axiom,
    ! [C2: set_nat,B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),B4)
     => aa_set_set_nat_o(member_set_nat(C2),sup_sup_set_set_nat(A4,B4)) ) ).

% UnI2
tff(fact_893_UnI2,axiom,
    ! [C2: int,B4: set_int,A4: set_int] :
      ( aa_set_int_o(member_int(C2),B4)
     => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) ) ).

% UnI2
tff(fact_894_UnI2,axiom,
    ! [C2: produc4166570645942440679at_nat,B4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( member6689249552917799696at_nat(C2,B4)
     => member6689249552917799696at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) ) ).

% UnI2
tff(fact_895_UnI2,axiom,
    ! [C2: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),B4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ) ).

% UnI2
tff(fact_896_UnI2,axiom,
    ! [C2: nat,B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),B4)
     => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ) ).

% UnI2
tff(fact_897_bex__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,Pa: fun_Pr1099658227000185396_nat_o] :
      ( ? [X4: produc4166570645942440679at_nat] :
          ( member6689249552917799696at_nat(X4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4))
          & aa_Pro8143006071014924973_nat_o(Pa,X4) )
    <=> ( ? [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,A4)
            & aa_Pro8143006071014924973_nat_o(Pa,X4) )
        | ? [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,B4)
            & aa_Pro8143006071014924973_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_898_bex__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      ( ? [X4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
          & aa_Pro5947980084102686253_nat_o(Pa,X4) )
    <=> ( ? [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A4)
            & aa_Pro5947980084102686253_nat_o(Pa,X4) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B4)
            & aa_Pro5947980084102686253_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_899_bex__Un,axiom,
    ! [A4: set_nat,B4: set_nat,Pa: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
          & aa_nat_o(Pa,X4) )
    <=> ( ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_nat_o(Pa,X4) )
        | ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
            & aa_nat_o(Pa,X4) ) ) ) ).

% bex_Un
tff(fact_900_ball__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,Pa: fun_Pr1099658227000185396_nat_o] :
      ( ! [X4: produc4166570645942440679at_nat] :
          ( member6689249552917799696at_nat(X4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4))
         => aa_Pro8143006071014924973_nat_o(Pa,X4) )
    <=> ( ! [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,A4)
           => aa_Pro8143006071014924973_nat_o(Pa,X4) )
        & ! [X4: produc4166570645942440679at_nat] :
            ( member6689249552917799696at_nat(X4,B4)
           => aa_Pro8143006071014924973_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_901_ball__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      ( ! [X4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
         => aa_Pro5947980084102686253_nat_o(Pa,X4) )
    <=> ( ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A4)
           => aa_Pro5947980084102686253_nat_o(Pa,X4) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B4)
           => aa_Pro5947980084102686253_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_902_ball__Un,axiom,
    ! [A4: set_nat,B4: set_nat,Pa: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
         => aa_nat_o(Pa,X4) )
    <=> ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_nat_o(Pa,X4) )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
           => aa_nat_o(Pa,X4) ) ) ) ).

% ball_Un
tff(fact_903_Un__assoc,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)),C3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) ).

% Un_assoc
tff(fact_904_Un__assoc,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) ).

% Un_assoc
tff(fact_905_Un__assoc,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) ).

% Un_assoc
tff(fact_906_Un__absorb,axiom,
    ! [A4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),A4) = A4 ).

% Un_absorb
tff(fact_907_Un__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),A4) = A4 ).

% Un_absorb
tff(fact_908_Un__absorb,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),A4) = A4 ).

% Un_absorb
tff(fact_909_Un__commute,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),A4) ).

% Un_commute
tff(fact_910_Un__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),A4) ).

% Un_commute
tff(fact_911_Un__commute,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A4) ).

% Un_commute
tff(fact_912_Un__left__absorb,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) ).

% Un_left_absorb
tff(fact_913_Un__left__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) ).

% Un_left_absorb
tff(fact_914_Un__left__absorb,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) ).

% Un_left_absorb
tff(fact_915_Un__left__commute,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),C3)) ).

% Un_left_commute
tff(fact_916_Un__left__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),C3)) ).

% Un_left_commute
tff(fact_917_Un__left__commute,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),C3)) ).

% Un_left_commute
tff(fact_918_Collect__disj__eq,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_au(Pa),Qa)) = sup_sup_set_list_nat(collect_list_nat(Pa),collect_list_nat(Qa)) ).

% Collect_disj_eq
tff(fact_919_Collect__disj__eq,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] : collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_av(Pa),Qa)) = sup_su2047564715030645325nt_int(collec5210948495886036740nt_int(Pa),collec5210948495886036740nt_int(Qa)) ).

% Collect_disj_eq
tff(fact_920_Collect__disj__eq,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_aw(Pa),Qa)) = sup_sup_set_set_nat(collect_set_nat(Pa),collect_set_nat(Qa)) ).

% Collect_disj_eq
tff(fact_921_Collect__disj__eq,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(Pa),Qa)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(Pa)),collect_int(Qa)) ).

% Collect_disj_eq
tff(fact_922_Collect__disj__eq,axiom,
    ! [Pa: fun_Pr1099658227000185396_nat_o,Qa: fun_Pr1099658227000185396_nat_o] : collec5204685387357076818at_nat(aa_fun9125434449681309094_nat_o(aTP_Lamm_ay(Pa),Qa)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,collec5204685387357076818at_nat(Pa)),collec5204685387357076818at_nat(Qa)) ).

% Collect_disj_eq
tff(fact_923_Collect__disj__eq,axiom,
    ! [Pa: fun_Pr3059359313727237556_nat_o,Qa: fun_Pr3059359313727237556_nat_o] : collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_az(Pa),Qa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(Pa)),collec6321179662152712658at_nat(Qa)) ).

% Collect_disj_eq
tff(fact_924_Collect__disj__eq,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ba(Pa),Qa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(Pa)),collect_nat(Qa)) ).

% Collect_disj_eq
tff(fact_925_Un__def,axiom,
    ! [A4: set_o,B4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_bb(A4),B4)) ).

% Un_def
tff(fact_926_Un__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : sup_sup_set_list_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_bc(A4),B4)) ).

% Un_def
tff(fact_927_Un__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] : sup_su2047564715030645325nt_int(A4,B4) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_bd(A4),B4)) ).

% Un_def
tff(fact_928_Un__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : sup_sup_set_set_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_be(A4),B4)) ).

% Un_def
tff(fact_929_Un__def,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bf(A4),B4)) ).

% Un_def
tff(fact_930_Un__def,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = collec5204685387357076818at_nat(aa_set1219845518824332607_nat_o(aTP_Lamm_bg(A4),B4)) ).

% Un_def
tff(fact_931_Un__def,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_bh(A4),B4)) ).

% Un_def
tff(fact_932_Un__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bi(A4),B4)) ).

% Un_def
tff(fact_933_Un__mono,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,D4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A4,C3)
     => ( ord_le8081472938463900775at_nat(B4,D4)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C3),D4)) ) ) ).

% Un_mono
tff(fact_934_Un__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),C3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),D4)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C3),D4)) ) ) ).

% Un_mono
tff(fact_935_Un__mono,axiom,
    ! [A4: set_nat,C3: set_nat,B4: set_nat,D4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),D4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C3),D4)) ) ) ).

% Un_mono
tff(fact_936_Un__mono,axiom,
    ! [A4: set_int,C3: set_int,B4: set_int,D4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),C3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),D4)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C3),D4)) ) ) ).

% Un_mono
tff(fact_937_Un__least,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A4,C3)
     => ( ord_le8081472938463900775at_nat(B4,C3)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4),C3) ) ) ).

% Un_least
tff(fact_938_Un__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),C3)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),C3)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C3) ) ) ).

% Un_least
tff(fact_939_Un__least,axiom,
    ! [A4: set_nat,C3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C3) ) ) ).

% Un_least
tff(fact_940_Un__least,axiom,
    ! [A4: set_int,C3: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),C3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)),C3) ) ) ).

% Un_least
tff(fact_941_Un__upper1,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(A4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) ).

% Un_upper1
tff(fact_942_Un__upper1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ).

% Un_upper1
tff(fact_943_Un__upper1,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ).

% Un_upper1
tff(fact_944_Un__upper1,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) ).

% Un_upper1
tff(fact_945_Un__upper2,axiom,
    ! [B4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(B4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) ).

% Un_upper2
tff(fact_946_Un__upper2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ).

% Un_upper2
tff(fact_947_Un__upper2,axiom,
    ! [B4: set_nat,A4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ).

% Un_upper2
tff(fact_948_Un__upper2,axiom,
    ! [B4: set_int,A4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) ).

% Un_upper2
tff(fact_949_Un__absorb1,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A4,B4)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_950_Un__absorb1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_951_Un__absorb1,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_952_Un__absorb1,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_953_Un__absorb2,axiom,
    ! [B4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B4,A4)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = A4 ) ) ).

% Un_absorb2
tff(fact_954_Un__absorb2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = A4 ) ) ).

% Un_absorb2
tff(fact_955_Un__absorb2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = A4 ) ) ).

% Un_absorb2
tff(fact_956_Un__absorb2,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = A4 ) ) ).

% Un_absorb2
tff(fact_957_subset__UnE,axiom,
    ! [C3: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4))
     => ~ ! [A5: set_Pr8551490117392284871at_nat] :
            ( ord_le8081472938463900775at_nat(A5,A4)
           => ! [B5: set_Pr8551490117392284871at_nat] :
                ( ord_le8081472938463900775at_nat(B5,B4)
               => ( C3 != aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_958_subset__UnE,axiom,
    ! [C3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A5),A4)
           => ! [B5: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B5),B4)
               => ( C3 != aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_959_subset__UnE,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ~ ! [A5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),A4)
           => ! [B5: set_nat] :
                ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B5),B4)
               => ( C3 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_960_subset__UnE,axiom,
    ! [C3: set_int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))
     => ~ ! [A5: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A5),A4)
           => ! [B5: set_int] :
                ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B5),B4)
               => ( C3 != aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A5),B5) ) ) ) ) ).

% subset_UnE
tff(fact_961_subset__Un__eq,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A4,B4)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_962_subset__Un__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_963_subset__Un__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_964_subset__Un__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_965_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K2) ) ).

% add_lessD1
tff(fact_966_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ) ).

% add_less_mono
tff(fact_967_not__add__less1,axiom,
    ! [I2: nat,J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),I2) ).

% not_add_less1
tff(fact_968_not__add__less2,axiom,
    ! [J2: nat,I2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I2)),I2) ).

% not_add_less2
tff(fact_969_add__less__mono1,axiom,
    ! [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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) ) ).

% add_less_mono1
tff(fact_970_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M: 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,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M)) ) ).

% trans_less_add1
tff(fact_971_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M: 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,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J2)) ) ).

% trans_less_add2
tff(fact_972_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),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,K2),N2) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% less_add_eq_less
tff(fact_973_add__leE,axiom,
    ! [M: nat,K2: nat,N2: 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),K2)),N2)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2) ) ) ).

% add_leE
tff(fact_974_le__add1,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) ).

% le_add1
tff(fact_975_le__add2,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% le_add2
tff(fact_976_add__leD1,axiom,
    ! [M: nat,K2: nat,N2: 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),K2)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% add_leD1
tff(fact_977_add__leD2,axiom,
    ! [M: nat,K2: nat,N2: 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),K2)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2) ) ).

% add_leD2
tff(fact_978_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
     => ? [N5: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N5) ) ).

% le_Suc_ex
tff(fact_979_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ) ).

% add_le_mono
tff(fact_980_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) ) ).

% add_le_mono1
tff(fact_981_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M)) ) ).

% trans_le_add1
tff(fact_982_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J2)) ) ).

% trans_le_add2
tff(fact_983_nat__le__iff__add,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
    <=> ? [K3: nat] : N2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_984_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R: b,Ha: heap_e7401611519738050253t_unit,N6: nat] : heap_T7616092557645711336rame_b(N2,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R),produc584006145561248582it_nat(Ha,N6)))) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R),produc584006145561248582it_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ).

% timeFrame.simps(1)
tff(fact_985_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R: a,Ha: heap_e7401611519738050253t_unit,N6: nat] : heap_T7616092557645711335rame_a(N2,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(Ha,N6)))) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R),produc584006145561248582it_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ).

% timeFrame.simps(1)
tff(fact_986_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R: product_unit,Ha: heap_e7401611519738050253t_unit,N6: nat] : heap_T3616969660504097270t_unit(N2,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(Ha,N6)))) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R,produc584006145561248582it_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N6)))) ).

% timeFrame.simps(1)
tff(fact_987_mono__nat__linear__lb,axiom,
    ! [Fa: fun_nat_nat,M: nat,K2: nat] :
      ( ! [M5: nat,N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,M5)),aa_nat_nat(Fa,N5)) )
     => 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(Fa,M)),K2)),aa_nat_nat(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2))) ) ).

% mono_nat_linear_lb
tff(fact_988_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_989_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_990_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_991_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_992_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_993_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_994_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_995_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_996_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_997_verit__comp__simplify1_I1_J,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),A) ).

% verit_comp_simplify1(1)
tff(fact_998_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_999_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1000_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_1001_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_1002_in__mono,axiom,
    ! [A4: set_o,B4: set_o,X: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( aa_set_o_o(member_o((X)),A4)
       => aa_set_o_o(member_o((X)),B4) ) ) ).

% in_mono
tff(fact_1003_in__mono,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B4) ) ) ).

% in_mono
tff(fact_1004_in__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,X: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
     => ( aa_set_set_nat_o(member_set_nat(X),A4)
       => aa_set_set_nat_o(member_set_nat(X),B4) ) ) ).

% in_mono
tff(fact_1005_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(X),A4)
       => aa_set_nat_o(member_nat(X),B4) ) ) ).

% in_mono
tff(fact_1006_in__mono,axiom,
    ! [A4: set_int,B4: set_int,X: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(member_int(X),A4)
       => aa_set_int_o(member_int(X),B4) ) ) ).

% in_mono
tff(fact_1007_subsetD,axiom,
    ! [A4: set_o,B4: set_o,C2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( aa_set_o_o(member_o((C2)),A4)
       => aa_set_o_o(member_o((C2)),B4) ) ) ).

% subsetD
tff(fact_1008_subsetD,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int,C2: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% subsetD
tff(fact_1009_subsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
     => ( aa_set_set_nat_o(member_set_nat(C2),A4)
       => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% subsetD
tff(fact_1010_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(C2),A4)
       => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% subsetD
tff(fact_1011_subsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(member_int(C2),A4)
       => aa_set_int_o(member_int(C2),B4) ) ) ).

% subsetD
tff(fact_1012_psubsetE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
         => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4) ) ) ).

% psubsetE
tff(fact_1013_equalityE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4) ) ) ).

% equalityE
tff(fact_1014_subset__eq,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A4)
         => aa_set_o_o(member_o((X4)),B4) ) ) ).

% subset_eq
tff(fact_1015_subset__eq,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)
    <=> ! [X4: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A4)
         => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),B4) ) ) ).

% subset_eq
tff(fact_1016_subset__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A4)
         => aa_set_set_nat_o(member_set_nat(X4),B4) ) ) ).

% subset_eq
tff(fact_1017_subset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_eq
tff(fact_1018_subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_eq
tff(fact_1019_equalityD1,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4) ) ).

% equalityD1
tff(fact_1020_equalityD2,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4) ) ).

% equalityD2
tff(fact_1021_psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
        & ( A4 != B4 ) ) ) ).

% psubset_eq
tff(fact_1022_subset__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
    <=> ! [T4: $o] :
          ( aa_set_o_o(member_o((T4)),A4)
         => aa_set_o_o(member_o((T4)),B4) ) ) ).

% subset_iff
tff(fact_1023_subset__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)
    <=> ! [T4: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(T4),A4)
         => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(T4),B4) ) ) ).

% subset_iff
tff(fact_1024_subset__iff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
    <=> ! [T4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T4),A4)
         => aa_set_set_nat_o(member_set_nat(T4),B4) ) ) ).

% subset_iff
tff(fact_1025_subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ! [T4: nat] :
          ( aa_set_nat_o(member_nat(T4),A4)
         => aa_set_nat_o(member_nat(T4),B4) ) ) ).

% subset_iff
tff(fact_1026_subset__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
    <=> ! [T4: int] :
          ( aa_set_int_o(member_int(T4),A4)
         => aa_set_int_o(member_int(T4),B4) ) ) ).

% subset_iff
tff(fact_1027_subset__refl,axiom,
    ! [A4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),A4) ).

% subset_refl
tff(fact_1028_Collect__mono,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(Pa,X3)
         => aa_list_nat_o(Qa,X3) )
     => ord_le6045566169113846134st_nat(collect_list_nat(Pa),collect_list_nat(Qa)) ) ).

% Collect_mono
tff(fact_1029_Collect__mono,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( aa_set6170482117830564987_int_o(Pa,X3)
         => aa_set6170482117830564987_int_o(Qa,X3) )
     => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(Pa)),collec5210948495886036740nt_int(Qa)) ) ).

% Collect_mono
tff(fact_1030_Collect__mono,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(Pa,X3)
         => aa_set_nat_o(Qa,X3) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(Pa)),collect_set_nat(Qa)) ) ).

% Collect_mono
tff(fact_1031_Collect__mono,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(Pa,X3)
         => aa_nat_o(Qa,X3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(Pa)),collect_nat(Qa)) ) ).

% Collect_mono
tff(fact_1032_Collect__mono,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(Pa,X3)
         => aa_int_o(Qa,X3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(Pa)),collect_int(Qa)) ) ).

% Collect_mono
tff(fact_1033_subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),C3) ) ) ).

% subset_trans
tff(fact_1034_set__eq__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4) ) ) ).

% set_eq_subset
tff(fact_1035_Collect__mono__iff,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(Pa),collect_list_nat(Qa))
    <=> ! [X4: list_nat] :
          ( aa_list_nat_o(Pa,X4)
         => aa_list_nat_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1036_Collect__mono__iff,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(Pa)),collec5210948495886036740nt_int(Qa))
    <=> ! [X4: set_Pr958786334691620121nt_int] :
          ( aa_set6170482117830564987_int_o(Pa,X4)
         => aa_set6170482117830564987_int_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1037_Collect__mono__iff,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(Pa)),collect_set_nat(Qa))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(Pa,X4)
         => aa_set_nat_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1038_Collect__mono__iff,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(Pa)),collect_nat(Qa))
    <=> ! [X4: nat] :
          ( aa_nat_o(Pa,X4)
         => aa_nat_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1039_Collect__mono__iff,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(Pa)),collect_int(Qa))
    <=> ! [X4: int] :
          ( aa_int_o(Pa,X4)
         => aa_int_o(Qa,X4) ) ) ).

% Collect_mono_iff
tff(fact_1040_psubset__imp__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4) ) ).

% psubset_imp_subset
tff(fact_1041_psubset__subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),C3) ) ) ).

% psubset_subset_trans
tff(fact_1042_subset__not__subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
        & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4) ) ) ).

% subset_not_subset_eq
tff(fact_1043_subset__psubset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B4),C3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),C3) ) ) ).

% subset_psubset_trans
tff(fact_1044_subset__iff__psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
        | ( A4 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1045_Collect__subset,axiom,
    ! [A4: set_o,Pa: fun_o_o] : aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))),A4) ).

% Collect_subset
tff(fact_1046_Collect__subset,axiom,
    ! [A4: set_list_nat,Pa: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(A4),Pa)),A4) ).

% Collect_subset
tff(fact_1047_Collect__subset,axiom,
    ! [A4: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] : aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(A4),Pa))),A4) ).

% Collect_subset
tff(fact_1048_Collect__subset,axiom,
    ! [A4: set_set_nat,Pa: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_al(A4),Pa))),A4) ).

% Collect_subset
tff(fact_1049_Collect__subset,axiom,
    ! [A4: set_nat,Pa: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))),A4) ).

% Collect_subset
tff(fact_1050_Collect__subset,axiom,
    ! [A4: set_int,Pa: fun_int_o] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))),A4) ).

% Collect_subset
tff(fact_1051_less__eq__set__def,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
    <=> ord_less_eq_fun_o_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_eq_set_def
tff(fact_1052_less__eq__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)
    <=> ord_le2758740137681283170_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4)) ) ).

% less_eq_set_def
tff(fact_1053_less__eq__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4)) ) ).

% less_eq_set_def
tff(fact_1054_less__eq__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_eq_set_def
tff(fact_1055_less__eq__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4)) ) ).

% less_eq_set_def
tff(fact_1056_le__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y),Z3)
    <=> ( ord_le8081472938463900775at_nat(X,Z3)
        & ord_le8081472938463900775at_nat(Y,Z3) ) ) ).

% le_sup_iff
tff(fact_1057_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z3)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Z3)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),Z3) ) ) ).

% le_sup_iff
tff(fact_1058_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z3)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z3)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z3) ) ) ).

% le_sup_iff
tff(fact_1059_le__sup__iff,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)),Z3)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z3)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),Z3) ) ) ).

% le_sup_iff
tff(fact_1060_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),Z3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z3) ) ) ).

% le_sup_iff
tff(fact_1061_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z3)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3) ) ) ).

% le_sup_iff
tff(fact_1062_le__sup__iff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3) ) ) ).

% le_sup_iff
tff(fact_1063_sup_Obounded__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C2),A)
    <=> ( ord_le8081472938463900775at_nat(B,A)
        & ord_le8081472938463900775at_nat(C2,A) ) ) ).

% sup.bounded_iff
tff(fact_1064_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)),A)
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1065_sup_Obounded__iff,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1066_sup_Obounded__iff,axiom,
    ! [B: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C2)),A)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1067_sup_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1068_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1069_sup_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% sup.bounded_iff
tff(fact_1070_add__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_1071_add__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_1072_add__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_1073_add__less__cancel__right,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_1074_add__less__cancel__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_1075_add__less__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_1076_add__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_1077_add__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_1078_add__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_1079_add__le__cancel__right,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_1080_add__le__cancel__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_1081_add__le__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_1082_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_1083_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_1084_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_1085_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_1086_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_1087_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_1088_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1089_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1090_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1091_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1092_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1093_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1094_sup_Oidem,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),A) = A ).

% sup.idem
tff(fact_1095_sup_Oidem,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),A) = A ).

% sup.idem
tff(fact_1096_sup_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).

% sup.idem
tff(fact_1097_sup_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),A) = A ).

% sup.idem
tff(fact_1098_sup_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,A),A) = A ).

% sup.idem
tff(fact_1099_add__right__cancel,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),A) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1100_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1101_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1102_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_1103_add__left__cancel,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1104_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1105_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1106_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_1107_sup_Oright__idem,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)),B) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ).

% sup.right_idem
tff(fact_1108_sup_Oright__idem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ).

% sup.right_idem
tff(fact_1109_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.right_idem
tff(fact_1110_sup_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ).

% sup.right_idem
tff(fact_1111_sup_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ).

% sup.right_idem
tff(fact_1112_sup__left__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) ).

% sup_left_idem
tff(fact_1113_sup__left__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ).

% sup_left_idem
tff(fact_1114_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% sup_left_idem
tff(fact_1115_sup__left__idem,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) ).

% sup_left_idem
tff(fact_1116_sup__left__idem,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) ).

% sup_left_idem
tff(fact_1117_sup_Oleft__idem,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ).

% sup.left_idem
tff(fact_1118_sup_Oleft__idem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ).

% sup.left_idem
tff(fact_1119_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.left_idem
tff(fact_1120_sup_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ).

% sup.left_idem
tff(fact_1121_sup_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ).

% sup.left_idem
tff(fact_1122_sup__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),X) = X ).

% sup_idem
tff(fact_1123_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),X) = X ).

% sup_idem
tff(fact_1124_sup__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),X) = X ).

% sup_idem
tff(fact_1125_sup__idem,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),X) = X ).

% sup_idem
tff(fact_1126_sup__idem,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),X) = X ).

% sup_idem
tff(fact_1127_mod__or__dist,axiom,
    ! [Pa: assn,Qa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
        | aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% mod_or_dist
tff(fact_1128_sup__set__def,axiom,
    ! [A4: set_o,B4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4) = collect_o(sup_sup_fun_o_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% sup_set_def
tff(fact_1129_sup__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : sup_sup_set_list_nat(A4,B4) = collect_list_nat(sup_su6906368799827444965_nat_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ).

% sup_set_def
tff(fact_1130_sup__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] : sup_su2047564715030645325nt_int(A4,B4) = collec5210948495886036740nt_int(sup_su2829768162508406574_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% sup_set_def
tff(fact_1131_sup__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : sup_sup_set_set_nat(A4,B4) = collect_set_nat(sup_su7503774895437956671_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% sup_set_def
tff(fact_1132_sup__set__def,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = collect_int(sup_sup_fun_int_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% sup_set_def
tff(fact_1133_sup__set__def,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = collec5204685387357076818at_nat(sup_su5351388498073598816_nat_o(aTP_Lamm_bj(A4),aTP_Lamm_bj(B4))) ).

% sup_set_def
tff(fact_1134_sup__set__def,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = collec6321179662152712658at_nat(sup_su2398265404452057824_nat_o(aTP_Lamm_bk(A4),aTP_Lamm_bk(B4))) ).

% sup_set_def
tff(fact_1135_sup__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = collect_nat(sup_sup_fun_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% sup_set_def
tff(fact_1136_add__right__imp__eq,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),A) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1137_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1138_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1139_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_1140_add__left__imp__eq,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1141_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: 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),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1142_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1143_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_1144_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),C2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1145_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: 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),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1146_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1147_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1148_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_1149_ab__semigroup__add__class_Oadd_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) ).

% ab_semigroup_add_class.add.commute
tff(fact_1150_ab__semigroup__add__class_Oadd_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) ).

% ab_semigroup_add_class.add.commute
tff(fact_1151_ab__semigroup__add__class_Oadd_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) ).

% ab_semigroup_add_class.add.commute
tff(fact_1152_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_1153_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_1154_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: 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),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_1155_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_1156_add_Oassoc,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B)),C2) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),C2)) ).

% add.assoc
tff(fact_1157_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% add.assoc
tff(fact_1158_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% add.assoc
tff(fact_1159_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% add.assoc
tff(fact_1160_group__cancel_Oadd2,axiom,
    ! [B4: multis2468970476368604999at_nat,K2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( B4 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(K2),B) )
     => ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B4) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(K2),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_1161_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1162_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1163_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1164_group__cancel_Oadd1,axiom,
    ! [A4: multis2468970476368604999at_nat,K2: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A4 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(K2),A) )
     => ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A4),B) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(K2),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_1165_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1166_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1167_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1168_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1169_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1170_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1171_sup__left__commute,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z3)) ).

% sup_left_commute
tff(fact_1172_sup__left__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z3)) ).

% sup_left_commute
tff(fact_1173_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3)) ).

% sup_left_commute
tff(fact_1174_sup__left__commute,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z3)) ).

% sup_left_commute
tff(fact_1175_sup__left__commute,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z3)) ).

% sup_left_commute
tff(fact_1176_sup_Oleft__commute,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),C2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C2)) ).

% sup.left_commute
tff(fact_1177_sup_Oleft__commute,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) ).

% sup.left_commute
tff(fact_1178_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% sup.left_commute
tff(fact_1179_sup_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)) ).

% sup.left_commute
tff(fact_1180_sup_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),C2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)) ).

% sup.left_commute
tff(fact_1181_sup__commute,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X) ).

% sup_commute
tff(fact_1182_sup__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ).

% sup_commute
tff(fact_1183_sup__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% sup_commute
tff(fact_1184_sup__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X) ).

% sup_commute
tff(fact_1185_sup__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),X) ).

% sup_commute
tff(fact_1186_sup_Ocommute,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),A) ).

% sup.commute
tff(fact_1187_sup_Ocommute,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),A) ).

% sup.commute
tff(fact_1188_sup_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).

% sup.commute
tff(fact_1189_sup_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),A) ).

% sup.commute
tff(fact_1190_sup_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,B),A) ).

% sup.commute
tff(fact_1191_sup__assoc,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),Z3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) ).

% sup_assoc
tff(fact_1192_sup__assoc,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) ).

% sup_assoc
tff(fact_1193_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) ).

% sup_assoc
tff(fact_1194_sup__assoc,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z3) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)) ).

% sup_assoc
tff(fact_1195_sup__assoc,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z3) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)) ).

% sup_assoc
tff(fact_1196_sup_Oassoc,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)),C2) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C2)) ).

% sup.assoc
tff(fact_1197_sup_Oassoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) ).

% sup.assoc
tff(fact_1198_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% sup.assoc
tff(fact_1199_sup_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)) ).

% sup.assoc
tff(fact_1200_sup_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)) ).

% sup.assoc
tff(fact_1201_inf__sup__aci_I5_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_1202_inf__sup__aci_I5_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_1203_inf__sup__aci_I5_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_1204_inf__sup__aci_I5_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_1205_inf__sup__aci_I5_J,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),X) ).

% inf_sup_aci(5)
tff(fact_1206_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),Z3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) ).

% inf_sup_aci(6)
tff(fact_1207_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) ).

% inf_sup_aci(6)
tff(fact_1208_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) ).

% inf_sup_aci(6)
tff(fact_1209_inf__sup__aci_I6_J,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z3) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)) ).

% inf_sup_aci(6)
tff(fact_1210_inf__sup__aci_I6_J,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z3) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)) ).

% inf_sup_aci(6)
tff(fact_1211_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z3)) ).

% inf_sup_aci(7)
tff(fact_1212_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z3)) ).

% inf_sup_aci(7)
tff(fact_1213_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3)) ).

% inf_sup_aci(7)
tff(fact_1214_inf__sup__aci_I7_J,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z3)) ).

% inf_sup_aci(7)
tff(fact_1215_inf__sup__aci_I7_J,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z3)) ).

% inf_sup_aci(7)
tff(fact_1216_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_1217_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_1218_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_1219_inf__sup__aci_I8_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_1220_inf__sup__aci_I8_J,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) ).

% inf_sup_aci(8)
tff(fact_1221_add__le__imp__le__right,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_1222_add__le__imp__le__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_1223_add__le__imp__le__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_1224_add__le__imp__le__left,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_1225_add__le__imp__le__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_1226_add__le__imp__le__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_1227_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_1228_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ) ).

% add_right_mono
tff(fact_1229_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_right_mono
tff(fact_1230_add__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_right_mono
tff(fact_1231_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ).

% less_eqE
tff(fact_1232_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)) ) ).

% add_left_mono
tff(fact_1233_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_left_mono
tff(fact_1234_add__left__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_left_mono
tff(fact_1235_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_mono
tff(fact_1236_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_mono
tff(fact_1237_add__mono,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_mono
tff(fact_1238_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1239_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1240_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1241_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1242_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1243_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1244_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J2)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1245_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1246_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1247_add__less__imp__less__right,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1248_add__less__imp__less__right,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_1249_add__less__imp__less__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_1250_add__less__imp__less__left,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1251_add__less__imp__less__left,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_1252_add__less__imp__less__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_1253_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ) ).

% add_strict_right_mono
tff(fact_1254_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_strict_right_mono
tff(fact_1255_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_strict_right_mono
tff(fact_1256_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)) ) ).

% add_strict_left_mono
tff(fact_1257_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_strict_left_mono
tff(fact_1258_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_strict_left_mono
tff(fact_1259_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_1260_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_1261_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_strict_mono
tff(fact_1262_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J2)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1263_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1264_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J2)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1265_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1266_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1267_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1268_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1269_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1270_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1271_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C2,B)
     => ord_le8081472938463900775at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1272_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),B)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1273_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1274_sup_OcoboundedI2,axiom,
    ! [C2: set_int,B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),B)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_1275_sup_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1276_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1277_sup_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_1278_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C2,A)
     => ord_le8081472938463900775at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1279_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1280_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1281_sup_OcoboundedI1,axiom,
    ! [C2: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_1282_sup_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1283_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1284_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_1285_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,B)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1286_sup_Oabsorb__iff2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1287_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1288_sup_Oabsorb__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1289_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1290_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1291_sup_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1292_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1293_sup_Oabsorb__iff1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1294_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1295_sup_Oabsorb__iff1,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1296_sup_Oabsorb__iff1,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(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1297_sup_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1298_sup_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1299_sup_Ocobounded2,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(B,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ).

% sup.cobounded2
tff(fact_1300_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded2
tff(fact_1301_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded2
tff(fact_1302_sup_Ocobounded2,axiom,
    ! [B: set_int,A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ).

% sup.cobounded2
tff(fact_1303_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded2
tff(fact_1304_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded2
tff(fact_1305_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded2
tff(fact_1306_sup_Ocobounded1,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(A,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ).

% sup.cobounded1
tff(fact_1307_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% sup.cobounded1
tff(fact_1308_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded1
tff(fact_1309_sup_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ).

% sup.cobounded1
tff(fact_1310_sup_Ocobounded1,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(sup_sup_rat,A),B)) ).

% sup.cobounded1
tff(fact_1311_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded1
tff(fact_1312_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded1
tff(fact_1313_sup_Oorder__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
    <=> ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_1314_sup_Oorder__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
    <=> ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_1315_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_1316_sup_Oorder__iff,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) ) ) ).

% sup.order_iff
tff(fact_1317_sup_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_1318_sup_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_1319_sup_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_1320_sup_OboundedI,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( ord_le8081472938463900775at_nat(C2,A)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C2),A) ) ) ).

% sup.boundedI
tff(fact_1321_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_1322_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_1323_sup_OboundedI,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_1324_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_1325_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_1326_sup_OboundedI,axiom,
    ! [B: int,A: int,C2: 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,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A) ) ) ).

% sup.boundedI
tff(fact_1327_sup_OboundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C2),A)
     => ~ ( ord_le8081472938463900775at_nat(B,A)
         => ~ ord_le8081472938463900775at_nat(C2,A) ) ) ).

% sup.boundedE
tff(fact_1328_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)),A)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
         => ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A) ) ) ).

% sup.boundedE
tff(fact_1329_sup_OboundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A) ) ) ).

% sup.boundedE
tff(fact_1330_sup_OboundedE,axiom,
    ! [B: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C2)),A)
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A) ) ) ).

% sup.boundedE
tff(fact_1331_sup_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% sup.boundedE
tff(fact_1332_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% sup.boundedE
tff(fact_1333_sup_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% sup.boundedE
tff(fact_1334_sup__absorb2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,Y)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1335_sup__absorb2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Y)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1336_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1337_sup__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1338_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1339_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1340_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1341_sup__absorb1,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Y,X)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1342_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1343_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1344_sup__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1345_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1346_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1347_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1348_sup_Oabsorb2,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,B)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1349_sup_Oabsorb2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1350_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1351_sup_Oabsorb2,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1352_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1353_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1354_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1355_sup_Oabsorb1,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1356_sup_Oabsorb1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1357_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1358_sup_Oabsorb1,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1359_sup_Oabsorb1,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(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1360_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1361_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1362_sup__unique,axiom,
    ! [Fa: fun_se6682931024248017501at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(Fa,X3),Y3))
     => ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(Fa,X3),Y3))
       => ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
              ( ord_le8081472938463900775at_nat(Y3,X3)
             => ( ord_le8081472938463900775at_nat(Z2,X3)
               => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(Fa,Y3),Z2),X3) ) )
         => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1363_sup__unique,axiom,
    ! [Fa: fun_se177437144654672861at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(Fa,X3),Y3))
     => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(Fa,X3),Y3))
       => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y3),X3)
             => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z2),X3)
               => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(Fa,Y3),Z2)),X3) ) )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1364_sup__unique,axiom,
    ! [Fa: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,X3),Y3))
     => ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,X3),Y3))
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y3),X3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z2),X3)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,Y3),Z2)),X3) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1365_sup__unique,axiom,
    ! [Fa: fun_se6573205548028396576et_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,X3),Y3))
     => ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y3),aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,X3),Y3))
       => ( ! [X3: set_int,Y3: set_int,Z2: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y3),X3)
             => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z2),X3)
               => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,Y3),Z2)),X3) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1366_sup__unique,axiom,
    ! [Fa: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(Fa,X3),Y3))
     => ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),aa_rat_rat(aa_rat_fun_rat_rat(Fa,X3),Y3))
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),X3)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(Fa,Y3),Z2)),X3) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1367_sup__unique,axiom,
    ! [Fa: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(Fa,X3),Y3))
     => ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),aa_nat_nat(aa_nat_fun_nat_nat(Fa,X3),Y3))
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(Fa,Y3),Z2)),X3) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1368_sup__unique,axiom,
    ! [Fa: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(Fa,X3),Y3))
     => ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),aa_int_int(aa_int_fun_int_int(Fa,X3),Y3))
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),X3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(Fa,Y3),Z2)),X3) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(Fa,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1369_sup_OorderI,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) )
     => ord_le8081472938463900775at_nat(B,A) ) ).

% sup.orderI
tff(fact_1370_sup_OorderI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A) ) ).

% sup.orderI
tff(fact_1371_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% sup.orderI
tff(fact_1372_sup_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A) ) ).

% sup.orderI
tff(fact_1373_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% sup.orderI
tff(fact_1374_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% sup.orderI
tff(fact_1375_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% sup.orderI
tff(fact_1376_sup_OorderE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(B,A)
     => ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_1377_sup_OorderE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),A)
     => ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ) ) ).

% sup.orderE
tff(fact_1378_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_1379_sup_OorderE,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) ) ) ).

% sup.orderE
tff(fact_1380_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_1381_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_1382_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_1383_le__iff__sup,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,Y)
    <=> ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1384_le__iff__sup,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),Y)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1385_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1386_le__iff__sup,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1387_le__iff__sup,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1388_le__iff__sup,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1389_le__iff__sup,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1390_sup__least,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(Y,X)
     => ( ord_le8081472938463900775at_nat(Z3,X)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3),X) ) ) ).

% sup_least
tff(fact_1391_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),X)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Z3),X)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)),X) ) ) ).

% sup_least
tff(fact_1392_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z3),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)),X) ) ) ).

% sup_least
tff(fact_1393_sup__least,axiom,
    ! [Y: set_int,X: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Z3),X)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z3)),X) ) ) ).

% sup_least
tff(fact_1394_sup__least,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z3)),X) ) ) ).

% sup_least
tff(fact_1395_sup__least,axiom,
    ! [Y: nat,X: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)),X) ) ) ).

% sup_least
tff(fact_1396_sup__least,axiom,
    ! [Y: int,X: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)),X) ) ) ).

% sup_least
tff(fact_1397_sup__mono,axiom,
    ! [A: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,C2)
     => ( ord_le8081472938463900775at_nat(B,D2)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),D2)) ) ) ).

% sup_mono
tff(fact_1398_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),C2)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),D2)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D2)) ) ) ).

% sup_mono
tff(fact_1399_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D2)) ) ) ).

% sup_mono
tff(fact_1400_sup__mono,axiom,
    ! [A: set_int,C2: set_int,B: set_int,D2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D2)) ) ) ).

% sup_mono
tff(fact_1401_sup__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C2),D2)) ) ) ).

% sup_mono
tff(fact_1402_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D2)) ) ) ).

% sup_mono
tff(fact_1403_sup__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D2)) ) ) ).

% sup_mono
tff(fact_1404_sup_Omono,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(C2,A)
     => ( ord_le8081472938463900775at_nat(D2,B)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C2),D2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_1405_sup_Omono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C2),A)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(D2),B)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% sup.mono
tff(fact_1406_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_1407_sup_Omono,axiom,
    ! [C2: set_int,A: set_int,D2: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,D2),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,C2),D2)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ) ).

% sup.mono
tff(fact_1408_sup_Omono,axiom,
    ! [C2: rat,A: rat,D2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C2),D2)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ) ).

% sup.mono
tff(fact_1409_sup_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_1410_sup_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C2),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_1411_le__supI2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,B)
     => ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% le_supI2
tff(fact_1412_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),B)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI2
tff(fact_1413_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI2
tff(fact_1414_le__supI2,axiom,
    ! [X: set_int,B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),B)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% le_supI2
tff(fact_1415_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI2
tff(fact_1416_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI2
tff(fact_1417_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI2
tff(fact_1418_le__supI1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(X,A)
     => ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% le_supI1
tff(fact_1419_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),A)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% le_supI1
tff(fact_1420_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI1
tff(fact_1421_le__supI1,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% le_supI1
tff(fact_1422_le__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI1
tff(fact_1423_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI1
tff(fact_1424_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI1
tff(fact_1425_sup__ge2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% sup_ge2
tff(fact_1426_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge2
tff(fact_1427_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge2
tff(fact_1428_sup__ge2,axiom,
    ! [Y: set_int,X: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% sup_ge2
tff(fact_1429_sup__ge2,axiom,
    ! [Y: rat,X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% sup_ge2
tff(fact_1430_sup__ge2,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge2
tff(fact_1431_sup__ge2,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge2
tff(fact_1432_sup__ge1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% sup_ge1
tff(fact_1433_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge1
tff(fact_1434_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge1
tff(fact_1435_sup__ge1,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% sup_ge1
tff(fact_1436_sup__ge1,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% sup_ge1
tff(fact_1437_sup__ge1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge1
tff(fact_1438_sup__ge1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge1
tff(fact_1439_le__supI,axiom,
    ! [A: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A,X)
     => ( ord_le8081472938463900775at_nat(B,X)
       => ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B),X) ) ) ).

% le_supI
tff(fact_1440_le__supI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_1441_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_1442_le__supI,axiom,
    ! [A: set_int,X: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),X)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),X) ) ) ).

% le_supI
tff(fact_1443_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X) ) ) ).

% le_supI
tff(fact_1444_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_1445_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X) ) ) ).

% le_supI
tff(fact_1446_le__supE,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B),X)
     => ~ ( ord_le8081472938463900775at_nat(A,X)
         => ~ ord_le8081472938463900775at_nat(B,X) ) ) ).

% le_supE
tff(fact_1447_le__supE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),X)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A),X)
         => ~ aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B),X) ) ) ).

% le_supE
tff(fact_1448_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X) ) ) ).

% le_supE
tff(fact_1449_le__supE,axiom,
    ! [A: set_int,B: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)),X)
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),X) ) ) ).

% le_supE
tff(fact_1450_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X) ) ) ).

% le_supE
tff(fact_1451_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X) ) ) ).

% le_supE
tff(fact_1452_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X) ) ) ).

% le_supE
tff(fact_1453_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1454_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1455_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1456_inf__sup__ord_I3_J,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1457_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1458_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1459_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1460_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(Y,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1461_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1462_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1463_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1464_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1465_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1466_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1467_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(C2,B)
     => ord_le7642048601412989811at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1468_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C2),B)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1469_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1470_sup_Ostrict__coboundedI2,axiom,
    ! [C2: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1471_sup_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1472_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1473_sup_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1474_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(C2,A)
     => ord_le7642048601412989811at_nat(C2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1475_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C2),A)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1476_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1477_sup_Ostrict__coboundedI1,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1478_sup_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1479_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1480_sup_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1481_sup_Ostrict__order__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(B,A)
    <=> ( ( A = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1482_sup_Ostrict__order__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
    <=> ( ( A = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1483_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1484_sup_Ostrict__order__iff,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1485_sup_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1486_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1487_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1488_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B),C2),A)
     => ~ ( ord_le7642048601412989811at_nat(B,A)
         => ~ ord_le7642048601412989811at_nat(C2,A) ) ) ).

% sup.strict_boundedE
tff(fact_1489_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)),A)
     => ~ ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
         => ~ aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1490_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1491_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2)),A)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1492_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C2)),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,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1493_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1494_sup_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% sup.strict_boundedE
tff(fact_1495_sup_Oabsorb4,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(A,B)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1496_sup_Oabsorb4,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A),B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1497_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1498_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1499_sup_Oabsorb4,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(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1500_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1501_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1502_sup_Oabsorb3,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(B,A)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1503_sup_Oabsorb3,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(B),A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1504_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1505_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1506_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1507_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1508_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1509_less__supI2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(X,B)
     => ord_le7642048601412989811at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% less_supI2
tff(fact_1510_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),B)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% less_supI2
tff(fact_1511_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI2
tff(fact_1512_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI2
tff(fact_1513_less__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI2
tff(fact_1514_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI2
tff(fact_1515_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI2
tff(fact_1516_less__supI1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ord_le7642048601412989811at_nat(X,A)
     => ord_le7642048601412989811at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ).

% less_supI1
tff(fact_1517_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),A)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% less_supI1
tff(fact_1518_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI1
tff(fact_1519_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI1
tff(fact_1520_less__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI1
tff(fact_1521_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI1
tff(fact_1522_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI1
tff(fact_1523_pred__subset__eq,axiom,
    ! [Ra: set_o,S: set_o] :
      ( ord_less_eq_fun_o_o(aTP_Lamm_a(Ra),aTP_Lamm_a(S))
    <=> aa_set_o_o(ord_less_eq_set_o(Ra),S) ) ).

% pred_subset_eq
tff(fact_1524_pred__subset__eq,axiom,
    ! [Ra: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
      ( ord_le2758740137681283170_int_o(aTP_Lamm_ab(Ra),aTP_Lamm_ab(S))
    <=> aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(Ra),S) ) ).

% pred_subset_eq
tff(fact_1525_pred__subset__eq,axiom,
    ! [Ra: set_set_nat,S: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(Ra),aTP_Lamm_ac(S))
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(Ra),S) ) ).

% pred_subset_eq
tff(fact_1526_pred__subset__eq,axiom,
    ! [Ra: set_nat,S: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(Ra),aTP_Lamm_ad(S))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Ra),S) ) ).

% pred_subset_eq
tff(fact_1527_pred__subset__eq,axiom,
    ! [Ra: set_int,S: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(Ra),aTP_Lamm_ae(S))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ra),S) ) ).

% pred_subset_eq
tff(fact_1528_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] :
      ( ( heap_T7616092557645711336rame_b(X,Xa) = Y )
     => ( ! [R2: b,H: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H,N4))) )
           => ( Y != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) ) )
       => ~ ( ( Xa = none_P6779099274072355161it_nat )
           => ( Y != none_P6779099274072355161it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_1529_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( heap_T7616092557645711335rame_a(X,Xa) = Y )
     => ( ! [R2: a,H: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H,N4))) )
           => ( Y != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) ) )
       => ~ ( ( Xa = none_P2651198173097904984it_nat )
           => ( Y != none_P2651198173097904984it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_1530_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( heap_T3616969660504097270t_unit(X,Xa) = Y )
     => ( ! [R2: product_unit,H: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H,N4))) )
           => ( Y != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) ) )
       => ~ ( ( Xa = none_P9117596204409417319it_nat )
           => ( Y != none_P9117596204409417319it_nat ) ) ) ) ).

% timeFrame.elims
tff(fact_1531_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5174734335190411425at_nat(sup_su3035147773818789531at_nat,aTP_Lamm_bl,aTP_Lamm_bm) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1532_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic6976360593897529633at_nat(sup_su5525570899277871387at_nat,aTP_Lamm_bn,aTP_Lamm_bo) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1533_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat(sup_sup_set_nat,aTP_Lamm_bp,aTP_Lamm_bq) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1534_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn(sup_sup_assn,aTP_Lamm_br,aTP_Lamm_bs) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1535_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic8154731777541915528et_int(sup_sup_set_int,aTP_Lamm_bt,aTP_Lamm_bu) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1536_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic5374021519246970238et_rat(sup_sup_rat,aTP_Lamm_bv,aTP_Lamm_bw) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1537_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat(sup_sup_nat,aTP_Lamm_bx,aTP_Lamm_by) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1538_Sup__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int(sup_sup_int,aTP_Lamm_bz,aTP_Lamm_ca) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_1539_sup__option__def,axiom,
    ! [X: option2498585697089621389at_nat,Y: option2498585697089621389at_nat] : sup_su5873096292857799009at_nat(X,Y) = case_o8757249484648823045at_nat(Y,aa_opt1741456117285354266at_nat(aTP_Lamm_cc(X),Y),X) ).

% sup_option_def
tff(fact_1540_sup__option__def,axiom,
    ! [X: option1583680563626158861at_nat,Y: option1583680563626158861at_nat] : sup_su2273273666271716065at_nat(X,Y) = case_o9080059180780454917at_nat(Y,aa_opt1027706798495913626at_nat(aTP_Lamm_ce(X),Y),X) ).

% sup_option_def
tff(fact_1541_sup__option__def,axiom,
    ! [X: option_set_nat,Y: option_set_nat] : sup_su3598758113090618626et_nat(X,Y) = case_o4054078431260844265et_nat(Y,aa_opt5648106117674462701et_nat(aTP_Lamm_cg(X),Y),X) ).

% sup_option_def
tff(fact_1542_sup__option__def,axiom,
    ! [X: option_nat,Y: option_nat] : sup_sup_option_nat(X,Y) = case_o7429725398727453821at_nat(Y,aa_opt1558076126722380235on_nat(aTP_Lamm_ci(X),Y),X) ).

% sup_option_def
tff(fact_1543_sup__option__def,axiom,
    ! [X: option_int,Y: option_int] : sup_sup_option_int(X,Y) = case_o390784466056649525nt_int(Y,aa_opt334079648442228447on_int(aTP_Lamm_ck(X),Y),X) ).

% sup_option_def
tff(fact_1544_sup__Un__eq,axiom,
    ! [Ra: set_o,S: set_o,X5: $o] :
      ( aa_o_o(sup_sup_fun_o_o(aTP_Lamm_a(Ra),aTP_Lamm_a(S)),(X5))
    <=> aa_set_o_o(member_o((X5)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1545_sup__Un__eq,axiom,
    ! [Ra: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int,X5: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(sup_su2829768162508406574_int_o(aTP_Lamm_ab(Ra),aTP_Lamm_ab(S)),X5)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X5),sup_su2047564715030645325nt_int(Ra,S)) ) ).

% sup_Un_eq
tff(fact_1546_sup__Un__eq,axiom,
    ! [Ra: set_set_nat,S: set_set_nat,X5: set_nat] :
      ( aa_set_nat_o(sup_su7503774895437956671_nat_o(aTP_Lamm_ac(Ra),aTP_Lamm_ac(S)),X5)
    <=> aa_set_set_nat_o(member_set_nat(X5),sup_sup_set_set_nat(Ra,S)) ) ).

% sup_Un_eq
tff(fact_1547_sup__Un__eq,axiom,
    ! [Ra: set_int,S: set_int,X5: int] :
      ( aa_int_o(sup_sup_fun_int_o(aTP_Lamm_ae(Ra),aTP_Lamm_ae(S)),X5)
    <=> aa_set_int_o(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1548_sup__Un__eq,axiom,
    ! [Ra: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat,X5: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(sup_su5351388498073598816_nat_o(aTP_Lamm_bj(Ra),aTP_Lamm_bj(S)),X5)
    <=> member6689249552917799696at_nat(X5,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1549_sup__Un__eq,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X5: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(sup_su2398265404452057824_nat_o(aTP_Lamm_bk(Ra),aTP_Lamm_bk(S)),X5)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1550_sup__Un__eq,axiom,
    ! [Ra: set_nat,S: set_nat,X5: nat] :
      ( aa_nat_o(sup_sup_fun_nat_o(aTP_Lamm_ad(Ra),aTP_Lamm_ad(S)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Ra),S)) ) ).

% sup_Un_eq
tff(fact_1551_sup__Un__eq2,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,X5: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(sup_su7393609867835711046_int_o(aTP_Lamm_cl(Ra),aTP_Lamm_cl(S)),X5),Xa2)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa2)),sup_su6024340866399070445nt_int(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1552_sup__Un__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(sup_su4970993554803434671_nat_o(aTP_Lamm_cm(Ra),aTP_Lamm_cm(S)),X5),Xa2)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X5),Xa2),sup_su2387659076810923138et_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1553_sup__Un__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(sup_su6780828561269846801_nat_o(aTP_Lamm_cn(Ra),aTP_Lamm_cn(S)),X5),Xa2)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X5),Xa2),sup_su5858552540197408416et_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1554_sup__Un__eq2,axiom,
    ! [Ra: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat,X5: b,Xa2: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(sup_su3888246858588632405_nat_o(aTP_Lamm_co(Ra),aTP_Lamm_co(S)),X5),Xa2)
    <=> member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X5),Xa2),sup_su3714129841433130204it_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1555_sup__Un__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat,X5: a,Xa2: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(sup_su5814904253042911126_nat_o(aTP_Lamm_cp(Ra),aTP_Lamm_cp(S)),X5),Xa2)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X5),Xa2),sup_su3211442154794319707it_nat(Ra,S)) ) ).

% sup_Un_eq2
tff(fact_1556_sup__Un__eq2,axiom,
    ! [Ra: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat,X5: multis2468970476368604999at_nat,Xa2: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(sup_su9034197199115871746_nat_o(aTP_Lamm_cq(Ra),aTP_Lamm_cq(S)),X5),Xa2)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(X5,Xa2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Ra),S)) ) ).

% sup_Un_eq2
tff(fact_1557_sup__Un__eq2,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,X5: set_Pr1261947904930325089at_nat,Xa2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aTP_Lamm_cr(Ra),aTP_Lamm_cr(S)),X5),Xa2)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Xa2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Ra),S)) ) ).

% sup_Un_eq2
tff(fact_1558_pred__subset__eq2,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_cl(Ra),aTP_Lamm_cl(S))
    <=> aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(Ra),S) ) ).

% pred_subset_eq2
tff(fact_1559_pred__subset__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
      ( ord_le3580378406483493347_nat_o(aTP_Lamm_cm(Ra),aTP_Lamm_cm(S))
    <=> ord_le3919382008627836342et_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1560_pred__subset__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
      ( ord_le2834457679758710341_nat_o(aTP_Lamm_cn(Ra),aTP_Lamm_cn(S))
    <=> ord_le5189643068899110356et_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1561_pred__subset__eq2,axiom,
    ! [Ra: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat] :
      ( ord_le4682729914586826889_nat_o(aTP_Lamm_co(Ra),aTP_Lamm_co(S))
    <=> ord_le4980869671742669840it_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1562_pred__subset__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ord_le6609387309041105610_nat_o(aTP_Lamm_cp(Ra),aTP_Lamm_cp(S))
    <=> ord_le4478181985103859343it_nat(Ra,S) ) ).

% pred_subset_eq2
tff(fact_1563_pred__equals__eq2,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa3: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),Ra)
        <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1564_pred__equals__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Xa3),Ra)
        <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Xa3),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1565_pred__equals__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Xa3),Ra)
        <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Xa3),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1566_pred__equals__eq2,axiom,
    ! [Ra: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat] :
      ( ! [X4: b,Xa3: produc6653097349344004940it_nat] :
          ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X4),Xa3),Ra)
        <=> member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X4),Xa3),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1567_pred__equals__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ! [X4: a,Xa3: produc6653097349344004940it_nat] :
          ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Xa3),Ra)
        <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Xa3),S) )
    <=> ( Ra = S ) ) ).

% pred_equals_eq2
tff(fact_1568_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,Fa: fun_he4944555133437423336it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T8440541562793052209t_unit(Pa,Fa)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_hea256619718444519449it_nat(Fa,Ha)) ) ) ).

% execute_guard(2)
tff(fact_1569_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,Fa: fun_he168995381318928090it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_guard_b(Pa,Fa)),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_hea550092171636296971it_nat(Fa,Ha)) ) ) ).

% execute_guard(2)
tff(fact_1570_execute__guard_I2_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,Fa: fun_he8889679731534893401it_nat] :
      ( aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_guard_a(Pa,Fa)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_hea5645563107516622602it_nat(Fa,Ha)) ) ) ).

% execute_guard(2)
tff(fact_1571_some__opt__eq__trivial,axiom,
    ! [X: produc7388388658123137530it_nat] : eps_Op5869321083069970014it_nat(aTP_Lamm_cs(X)) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) ).

% some_opt_eq_trivial
tff(fact_1572_some__opt__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] : eps_Op1741419982095519837it_nat(aTP_Lamm_ct(X)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) ).

% some_opt_eq_trivial
tff(fact_1573_some__opt__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] : eps_Op3393321821070424684it_nat(aTP_Lamm_cu(X)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) ).

% some_opt_eq_trivial
tff(fact_1574_some__opt__eq__trivial,axiom,
    ! [X: num] : eps_Opt_num(aa_num_fun_num_o(aTP_Lamm_cv,X)) = aa_num_option_num(some_num,X) ).

% some_opt_eq_trivial
tff(fact_1575_not__None__eq,axiom,
    ! [X: option4065278094766928714it_nat] :
      ( ( X != none_P6779099274072355161it_nat )
    <=> ? [Y4: produc7388388658123137530it_nat] : X = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y4) ) ).

% not_None_eq
tff(fact_1576_not__None__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ( X != none_P2651198173097904984it_nat )
    <=> ? [Y4: produc3260487557148687353it_nat] : X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y4) ) ).

% not_None_eq
tff(fact_1577_not__None__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( X != none_P9117596204409417319it_nat )
    <=> ? [Y4: produc8664842809031399944it_nat] : X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y4) ) ).

% not_None_eq
tff(fact_1578_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y4: num] : X = aa_num_option_num(some_num,Y4) ) ).

% not_None_eq
tff(fact_1579_not__Some__eq,axiom,
    ! [X: option4065278094766928714it_nat] :
      ( ! [Y4: produc7388388658123137530it_nat] : X != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y4)
    <=> ( X = none_P6779099274072355161it_nat ) ) ).

% not_Some_eq
tff(fact_1580_not__Some__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ! [Y4: produc3260487557148687353it_nat] : X != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y4)
    <=> ( X = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq
tff(fact_1581_not__Some__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ! [Y4: produc8664842809031399944it_nat] : X != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y4)
    <=> ( X = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq
tff(fact_1582_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y4: num] : X != aa_num_option_num(some_num,Y4)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_1583_some__opt__sym__eq__trivial,axiom,
    ! [X: produc7388388658123137530it_nat] : eps_Op5869321083069970014it_nat(fequal2310338401145355621it_nat(X)) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) ).

% some_opt_sym_eq_trivial
tff(fact_1584_some__opt__sym__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] : eps_Op1741419982095519837it_nat(fequal7405809337025681252it_nat(X)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) ).

% some_opt_sym_eq_trivial
tff(fact_1585_some__opt__sym__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] : eps_Op3393321821070424684it_nat(fequal1407654770548522611it_nat(X)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) ).

% some_opt_sym_eq_trivial
tff(fact_1586_some__opt__sym__eq__trivial,axiom,
    ! [X: num] : eps_Opt_num(aa_num_fun_num_o(fequal_num,X)) = aa_num_option_num(some_num,X) ).

% some_opt_sym_eq_trivial
tff(fact_1587_less__eq__option__None__code,axiom,
    ! [X: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),X) ).

% less_eq_option_None_code
tff(fact_1588_less__option__None,axiom,
    ! [X: option_num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,X),none_num) ).

% less_option_None
tff(fact_1589_some__opt__false__trivial,axiom,
    eps_Opt_num(aTP_Lamm_cw) = none_num ).

% some_opt_false_trivial
tff(fact_1590_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X4: int,Y4: int] : V != some_P4184893108420464158nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_1591_not__Some__eq2,axiom,
    ! [V: option4955102875562834416et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Y4: produc3658429121746597890et_nat] : V != some_P994283844982041979et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X4),Y4))
    <=> ( V = none_P7891021755743879167et_nat ) ) ).

% not_Some_eq2
tff(fact_1592_not__Some__eq2,axiom,
    ! [V: option5899113203645146382et_nat] :
      ( ! [X4: fun_Pr6460944699146269935_nat_o,Y4: produc8799533983703829792et_nat] : V != some_P3006888012174695577et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X4),Y4))
    <=> ( V = none_P2758821087663475997et_nat ) ) ).

% not_Some_eq2
tff(fact_1593_not__Some__eq2,axiom,
    ! [V: option4065278094766928714it_nat] :
      ( ! [X4: b,Y4: produc6653097349344004940it_nat] : V != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X4),Y4))
    <=> ( V = none_P6779099274072355161it_nat ) ) ).

% not_Some_eq2
tff(fact_1594_not__Some__eq2,axiom,
    ! [V: option3562590408128118217it_nat] :
      ( ! [X4: a,Y4: produc6653097349344004940it_nat] : V != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X4),Y4))
    <=> ( V = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq2
tff(fact_1595_not__Some__eq2,axiom,
    ! [V: option8956607266484857688it_nat] :
      ( ! [X4: product_unit,Y4: produc6653097349344004940it_nat] : V != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X4,Y4))
    <=> ( V = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq2
tff(fact_1596_less__eq__option__Some__None,axiom,
    ! [X: num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,X)),none_num) ).

% less_eq_option_Some_None
tff(fact_1597_less__option__None__Some__code,axiom,
    ! [X: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,X)) ).

% less_option_None_Some_code
tff(fact_1598_option_Osimps_I4_J,axiom,
    ! [F1: int,F2: fun_num_int] : aa_option_num_int(case_option_int_num(F1,F2),none_num) = F1 ).

% option.simps(4)
tff(fact_1599_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),none_num) = F1 ).

% option.simps(4)
tff(fact_1600_option_Osimps_I4_J,axiom,
    ! [F1: num,F2: fun_num_num] : aa_option_num_num(case_option_num_num(F1,F2),none_num) = F1 ).

% option.simps(4)
tff(fact_1601_execute__guard_I1_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,Fa: fun_he168995381318928090it_nat] :
      ( ~ aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_guard_b(Pa,Fa)),Ha) = none_P6779099274072355161it_nat ) ) ).

% execute_guard(1)
tff(fact_1602_execute__guard_I1_J,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,Fa: fun_he8889679731534893401it_nat] :
      ( ~ aa_hea7994211679605476167unit_o(Pa,Ha)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_guard_a(Pa,Fa)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_guard(1)
tff(fact_1603_execute__raise,axiom,
    ! [S2: list_char,X5: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_raise_b(S2)),X5) = none_P6779099274072355161it_nat ).

% execute_raise
tff(fact_1604_execute__raise,axiom,
    ! [S2: list_char,X5: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_raise_a(S2)),X5) = none_P2651198173097904984it_nat ).

% execute_raise
tff(fact_1605_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,F1: int,F2: fun_num_int,Option: option_num] : aa_int_int(Ha,aa_option_num_int(case_option_int_num(F1,F2),Option)) = aa_option_num_int(case_option_int_num(aa_int_int(Ha,F1),aa_fun329774391934724804um_int(aTP_Lamm_cx(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1606_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_option_num,F1: int,F2: fun_num_int,Option: option_num] : aa_int_option_num(Ha,aa_option_num_int(case_option_int_num(F1,F2),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_int_option_num(Ha,F1),aa_fun3858197967482829954on_num(aTP_Lamm_cy(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1607_option_Ocase__distrib,axiom,
    ! [Ha: fun_int_num,F1: int,F2: fun_num_int,Option: option_num] : aa_int_num(Ha,aa_option_num_int(case_option_int_num(F1,F2),Option)) = aa_option_num_num(case_option_num_num(aa_int_num(Ha,F1),aa_fun5215869348392759218um_num(aTP_Lamm_cz(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1608_option_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_int,F1: option_num,F2: fun_num_option_num,Option: option_num] : aa_option_num_int(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option)) = aa_option_num_int(case_option_int_num(aa_option_num_int(Ha,F1),aa_fun2256179333612926594um_int(aTP_Lamm_da(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1609_option_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F2: fun_num_option_num,Option: option_num] : aa_opt5369159329878751634on_num(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun3636977833914307392on_num(aTP_Lamm_db(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1610_option_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_num,F1: option_num,F2: fun_num_option_num,Option: option_num] : aa_option_num_num(Ha,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option)) = aa_option_num_num(case_option_num_num(aa_option_num_num(Ha,F1),aa_fun7142274290070961008um_num(aTP_Lamm_dc(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1611_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_int,F1: num,F2: fun_num_num,Option: option_num] : aa_num_int(Ha,aa_option_num_num(case_option_num_num(F1,F2),Option)) = aa_option_num_int(case_option_int_num(aa_num_int(Ha,F1),aa_fun3162900327506486194um_int(aTP_Lamm_dd(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1612_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_option_num,F1: num,F2: fun_num_num,Option: option_num] : aa_num_option_num(Ha,aa_option_num_num(case_option_num_num(F1,F2),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(Ha,F1),aa_fun612943910911948912on_num(aTP_Lamm_de(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1613_option_Ocase__distrib,axiom,
    ! [Ha: fun_num_num,F1: num,F2: fun_num_num,Option: option_num] : aa_num_num(Ha,aa_option_num_num(case_option_num_num(F1,F2),Option)) = aa_option_num_num(case_option_num_num(aa_num_num(Ha,F1),aa_fun8048995283964520608um_num(aTP_Lamm_df(Ha),F2)),Option) ).

% option.case_distrib
tff(fact_1614_option_Odistinct_I1_J,axiom,
    ! [X2: produc7388388658123137530it_nat] : none_P6779099274072355161it_nat != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2) ).

% option.distinct(1)
tff(fact_1615_option_Odistinct_I1_J,axiom,
    ! [X2: produc3260487557148687353it_nat] : none_P2651198173097904984it_nat != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2) ).

% option.distinct(1)
tff(fact_1616_option_Odistinct_I1_J,axiom,
    ! [X2: produc8664842809031399944it_nat] : none_P9117596204409417319it_nat != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2) ).

% option.distinct(1)
tff(fact_1617_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != aa_num_option_num(some_num,X2) ).

% option.distinct(1)
tff(fact_1618_option_OdiscI,axiom,
    ! [Option: option4065278094766928714it_nat,X2: produc7388388658123137530it_nat] :
      ( ( Option = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2) )
     => ( Option != none_P6779099274072355161it_nat ) ) ).

% option.discI
tff(fact_1619_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_1620_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_1621_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = aa_num_option_num(some_num,X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_1622_option_Oexhaust,axiom,
    ! [Y: option4065278094766928714it_nat] :
      ( ( Y != none_P6779099274072355161it_nat )
     => ~ ! [X22: produc7388388658123137530it_nat] : Y != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X22) ) ).

% option.exhaust
tff(fact_1623_option_Oexhaust,axiom,
    ! [Y: option3562590408128118217it_nat] :
      ( ( Y != none_P2651198173097904984it_nat )
     => ~ ! [X22: produc3260487557148687353it_nat] : Y != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X22) ) ).

% option.exhaust
tff(fact_1624_option_Oexhaust,axiom,
    ! [Y: option8956607266484857688it_nat] :
      ( ( Y != none_P9117596204409417319it_nat )
     => ~ ! [X22: produc8664842809031399944it_nat] : Y != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X22) ) ).

% option.exhaust
tff(fact_1625_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != aa_num_option_num(some_num,X22) ) ).

% option.exhaust
tff(fact_1626_split__option__ex,axiom,
    ! [Pa: fun_op3313016124244812839_nat_o] :
      ( ? [X_12: option4065278094766928714it_nat] : aa_opt3243322551152461968_nat_o(Pa,X_12)
    <=> ( aa_opt3243322551152461968_nat_o(Pa,none_P6779099274072355161it_nat)
        | ? [X4: produc7388388658123137530it_nat] : aa_opt3243322551152461968_nat_o(Pa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X4)) ) ) ).

% split_option_ex
tff(fact_1627_split__option__ex,axiom,
    ! [Pa: fun_op9133723461703387112_nat_o] :
      ( ? [X_12: option3562590408128118217it_nat] : aa_opt9064029888611036241_nat_o(Pa,X_12)
    <=> ( aa_opt9064029888611036241_nat_o(Pa,none_P2651198173097904984it_nat)
        | ? [X4: produc3260487557148687353it_nat] : aa_opt9064029888611036241_nat_o(Pa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X4)) ) ) ).

% split_option_ex
tff(fact_1628_split__option__ex,axiom,
    ! [Pa: fun_op479301348043730329_nat_o] :
      ( ? [X_12: option8956607266484857688it_nat] : aa_opt8360315729766332930_nat_o(Pa,X_12)
    <=> ( aa_opt8360315729766332930_nat_o(Pa,none_P9117596204409417319it_nat)
        | ? [X4: produc8664842809031399944it_nat] : aa_opt8360315729766332930_nat_o(Pa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X4)) ) ) ).

% split_option_ex
tff(fact_1629_split__option__ex,axiom,
    ! [Pa: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(Pa,X_12)
    <=> ( aa_option_num_o(Pa,none_num)
        | ? [X4: num] : aa_option_num_o(Pa,aa_num_option_num(some_num,X4)) ) ) ).

% split_option_ex
tff(fact_1630_split__option__all,axiom,
    ! [Pa: fun_op3313016124244812839_nat_o] :
      ( ! [X_12: option4065278094766928714it_nat] : aa_opt3243322551152461968_nat_o(Pa,X_12)
    <=> ( aa_opt3243322551152461968_nat_o(Pa,none_P6779099274072355161it_nat)
        & ! [X4: produc7388388658123137530it_nat] : aa_opt3243322551152461968_nat_o(Pa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X4)) ) ) ).

% split_option_all
tff(fact_1631_split__option__all,axiom,
    ! [Pa: fun_op9133723461703387112_nat_o] :
      ( ! [X_12: option3562590408128118217it_nat] : aa_opt9064029888611036241_nat_o(Pa,X_12)
    <=> ( aa_opt9064029888611036241_nat_o(Pa,none_P2651198173097904984it_nat)
        & ! [X4: produc3260487557148687353it_nat] : aa_opt9064029888611036241_nat_o(Pa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X4)) ) ) ).

% split_option_all
tff(fact_1632_split__option__all,axiom,
    ! [Pa: fun_op479301348043730329_nat_o] :
      ( ! [X_12: option8956607266484857688it_nat] : aa_opt8360315729766332930_nat_o(Pa,X_12)
    <=> ( aa_opt8360315729766332930_nat_o(Pa,none_P9117596204409417319it_nat)
        & ! [X4: produc8664842809031399944it_nat] : aa_opt8360315729766332930_nat_o(Pa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X4)) ) ) ).

% split_option_all
tff(fact_1633_split__option__all,axiom,
    ! [Pa: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(Pa,X_12)
    <=> ( aa_option_num_o(Pa,none_num)
        & ! [X4: num] : aa_option_num_o(Pa,aa_num_option_num(some_num,X4)) ) ) ).

% split_option_all
tff(fact_1634_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( X = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) )
       => ( ! [A3: num,B3: num] :
              ( ( X = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_num_option_num(some_num,B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1635_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,Pa: fun_op8852741442951501078_num_o,Y: option_num] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => aa_option_num_o(aa_opt1876318183327259391_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt1876318183327259391_num_o(Pa,X),Y) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: num] :
              ( ( X = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,A3) )
             => ( ( Y = aa_num_option_num(some_num,B3) )
               => aa_option_num_o(aa_opt1876318183327259391_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt1876318183327259391_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1636_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,Pa: fun_op8016730479996091735_num_o,Y: option_num] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) )
       => ( ! [A3: produc3260487557148687353it_nat,B3: num] :
              ( ( X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A3) )
             => ( ( Y = aa_num_option_num(some_num,B3) )
               => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt1040307220371850048_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1637_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,Pa: fun_op6018817604785332104_num_o,Y: option_num] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) )
       => ( ! [A3: produc8664842809031399944it_nat,B3: num] :
              ( ( X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A3) )
             => ( ( Y = aa_num_option_num(some_num,B3) )
               => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) ) )
         => aa_option_num_o(aa_opt3964967581558639985_num_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1638_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op8645536343338973462_nat_o,Y: option4065278094766928714it_nat] :
      ( ( ( X = none_num )
       => aa_opt3243322551152461968_nat_o(aa_opt6936930315403406079_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P6779099274072355161it_nat )
         => aa_opt3243322551152461968_nat_o(aa_opt6936930315403406079_nat_o(Pa,X),Y) )
       => ( ! [A3: num,B3: produc7388388658123137530it_nat] :
              ( ( X = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,B3) )
               => aa_opt3243322551152461968_nat_o(aa_opt6936930315403406079_nat_o(Pa,X),Y) ) )
         => aa_opt3243322551152461968_nat_o(aa_opt6936930315403406079_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1639_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op7041624247167716439_nat_o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_num )
       => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) )
       => ( ! [A3: num,B3: produc3260487557148687353it_nat] :
              ( ( X = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B3) )
               => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt3534265616007204544_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1640_combine__options__cases,axiom,
    ! [X: option_num,Pa: fun_op298641349230941576_nat_o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_num )
       => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) )
       => ( ! [A3: num,B3: produc8664842809031399944it_nat] :
              ( ( X = aa_num_option_num(some_num,A3) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B3) )
               => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt6324340006354107505_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1641_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,Pa: fun_op1120011469960559414_nat_o,Y: option4065278094766928714it_nat] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => aa_opt3243322551152461968_nat_o(aa_opt6874744473778840863_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P6779099274072355161it_nat )
         => aa_opt3243322551152461968_nat_o(aa_opt6874744473778840863_nat_o(Pa,X),Y) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: produc7388388658123137530it_nat] :
              ( ( X = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,A3) )
             => ( ( Y = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,B3) )
               => aa_opt3243322551152461968_nat_o(aa_opt6874744473778840863_nat_o(Pa,X),Y) ) )
         => aa_opt3243322551152461968_nat_o(aa_opt6874744473778840863_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1642_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,Pa: fun_op8739471410644078199_nat_o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => aa_opt9064029888611036241_nat_o(aa_opt3472079774382639328_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => aa_opt9064029888611036241_nat_o(aa_opt3472079774382639328_nat_o(Pa,X),Y) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: produc3260487557148687353it_nat] :
              ( ( X = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,A3) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B3) )
               => aa_opt9064029888611036241_nat_o(aa_opt3472079774382639328_nat_o(Pa,X),Y) ) )
         => aa_opt9064029888611036241_nat_o(aa_opt3472079774382639328_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1643_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,Pa: fun_op2273118692420253608_nat_o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => aa_opt8360315729766332930_nat_o(aa_opt8936879595723511441_nat_o(Pa,X),Y) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => aa_opt8360315729766332930_nat_o(aa_opt8936879595723511441_nat_o(Pa,X),Y) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: produc8664842809031399944it_nat] :
              ( ( X = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,A3) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B3) )
               => aa_opt8360315729766332930_nat_o(aa_opt8936879595723511441_nat_o(Pa,X),Y) ) )
         => aa_opt8360315729766332930_nat_o(aa_opt8936879595723511441_nat_o(Pa,X),Y) ) ) ) ).

% combine_options_cases
tff(fact_1644_less__eq__option__None__is__None,axiom,
    ! [X: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,X),none_num)
     => ( X = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_1645_less__eq__option__None,axiom,
    ! [X: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),X) ).

% less_eq_option_None
tff(fact_1646_option_Osimps_I5_J,axiom,
    ! [F1: int,F2: fun_num_int,X2: num] : aa_option_num_int(case_option_int_num(F1,F2),aa_num_option_num(some_num,X2)) = aa_num_int(F2,X2) ).

% option.simps(5)
tff(fact_1647_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num,X2: num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),aa_num_option_num(some_num,X2)) = aa_num_option_num(F2,X2) ).

% option.simps(5)
tff(fact_1648_option_Osimps_I5_J,axiom,
    ! [F1: num,F2: fun_num_num,X2: num] : aa_option_num_num(case_option_num_num(F1,F2),aa_num_option_num(some_num,X2)) = aa_num_num(F2,X2) ).

% option.simps(5)
tff(fact_1649_the__default_Osimps_I2_J,axiom,
    ! [X: num] : aa_option_num_num(the_default_num(X),none_num) = X ).

% the_default.simps(2)
tff(fact_1650_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,X: produc7388388658123137530it_nat] :
      ( ( eps_Op5869321083069970014it_nat(Pa) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) )
     => aa_Pro1537530206668979808_nat_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_1651_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat] :
      ( ( eps_Op1741419982095519837it_nat(Pa) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) )
     => aa_Pro6426884464393220769_nat_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_1652_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat(Pa) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) )
     => aa_Pro4744234731172327378_nat_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_1653_Eps__Opt__eq__Some__implies,axiom,
    ! [Pa: fun_num_o,X: num] :
      ( ( eps_Opt_num(Pa) = aa_num_option_num(some_num,X) )
     => aa_num_o(Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_1654_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,X: produc7388388658123137530it_nat] :
      ( ! [X7: produc7388388658123137530it_nat] :
          ( aa_Pro1537530206668979808_nat_o(Pa,X)
         => ( aa_Pro1537530206668979808_nat_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Op5869321083069970014it_nat(Pa) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) )
      <=> aa_Pro1537530206668979808_nat_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_1655_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat] :
      ( ! [X7: produc3260487557148687353it_nat] :
          ( aa_Pro6426884464393220769_nat_o(Pa,X)
         => ( aa_Pro6426884464393220769_nat_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Op1741419982095519837it_nat(Pa) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) )
      <=> aa_Pro6426884464393220769_nat_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_1656_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,X: produc8664842809031399944it_nat] :
      ( ! [X7: produc8664842809031399944it_nat] :
          ( aa_Pro4744234731172327378_nat_o(Pa,X)
         => ( aa_Pro4744234731172327378_nat_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Op3393321821070424684it_nat(Pa) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) )
      <=> aa_Pro4744234731172327378_nat_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_1657_Eps__Opt__eq__Some,axiom,
    ! [Pa: fun_num_o,X: num] :
      ( ! [X7: num] :
          ( aa_num_o(Pa,X)
         => ( aa_num_o(Pa,X7)
           => ( X7 = X ) ) )
     => ( ( eps_Opt_num(Pa) = aa_num_option_num(some_num,X) )
      <=> aa_num_o(Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_1658_execute__bind_I2_J,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,Ga: fun_b_8426044978055006235Heap_b] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = none_P6779099274072355161it_nat )
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_b_b(Fa,Ga)),Ha) = none_P6779099274072355161it_nat ) ) ).

% execute_bind(2)
tff(fact_1659_execute__bind_I2_J,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit,Ga: fun_b_8355010938008660250Heap_a] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = none_P6779099274072355161it_nat )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_b_a(Fa,Ga)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_bind(2)
tff(fact_1660_execute__bind_I2_J,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Ga: fun_a_6164325696594227994Heap_b] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = none_P2651198173097904984it_nat )
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_a_b(Fa,Ga)),Ha) = none_P6779099274072355161it_nat ) ) ).

% execute_bind(2)
tff(fact_1661_execute__bind_I2_J,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit,Ga: fun_a_6093291656547882009Heap_a] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = none_P2651198173097904984it_nat )
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(Fa,Ga)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_bind(2)
tff(fact_1662_success__def,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_b(Fa),Ha)
    <=> ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) != none_P6779099274072355161it_nat ) ) ).

% success_def
tff(fact_1663_success__def,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_Time_success_a(Fa),Ha)
    <=> ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) != none_P2651198173097904984it_nat ) ) ).

% success_def
tff(fact_1664_successI,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) != none_P6779099274072355161it_nat )
     => aa_hea7994211679605476167unit_o(heap_Time_success_b(Fa),Ha) ) ).

% successI
tff(fact_1665_successI,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) != none_P2651198173097904984it_nat )
     => aa_hea7994211679605476167unit_o(heap_Time_success_a(Fa),Ha) ) ).

% successI
tff(fact_1666_less__option__None__is__Some,axiom,
    ! [X: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),X)
     => ? [Z2: num] : X = aa_num_option_num(some_num,Z2) ) ).

% less_option_None_is_Some
tff(fact_1667_less__option__None__Some,axiom,
    ! [X: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,X)) ).

% less_option_None_Some
tff(fact_1668_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)),R)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)),S2) )
     => aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(R),S2) ) ).

% subrelI
tff(fact_1669_subrelI,axiom,
    ! [R: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3),R)
         => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3),S2) )
     => ord_le3919382008627836342et_nat(R,S2) ) ).

% subrelI
tff(fact_1670_subrelI,axiom,
    ! [R: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3),R)
         => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3),S2) )
     => ord_le5189643068899110356et_nat(R,S2) ) ).

% subrelI
tff(fact_1671_subrelI,axiom,
    ! [R: set_Pr7600907837789447088it_nat,S2: set_Pr7600907837789447088it_nat] :
      ( ! [X3: b,Y3: produc6653097349344004940it_nat] :
          ( member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3),R)
         => member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3),S2) )
     => ord_le4980869671742669840it_nat(R,S2) ) ).

% subrelI
tff(fact_1672_subrelI,axiom,
    ! [R: set_Pr7098220151150636591it_nat,S2: set_Pr7098220151150636591it_nat] :
      ( ! [X3: a,Y3: produc6653097349344004940it_nat] :
          ( member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3),R)
         => member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3),S2) )
     => ord_le4478181985103859343it_nat(R,S2) ) ).

% subrelI
tff(fact_1673_Heap__cases,axiom,
    ! [Fa: heap_T5738788834812785303t_unit,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X3: product_unit,H3: produc6653097349344004940it_nat] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) != aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X3,H3))
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(Fa),Ha) = none_P9117596204409417319it_nat ) ) ).

% Heap_cases
tff(fact_1674_Heap__cases,axiom,
    ! [Fa: heap_Time_Heap_b,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X3: b,H3: produc6653097349344004940it_nat] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) != aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),H3))
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha) = none_P6779099274072355161it_nat ) ) ).

% Heap_cases
tff(fact_1675_Heap__cases,axiom,
    ! [Fa: heap_Time_Heap_a,Ha: heap_e7401611519738050253t_unit] :
      ( ! [X3: a,H3: produc6653097349344004940it_nat] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) != aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),H3))
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha) = none_P2651198173097904984it_nat ) ) ).

% Heap_cases
tff(fact_1676_timeFrame_Ocases,axiom,
    ! [X: produc2207270350733924475it_nat] :
      ( ! [N5: nat,R2: b,H: heap_e7401611519738050253t_unit,N4: nat] : X != produc4613957359643799859it_nat(N5,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H,N4))))
     => ~ ! [N5: nat] : X != produc4613957359643799859it_nat(N5,none_P6779099274072355161it_nat) ) ).

% timeFrame.cases
tff(fact_1677_timeFrame_Ocases,axiom,
    ! [X: produc4453839368661128058it_nat] :
      ( ! [N5: nat,R2: a,H: heap_e7401611519738050253t_unit,N4: nat] : X != produc4111269673004989362it_nat(N5,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H,N4))))
     => ~ ! [N5: nat] : X != produc4111269673004989362it_nat(N5,none_P2651198173097904984it_nat) ) ).

% timeFrame.cases
tff(fact_1678_timeFrame_Ocases,axiom,
    ! [X: produc3911288613690379145it_nat] :
      ( ! [N5: nat,R2: product_unit,H: heap_e7401611519738050253t_unit,N4: nat] : X != produc638857205735767105it_nat(N5,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H,N4))))
     => ~ ! [N5: nat] : X != produc638857205735767105it_nat(N5,none_P9117596204409417319it_nat) ) ).

% timeFrame.cases
tff(fact_1679_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] :
      ( ( heap_T7616092557645711336rame_b(X,Xa) = Y )
     => ( accp_P4016523247509365636it_nat(heap_T8132184524487034139_rel_b,produc4613957359643799859it_nat(X,Xa))
       => ( ! [R2: b,H: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H,N4))) )
             => ( ( Y = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) )
               => ~ accp_P4016523247509365636it_nat(heap_T8132184524487034139_rel_b,produc4613957359643799859it_nat(X,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,R2),produc584006145561248582it_nat(H,N4))))) ) )
         => ~ ( ( Xa = none_P6779099274072355161it_nat )
             => ( ( Y = none_P6779099274072355161it_nat )
               => ~ accp_P4016523247509365636it_nat(heap_T8132184524487034139_rel_b,produc4613957359643799859it_nat(X,none_P6779099274072355161it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_1680_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( heap_T7616092557645711335rame_a(X,Xa) = Y )
     => ( accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(X,Xa))
       => ( ! [R2: a,H: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H,N4))) )
             => ( ( Y = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) )
               => ~ accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(X,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,R2),produc584006145561248582it_nat(H,N4))))) ) )
         => ~ ( ( Xa = none_P2651198173097904984it_nat )
             => ( ( Y = none_P2651198173097904984it_nat )
               => ~ accp_P6263092265436569219it_nat(heap_T8132184524487034138_rel_a,produc4111269673004989362it_nat(X,none_P2651198173097904984it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_1681_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( heap_T3616969660504097270t_unit(X,Xa) = Y )
     => ( accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(X,Xa))
       => ( ! [R2: product_unit,H: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H,N4))) )
             => ( ( Y = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),N4)))) )
               => ~ accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(X,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(R2,produc584006145561248582it_nat(H,N4))))) ) )
         => ~ ( ( Xa = none_P9117596204409417319it_nat )
             => ( ( Y = none_P9117596204409417319it_nat )
               => ~ accp_P414730952086964626it_nat(heap_T996182799752388649t_unit,produc638857205735767105it_nat(X,none_P9117596204409417319it_nat)) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_1682_disjE__realizer2,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num,Ra: fun_int_o,Fa: int,Ga: fun_num_int] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( (Pa)
         => aa_int_o(Ra,Fa) )
       => ( ! [Q3: num] :
              ( aa_num_o(Qa,Q3)
             => aa_int_o(Ra,aa_num_int(Ga,Q3)) )
         => aa_int_o(Ra,aa_option_num_int(case_option_int_num(Fa,Ga),X)) ) ) ) ).

% disjE_realizer2
tff(fact_1683_disjE__realizer2,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num,Ra: fun_option_num_o,Fa: option_num,Ga: fun_num_option_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( (Pa)
         => aa_option_num_o(Ra,Fa) )
       => ( ! [Q3: num] :
              ( aa_num_o(Qa,Q3)
             => aa_option_num_o(Ra,aa_num_option_num(Ga,Q3)) )
         => aa_option_num_o(Ra,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(Fa,Ga),X)) ) ) ) ).

% disjE_realizer2
tff(fact_1684_disjE__realizer2,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num,Ra: fun_num_o,Fa: num,Ga: fun_num_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( (Pa)
         => aa_num_o(Ra,Fa) )
       => ( ! [Q3: num] :
              ( aa_num_o(Qa,Q3)
             => aa_num_o(Ra,aa_num_num(Ga,Q3)) )
         => aa_num_o(Ra,aa_option_num_num(case_option_num_num(Fa,Ga),X)) ) ) ) ).

% disjE_realizer2
tff(fact_1685_set__to__map__def,axiom,
    ! [S: set_Pr958786334691620121nt_int,K2: int] : aa_int_option_int(set_to_map_int_int(S),K2) = eps_Opt_int(aa_int_fun_int_o(aTP_Lamm_cl(S),K2)) ).

% set_to_map_def
tff(fact_1686_set__to__map__def,axiom,
    ! [S: set_Pr1312406793078187350et_nat,K2: fun_Pr6460944699146269935_nat_o] : aa_fun5824947343344787589et_nat(set_to2037224588991580346et_nat(S),K2) = aa_fun5824947343344787589et_nat(eps_Op2013419657081471078et_nat,aa_fun2638059832795814268_nat_o(aTP_Lamm_cm(S),K2)) ).

% set_to_map_def
tff(fact_1687_set__to__map__def,axiom,
    ! [S: set_Pr4040394263813967220et_nat,K2: fun_Pr6460944699146269935_nat_o] : aa_fun1716447855799474851et_nat(set_to3162555076294590936et_nat(S),K2) = eps_Op5936278548196508548et_nat(aa_fun912680268133416158_nat_o(aTP_Lamm_cn(S),K2)) ).

% set_to_map_def
tff(fact_1688_set__to__map__def,axiom,
    ! [S: set_Pr7600907837789447088it_nat,K2: b] : aa_b_o2526299221643969375it_nat(set_to6136816377707935252it_nat(S),K2) = eps_Op8301357815426737072it_nat(aa_b_f7634180679818401570_nat_o(aTP_Lamm_co(S),K2)) ).

% set_to_map_def
tff(fact_1689_set__to__map__def,axiom,
    ! [S: set_Pr7098220151150636591it_nat,K2: a] : aa_a_o5231022909227610846it_nat(set_to2008915276733485075it_nat(S),K2) = eps_Op8301357815426737072it_nat(aa_a_f337466037417904483_nat_o(aTP_Lamm_cp(S),K2)) ).

% set_to_map_def
tff(fact_1690_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Fa: fun_he168995381318928090it_nat] : heap_Time_guard_b(Pa,Fa) = heap_Time_Heap_b2(aa_fun261163217688289804it_nat(aTP_Lamm_dg(Pa),Fa)) ).

% guard_def
tff(fact_1691_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Fa: fun_he8889679731534893401it_nat] : heap_Time_guard_a(Pa,Fa) = heap_Time_Heap_a2(aa_fun3370411987892275404it_nat(aTP_Lamm_dh(Pa),Fa)) ).

% guard_def
tff(fact_1692_guard__def,axiom,
    ! [Pa: fun_he8413527710729349838unit_o,Fa: fun_he4944555133437423336it_nat] : heap_T8440541562793052209t_unit(Pa,Fa) = heap_T6183433275982383450t_unit(aa_fun2986507860694160780it_nat(aTP_Lamm_di(Pa),Fa)) ).

% guard_def
tff(fact_1693_Heap_Oinject,axiom,
    ! [X: fun_he5234484606919320760it_nat,Ya: fun_he5234484606919320760it_nat] :
      ( ( heap_T6183433275982383450t_unit(X) = heap_T6183433275982383450t_unit(Ya) )
    <=> ( X = Ya ) ) ).

% Heap.inject
tff(fact_1694_Heap__execute,axiom,
    ! [Fa: heap_Time_Heap_b] : heap_Time_Heap_b2(heap_Time_execute_b(Fa)) = Fa ).

% Heap_execute
tff(fact_1695_Heap__execute,axiom,
    ! [Fa: heap_Time_Heap_a] : heap_Time_Heap_a2(heap_Time_execute_a(Fa)) = Fa ).

% Heap_execute
tff(fact_1696_Heap__execute,axiom,
    ! [Fa: heap_T5738788834812785303t_unit] : heap_T6183433275982383450t_unit(heap_T875086893843062177t_unit(Fa)) = Fa ).

% Heap_execute
tff(fact_1697_Heap_Oexhaust,axiom,
    ! [Y: heap_T5738788834812785303t_unit] :
      ~ ! [X3: fun_he5234484606919320760it_nat] : Y != heap_T6183433275982383450t_unit(X3) ).

% Heap.exhaust
tff(fact_1698_execute_Osimps,axiom,
    ! [Fa: fun_he8897818899970768042it_nat] : heap_Time_execute_b(heap_Time_Heap_b2(Fa)) = Fa ).

% execute.simps
tff(fact_1699_execute_Osimps,axiom,
    ! [Fa: fun_he1921015881043195817it_nat] : heap_Time_execute_a(heap_Time_Heap_a2(Fa)) = Fa ).

% execute.simps
tff(fact_1700_execute_Osimps,axiom,
    ! [Fa: fun_he5234484606919320760it_nat] : heap_T875086893843062177t_unit(heap_T6183433275982383450t_unit(Fa)) = Fa ).

% execute.simps
tff(fact_1701_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_dj),Option) ) ).

% option.disc_eq_case(2)
tff(fact_1702_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_cw),Option) ) ).

% option.disc_eq_case(1)
tff(fact_1703_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_Pr856651894144386551_nat_o,X: option4065278094766928714it_nat] :
      ( aa_opt3243322551152461968_nat_o(case_o7395119307266030407it_nat((Pa),Qa),X)
     => ( ( ( X = none_P6779099274072355161it_nat )
         => ~ (Pa) )
       => ~ ! [Y3: produc7388388658123137530it_nat] :
              ( ( X = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y3) )
             => ~ aa_Pro1537530206668979808_nat_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_1704_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_Pr5746006151868627512_nat_o,X: option3562590408128118217it_nat] :
      ( aa_opt9064029888611036241_nat_o(case_o3267218206291580230it_nat((Pa),Qa),X)
     => ( ( ( X = none_P2651198173097904984it_nat )
         => ~ (Pa) )
       => ~ ! [Y3: produc3260487557148687353it_nat] :
              ( ( X = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y3) )
             => ~ aa_Pro6426884464393220769_nat_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_1705_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_Pr8032424337255632745_nat_o,X: option8956607266484857688it_nat] :
      ( aa_opt8360315729766332930_nat_o(case_o2686588417244861013it_nat((Pa),Qa),X)
     => ( ( ( X = none_P9117596204409417319it_nat )
         => ~ (Pa) )
       => ~ ! [Y3: produc8664842809031399944it_nat] :
              ( ( X = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y3) )
             => ~ aa_Pro4744234731172327378_nat_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_1706_case__optionE,axiom,
    ! [Pa: $o,Qa: fun_num_o,X: option_num] :
      ( aa_option_num_o(case_option_o_num((Pa),Qa),X)
     => ( ( ( X = none_num )
         => ~ (Pa) )
       => ~ ! [Y3: num] :
              ( ( X = aa_num_option_num(some_num,Y3) )
             => ~ aa_num_o(Qa,Y3) ) ) ) ).

% case_optionE
tff(fact_1707_boolean__algebra__cancel_Osup2,axiom,
    ! [B4: set_Pr8551490117392284871at_nat,K2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( B4 = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K2),B) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1708_boolean__algebra__cancel_Osup2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,K2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( B4 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K2),B) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1709_boolean__algebra__cancel_Osup2,axiom,
    ! [B4: set_nat,K2: set_nat,B: set_nat,A: set_nat] :
      ( ( B4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K2),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1710_boolean__algebra__cancel_Osup2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1711_boolean__algebra__cancel_Osup2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(sup_sup_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B4) = aa_int_int(aa_int_fun_int_int(sup_sup_int,K2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_1712_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,K2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A4 = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K2),A) )
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,K2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1713_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,K2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A4 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K2),A) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,K2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1714_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: set_nat,K2: set_nat,A: set_nat,B: set_nat] :
      ( ( A4 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K2),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,K2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1715_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1716_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(sup_sup_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A4),B) = aa_int_int(aa_int_fun_int_int(sup_sup_int,K2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_1717_less__eq__option__def,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ord_le353528952715127954et_int(X,Y)
    <=> case_o223999843215110191et_int($true,aTP_Lamm_dk(Y),X) ) ).

% less_eq_option_def
tff(fact_1718_less__eq__option__def,axiom,
    ! [X: option_rat,Y: option_rat] :
      ( ord_le2406147912482264968on_rat(X,Y)
    <=> case_option_o_rat($true,aTP_Lamm_dl(Y),X) ) ).

% less_eq_option_def
tff(fact_1719_less__eq__option__def,axiom,
    ! [X: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,X),Y)
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_dm(Y)),X) ) ).

% less_eq_option_def
tff(fact_1720_less__eq__option__def,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ord_le5914376470875661696on_nat(X,Y)
    <=> case_option_o_nat($true,aTP_Lamm_dn(Y),X) ) ).

% less_eq_option_def
tff(fact_1721_less__eq__option__def,axiom,
    ! [X: option_int,Y: option_int] :
      ( ord_le1736525451366464988on_int(X,Y)
    <=> case_option_o_int($true,aTP_Lamm_do(Y),X) ) ).

% less_eq_option_def
tff(fact_1722_less__option__def,axiom,
    ! [X: option_assn,Y: option_assn] :
      ( ord_less_option_assn(X,Y)
    <=> case_option_o_assn($false,aTP_Lamm_dp(X),Y) ) ).

% less_option_def
tff(fact_1723_less__option__def,axiom,
    ! [X: option_rat,Y: option_rat] :
      ( ord_less_option_rat(X,Y)
    <=> case_option_o_rat($false,aTP_Lamm_dq(X),Y) ) ).

% less_option_def
tff(fact_1724_less__option__def,axiom,
    ! [X: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,X),Y)
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_ds(X)),Y) ) ).

% less_option_def
tff(fact_1725_less__option__def,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ord_less_option_nat(X,Y)
    <=> case_option_o_nat($false,aTP_Lamm_dt(X),Y) ) ).

% less_option_def
tff(fact_1726_less__option__def,axiom,
    ! [X: option_int,Y: option_int] :
      ( ord_less_option_int(X,Y)
    <=> case_option_o_int($false,aTP_Lamm_du(X),Y) ) ).

% less_option_def
tff(fact_1727_raise__def,axiom,
    ! [S2: list_char] : heap_T2927564422264180874t_unit(S2) = heap_T6183433275982383450t_unit(aTP_Lamm_dv) ).

% raise_def
tff(fact_1728_accp__subset,axiom,
    ! [R1: fun_Pr4799464687642144302_num_o,R22: fun_Pr4799464687642144302_num_o] :
      ( ord_le6421466325430216142_num_o(R1,R22)
     => ord_le3107050023622517222_num_o(accp_P3113834385874906142um_num(R22),accp_P3113834385874906142um_num(R1)) ) ).

% accp_subset
tff(fact_1729_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22)),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_1730_accp__subset,axiom,
    ! [R1: fun_Pr9018117161326028078_int_o,R22: fun_Pr9018117161326028078_int_o] :
      ( ord_le1416746762259324110_int_o(R1,R22)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,accp_P1096762738010456898nt_int(R22)),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_1731_accp__subset,axiom,
    ! [R1: fun_Pr3358449090565925198_nat_o,R22: fun_Pr3358449090565925198_nat_o] :
      ( ord_le6446978100474826990_nat_o(R1,R22)
     => ord_le3896530374514135251_nat_o(accp_P1394777952347926645et_nat(R22),accp_P1394777952347926645et_nat(R1)) ) ).

% accp_subset
tff(fact_1732_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(R1,R22)
     => ord_le7240017405068453697_nat_o(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_1733_accp__subset__induct,axiom,
    ! [D4: fun_Pr6331313853612757382_num_o,Ra: fun_Pr4799464687642144302_num_o,X: product_prod_num_num,Pa: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D4,accp_P3113834385874906142um_num(Ra))
     => ( ! [X3: product_prod_num_num,Z2: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D4,X3)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(Ra,Z2),X3)
             => aa_Pro7685007914727650815_num_o(D4,Z2) ) )
       => ( aa_Pro7685007914727650815_num_o(D4,X)
         => ( ! [X3: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D4,X3)
               => ( ! [Z5: product_prod_num_num] :
                      ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(Ra,Z5),X3)
                     => aa_Pro7685007914727650815_num_o(Pa,Z5) )
                 => aa_Pro7685007914727650815_num_o(Pa,X3) ) )
           => aa_Pro7685007914727650815_num_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_1734_accp__subset__induct,axiom,
    ! [D4: fun_Pr4796943543331071514_nat_o,Ra: fun_Pr2028404973800837934_nat_o,X: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(D4),accp_P4275260045618599050at_nat(Ra))
     => ( ! [X3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D4,X3)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,Z2),X3)
             => aa_Pro6150637604445964947_nat_o(D4,Z2) ) )
       => ( aa_Pro6150637604445964947_nat_o(D4,X)
         => ( ! [X3: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D4,X3)
               => ( ! [Z5: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,Z5),X3)
                     => aa_Pro6150637604445964947_nat_o(Pa,Z5) )
                 => aa_Pro6150637604445964947_nat_o(Pa,X3) ) )
           => aa_Pro6150637604445964947_nat_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_1735_accp__subset__induct,axiom,
    ! [D4: fun_Pr3238374608701176162_int_o,Ra: fun_Pr9018117161326028078_int_o,X: product_prod_int_int,Pa: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,D4),accp_P1096762738010456898nt_int(Ra))
     => ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D4,X3)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(Ra,Z2),X3)
             => aa_Pro4592068669816069595_int_o(D4,Z2) ) )
       => ( aa_Pro4592068669816069595_int_o(D4,X)
         => ( ! [X3: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D4,X3)
               => ( ! [Z5: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(Ra,Z5),X3)
                     => aa_Pro4592068669816069595_int_o(Pa,Z5) )
                 => aa_Pro4592068669816069595_int_o(Pa,X3) ) )
           => aa_Pro4592068669816069595_int_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_1736_accp__subset__induct,axiom,
    ! [D4: fun_Pr1227177680593602867_nat_o,Ra: fun_Pr3358449090565925198_nat_o,X: produc3112576180687729086et_nat,Pa: fun_Pr1227177680593602867_nat_o] :
      ( ord_le3896530374514135251_nat_o(D4,accp_P1394777952347926645et_nat(Ra))
     => ( ! [X3: produc3112576180687729086et_nat,Z2: produc3112576180687729086et_nat] :
            ( aa_Pro2671238023309943196_nat_o(D4,X3)
           => ( aa_Pro2671238023309943196_nat_o(aa_Pro4671852207856306999_nat_o(Ra,Z2),X3)
             => aa_Pro2671238023309943196_nat_o(D4,Z2) ) )
       => ( aa_Pro2671238023309943196_nat_o(D4,X)
         => ( ! [X3: produc3112576180687729086et_nat] :
                ( aa_Pro2671238023309943196_nat_o(D4,X3)
               => ( ! [Z5: produc3112576180687729086et_nat] :
                      ( aa_Pro2671238023309943196_nat_o(aa_Pro4671852207856306999_nat_o(Ra,Z5),X3)
                     => aa_Pro2671238023309943196_nat_o(Pa,Z5) )
                 => aa_Pro2671238023309943196_nat_o(Pa,X3) ) )
           => aa_Pro2671238023309943196_nat_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_1737_accp__subset__induct,axiom,
    ! [D4: fun_nat_o,Ra: fun_nat_fun_nat_o,X: nat,Pa: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D4,accp_nat(Ra))
     => ( ! [X3: nat,Z2: nat] :
            ( aa_nat_o(D4,X3)
           => ( aa_nat_o(aa_nat_fun_nat_o(Ra,Z2),X3)
             => aa_nat_o(D4,Z2) ) )
       => ( aa_nat_o(D4,X)
         => ( ! [X3: nat] :
                ( aa_nat_o(D4,X3)
               => ( ! [Z5: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(Ra,Z5),X3)
                     => aa_nat_o(Pa,Z5) )
                 => aa_nat_o(Pa,X3) ) )
           => aa_nat_o(Pa,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_1738_execute__assert_I2_J,axiom,
    ! [Pa: fun_b_o,X: b,Ha: heap_e7401611519738050253t_unit] :
      ( ~ aa_b_o(Pa,X)
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(heap_Time_assert_b(Pa),X)),Ha) = none_P6779099274072355161it_nat ) ) ).

% execute_assert(2)
tff(fact_1739_execute__assert_I2_J,axiom,
    ! [Pa: fun_a_o,X: a,Ha: heap_e7401611519738050253t_unit] :
      ( ~ aa_a_o(Pa,X)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(Pa),X)),Ha) = none_P2651198173097904984it_nat ) ) ).

% execute_assert(2)
tff(fact_1740_mod__h__bot__iff_I7_J,axiom,
    ! [Pa: assn,Qa: assn,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
        | aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(7)
tff(fact_1741_combine__options__def,axiom,
    ! [Fa: fun_Pr774283325640823912it_nat,X: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] : combin724284867043296059it_nat(Fa,X,Y) = case_o5901184372446974929it_nat(Y,aa_opt4401797230738824009it_nat(aTP_Lamm_dx(Fa),Y),X) ).

% combine_options_def
tff(fact_1742_combine__options__def,axiom,
    ! [Fa: fun_Pr7750636411365408485it_nat,X: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] : combin5819755802923621690it_nat(Fa,X,Y) = case_o6061097939050036047it_nat(Y,aa_opt996909609682481350it_nat(aTP_Lamm_dz(Fa),Y),X) ).

% combine_options_def
tff(fact_1743_combine__options__def,axiom,
    ! [Fa: fun_Pr1003294667014919570it_nat,X: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] : combin4318129983670048329it_nat(Fa,X,Y) = case_o2963978774867076333it_nat(Y,aa_opt7728438819757478003it_nat(aTP_Lamm_eb(Fa),Y),X) ).

% combine_options_def
tff(fact_1744_combine__options__def,axiom,
    ! [Fa: fun_num_fun_num_num,X: option_num,Y: option_num] : aa_opt5369159329878751634on_num(combine_options_num(Fa,X),Y) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(Y,aa_opt8899475237019873193on_num(aTP_Lamm_ed(Fa),Y)),X) ).

% combine_options_def
tff(fact_1745_reflcl__set__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Xa2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(sup_su52541223548345602_nat_o(aTP_Lamm_ee(R),fequal9188845342289841878at_nat),X5),Xa2)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X5,Xa2),sup_su718114333110466843at_nat(R,id_Pro2258643101195443293at_nat)) ) ).

% reflcl_set_eq
tff(fact_1746_reflcl__set__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(sup_su8083968640029086462_nat_o(aTP_Lamm_ef(R),fequal_nat),X5),Xa2)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,R),id_nat2)) ) ).

% reflcl_set_eq
tff(fact_1747_reflcl__set__eq,axiom,
    ! [R: set_Pr958786334691620121nt_int,X5: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(sup_su7393609867835711046_int_o(aTP_Lamm_cl(R),fequal_int),X5),Xa2)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa2)),sup_su6024340866399070445nt_int(R,id_int2)) ) ).

% reflcl_set_eq
tff(fact_1748_reflcl__set__eq,axiom,
    ! [R: set_Pr8551490117392284871at_nat,X5: multis2468970476368604999at_nat,Xa2: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(sup_su9034197199115871746_nat_o(aTP_Lamm_cq(R),fequal2154631582430647452at_nat),X5),Xa2)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(X5,Xa2),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,R),id_mul2649389997224486051at_nat)) ) ).

% reflcl_set_eq
tff(fact_1749_reflcl__set__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X5: set_Pr1261947904930325089at_nat,Xa2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aTP_Lamm_cr(R),fequal1430618250418080822at_nat),X5),Xa2)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Xa2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R),id_set1796276052702428605at_nat)) ) ).

% reflcl_set_eq
tff(fact_1750_in__measure,axiom,
    ! [X: int,Y: int,Fa: fun_int_nat] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),measure_int(Fa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Fa,X)),aa_int_nat(Fa,Y)) ) ).

% in_measure
tff(fact_1751_wait__bind__decon,axiom,
    ! [Pa: assn,M: heap_Time_Heap_a,Qa: fun_a_assn,N2: nat] :
      ( hoare_hoare_triple_a(Pa,M,Qa)
     => hoare_hoare_triple_a(Pa,heap_T757603679106148408unit_a(heap_Time_wait(N2),aTP_Lamm_eg(M)),Qa) ) ).

% wait_bind_decon
tff(fact_1752_wait__bind__decon,axiom,
    ! [Pa: assn,M: heap_Time_Heap_b,Qa: fun_b_assn,N2: nat] :
      ( hoare_hoare_triple_b(Pa,M,Qa)
     => hoare_hoare_triple_b(Pa,heap_T757603679106148409unit_b(heap_Time_wait(N2),aTP_Lamm_eh(M)),Qa) ) ).

% wait_bind_decon
tff(fact_1753_wait__bind__decon,axiom,
    ! [Pa: assn,M: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,N2: nat] :
      ( hoare_8945653483474564448t_unit(Pa,M,Qa)
     => hoare_8945653483474564448t_unit(Pa,heap_T2633723481742716231t_unit(heap_Time_wait(N2),aTP_Lamm_ei(M)),Qa) ) ).

% wait_bind_decon
tff(fact_1754_empty__Collect__eq,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(Pa) )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_1755_empty__Collect__eq,axiom,
    ! [Pa: fun_se7249947736802449410_int_o] :
      ( ( bot_bo1488462491386950373nt_int = collec5210948495886036740nt_int(Pa) )
    <=> ! [X4: set_Pr958786334691620121nt_int] : ~ aa_set6170482117830564987_int_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_1756_empty__Collect__eq,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(Pa) )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_1757_empty__Collect__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(Pa) )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_1758_empty__Collect__eq,axiom,
    ! [Pa: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(Pa) )
    <=> ! [X4: $o] : ~ aa_o_o(Pa,(X4)) ) ).

% empty_Collect_eq
tff(fact_1759_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(Pa) )
    <=> ! [X4: nat] : ~ aa_nat_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_1760_empty__Collect__eq,axiom,
    ! [Pa: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(Pa) )
    <=> ! [X4: int] : ~ aa_int_o(Pa,X4) ) ).

% empty_Collect_eq
tff(fact_1761_Collect__empty__eq,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ( collect_list_nat(Pa) = bot_bot_set_list_nat )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_1762_Collect__empty__eq,axiom,
    ! [Pa: fun_se7249947736802449410_int_o] :
      ( ( collec5210948495886036740nt_int(Pa) = bot_bo1488462491386950373nt_int )
    <=> ! [X4: set_Pr958786334691620121nt_int] : ~ aa_set6170482117830564987_int_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_1763_Collect__empty__eq,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ( collect_set_nat(Pa) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_1764_Collect__empty__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ( collec3392354462482085612at_nat(Pa) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_1765_Collect__empty__eq,axiom,
    ! [Pa: fun_o_o] :
      ( ( collect_o(Pa) = bot_bot_set_o )
    <=> ! [X4: $o] : ~ aa_o_o(Pa,(X4)) ) ).

% Collect_empty_eq
tff(fact_1766_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_o] :
      ( ( collect_nat(Pa) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_1767_Collect__empty__eq,axiom,
    ! [Pa: fun_int_o] :
      ( ( collect_int(Pa) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(Pa,X4) ) ).

% Collect_empty_eq
tff(fact_1768_all__not__in__conv,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ! [X4: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A4)
    <=> ( A4 = bot_bo1488462491386950373nt_int ) ) ).

% all_not_in_conv
tff(fact_1769_all__not__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ! [X4: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X4),A4)
    <=> ( A4 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_1770_all__not__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_1771_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ! [X4: $o] : ~ aa_set_o_o(member_o((X4)),A4)
    <=> ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_1772_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A4)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_1773_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A4)
    <=> ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_1774_empty__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),bot_bo1488462491386950373nt_int) ).

% empty_iff
tff(fact_1775_empty__iff,axiom,
    ! [C2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(C2),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_1776_empty__iff,axiom,
    ! [C2: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),bot_bo2099793752762293965at_nat) ).

% empty_iff
tff(fact_1777_empty__iff,axiom,
    ! [C2: $o] : ~ aa_set_o_o(member_o((C2)),bot_bot_set_o) ).

% empty_iff
tff(fact_1778_empty__iff,axiom,
    ! [C2: nat] : ~ aa_set_nat_o(member_nat(C2),bot_bot_set_nat) ).

% empty_iff
tff(fact_1779_empty__iff,axiom,
    ! [C2: int] : ~ aa_set_int_o(member_int(C2),bot_bot_set_int) ).

% empty_iff
tff(fact_1780_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),bot_bo2099793752762293965at_nat)
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_1781_subset__empty,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),bot_bot_set_o)
    <=> ( A4 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_1782_subset__empty,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),bot_bot_set_nat)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1783_subset__empty,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),bot_bot_set_int)
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1784_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A4) ).

% empty_subsetI
tff(fact_1785_empty__subsetI,axiom,
    ! [A4: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A4) ).

% empty_subsetI
tff(fact_1786_empty__subsetI,axiom,
    ! [A4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A4) ).

% empty_subsetI
tff(fact_1787_empty__subsetI,axiom,
    ! [A4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A4) ).

% empty_subsetI
tff(fact_1788_sup__bot__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),X) = X ).

% sup_bot_left
tff(fact_1789_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),X) = X ).

% sup_bot_left
tff(fact_1790_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),X) = X ).

% sup_bot_left
tff(fact_1791_sup__bot__left,axiom,
    ! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),X) = X ).

% sup_bot_left
tff(fact_1792_sup__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X) = X ).

% sup_bot_left
tff(fact_1793_sup__bot__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),X) = X ).

% sup_bot_left
tff(fact_1794_sup__bot__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),bot_bo8422036546324065075at_nat) = X ).

% sup_bot_right
tff(fact_1795_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).

% sup_bot_right
tff(fact_1796_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ).

% sup_bot_right
tff(fact_1797_sup__bot__right,axiom,
    ! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),bot_bot_set_o) = X ).

% sup_bot_right
tff(fact_1798_sup__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% sup_bot_right
tff(fact_1799_sup__bot__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).

% sup_bot_right
tff(fact_1800_bot__eq__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( bot_bo8422036546324065075at_nat = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) )
    <=> ( ( X = bot_bo8422036546324065075at_nat )
        & ( Y = bot_bo8422036546324065075at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1801_bot__eq__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1802_bot__eq__sup__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1803_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_1804_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_1805_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_1806_sup__eq__bot__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y) = bot_bo8422036546324065075at_nat )
    <=> ( ( X = bot_bo8422036546324065075at_nat )
        & ( Y = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1807_sup__eq__bot__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = bot_bo228742789529271731at_nat )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1808_sup__eq__bot__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1809_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = bot_bot_set_o )
    <=> ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_1810_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = bot_bot_set_nat )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_1811_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = bot_bot_set_int )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_1812_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) = bot_bo8422036546324065075at_nat )
    <=> ( ( A = bot_bo8422036546324065075at_nat )
        & ( B = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1813_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1814_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1815_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1816_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1817_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1818_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_1819_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_1820_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_1821_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),A) = A ).

% sup_bot.left_neutral
tff(fact_1822_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_1823_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A) = A ).

% sup_bot.left_neutral
tff(fact_1824_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( bot_bo8422036546324065075at_nat = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),B) )
    <=> ( ( A = bot_bo8422036546324065075at_nat )
        & ( B = bot_bo8422036546324065075at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1825_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1826_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1827_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B) )
    <=> ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1828_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1829_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1830_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A),bot_bo8422036546324065075at_nat) = A ).

% sup_bot.right_neutral
tff(fact_1831_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),bot_bo228742789529271731at_nat) = A ).

% sup_bot.right_neutral
tff(fact_1832_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),bot_bo2099793752762293965at_nat) = A ).

% sup_bot.right_neutral
tff(fact_1833_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),bot_bot_set_o) = A ).

% sup_bot.right_neutral
tff(fact_1834_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_1835_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_1836_Un__empty,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = bot_bo8422036546324065075at_nat )
    <=> ( ( A4 = bot_bo8422036546324065075at_nat )
        & ( B4 = bot_bo8422036546324065075at_nat ) ) ) ).

% Un_empty
tff(fact_1837_Un__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = bot_bo228742789529271731at_nat )
    <=> ( ( A4 = bot_bo228742789529271731at_nat )
        & ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
tff(fact_1838_Un__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> ( ( A4 = bot_bo2099793752762293965at_nat )
        & ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
tff(fact_1839_Un__empty,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4) = bot_bot_set_o )
    <=> ( ( A4 = bot_bot_set_o )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_1840_Un__empty,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = bot_bot_set_nat )
    <=> ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_1841_Un__empty,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = bot_bot_set_int )
    <=> ( ( A4 = bot_bot_set_int )
        & ( B4 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_1842_pair__in__Id__conv,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(A,B),id_Pro2258643101195443293at_nat)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_1843_pair__in__Id__conv,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( member6689249552917799696at_nat(produc4348348721325984599at_nat(A,B),id_mul2649389997224486051at_nat)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_1844_pair__in__Id__conv,axiom,
    ! [A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),id_nat2)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_1845_pair__in__Id__conv,axiom,
    ! [A: int,B: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),id_int2)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_1846_IdI,axiom,
    ! [A: product_prod_nat_nat] : member8206827879206165904at_nat(produc6161850002892822231at_nat(A,A),id_Pro2258643101195443293at_nat) ).

% IdI
tff(fact_1847_IdI,axiom,
    ! [A: multis2468970476368604999at_nat] : member6689249552917799696at_nat(produc4348348721325984599at_nat(A,A),id_mul2649389997224486051at_nat) ).

% IdI
tff(fact_1848_IdI,axiom,
    ! [A: nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),A)),id_nat2) ).

% IdI
tff(fact_1849_IdI,axiom,
    ! [A: int] : aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),A)),id_int2) ).

% IdI
tff(fact_1850_combine__options__simps_I3_J,axiom,
    ! [Fa: fun_Pr774283325640823912it_nat,A: produc7388388658123137530it_nat,B: produc7388388658123137530it_nat] : combin724284867043296059it_nat(Fa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,A),aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,B)) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro88785284265212210it_nat(aa_Pro3306356554137902633it_nat(Fa,A),B)) ).

% combine_options_simps(3)
tff(fact_1851_combine__options__simps_I3_J,axiom,
    ! [Fa: fun_Pr7750636411365408485it_nat,A: produc3260487557148687353it_nat,B: produc3260487557148687353it_nat] : combin5819755802923621690it_nat(Fa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,A),aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,B)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro4202338132779708464it_nat(aa_Pro2263720180353132454it_nat(Fa,A),B)) ).

% combine_options_simps(3)
tff(fact_1852_combine__options__simps_I3_J,axiom,
    ! [Fa: fun_Pr1003294667014919570it_nat,A: produc8664842809031399944it_nat,B: produc8664842809031399944it_nat] : combin4318129983670048329it_nat(Fa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,A),aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,B)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_Pro550487673543749710it_nat(aa_Pro4096753284154920275it_nat(Fa,A),B)) ).

% combine_options_simps(3)
tff(fact_1853_combine__options__simps_I3_J,axiom,
    ! [Fa: fun_num_fun_num_num,A: num,B: num] : aa_opt5369159329878751634on_num(combine_options_num(Fa,aa_num_option_num(some_num,A)),aa_num_option_num(some_num,B)) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Fa,A),B)) ).

% combine_options_simps(3)
tff(fact_1854_combine__options__simps_I1_J,axiom,
    ! [Fa: fun_num_fun_num_num,Y: option_num] : aa_opt5369159329878751634on_num(combine_options_num(Fa,none_num),Y) = Y ).

% combine_options_simps(1)
tff(fact_1855_combine__options__simps_I2_J,axiom,
    ! [Fa: fun_num_fun_num_num,X: option_num] : aa_opt5369159329878751634on_num(combine_options_num(Fa,X),none_num) = X ).

% combine_options_simps(2)
tff(fact_1856_bijective__Id,axiom,
    biject3503750217840948301at_nat(id_Pro2258643101195443293at_nat) ).

% bijective_Id
tff(fact_1857_bijective__Id,axiom,
    biject5714339216877808333at_nat(id_mul2649389997224486051at_nat) ).

% bijective_Id
tff(fact_1858_bijective__Id,axiom,
    aa_set3702741264012733747_nat_o(bijective_nat_nat,id_nat2) ).

% bijective_Id
tff(fact_1859_memb__imp__not__empty,axiom,
    ! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),S)
     => ( S != bot_bo1488462491386950373nt_int ) ) ).

% memb_imp_not_empty
tff(fact_1860_memb__imp__not__empty,axiom,
    ! [X: set_nat,S: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),S)
     => ( S != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_1861_memb__imp__not__empty,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% memb_imp_not_empty
tff(fact_1862_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
tff(fact_1863_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(X),S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_1864_memb__imp__not__empty,axiom,
    ! [X: int,S: set_int] :
      ( aa_set_int_o(member_int(X),S)
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_1865_set__notEmptyE,axiom,
    ! [S: set_se6260736226359567993nt_int] :
      ( ( S != bot_bo1488462491386950373nt_int )
     => ~ ! [X3: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),S) ) ).

% set_notEmptyE
tff(fact_1866_set__notEmptyE,axiom,
    ! [S: set_set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ~ ! [X3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_1867_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_1868_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] : ~ aa_set_o_o(member_o((X3)),S) ) ).

% set_notEmptyE
tff(fact_1869_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_1870_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] : ~ aa_set_int_o(member_int(X3),S) ) ).

% set_notEmptyE
tff(fact_1871_ex__in__conv,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ? [X4: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A4)
    <=> ( A4 != bot_bo1488462491386950373nt_int ) ) ).

% ex_in_conv
tff(fact_1872_ex__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ? [X4: set_nat] : aa_set_set_nat_o(member_set_nat(X4),A4)
    <=> ( A4 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1873_ex__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ? [X4: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
    <=> ( A4 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_1874_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ? [X4: $o] : aa_set_o_o(member_o((X4)),A4)
    <=> ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_1875_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A4)
    <=> ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1876_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A4)
    <=> ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1877_equals0I,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ! [Y3: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y3),A4)
     => ( A4 = bot_bo1488462491386950373nt_int ) ) ).

% equals0I
tff(fact_1878_equals0I,axiom,
    ! [A4: set_set_nat] :
      ( ! [Y3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y3),A4)
     => ( A4 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1879_equals0I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A4)
     => ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_1880_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y3: $o] : ~ aa_set_o_o(member_o((Y3)),A4)
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_1881_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A4)
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1882_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A4)
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1883_equals0D,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( A4 = bot_bo1488462491386950373nt_int )
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4) ) ).

% equals0D
tff(fact_1884_equals0D,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( A4 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A4) ) ).

% equals0D
tff(fact_1885_equals0D,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4) ) ).

% equals0D
tff(fact_1886_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A4) ) ).

% equals0D
tff(fact_1887_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A4) ) ).

% equals0D
tff(fact_1888_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A4) ) ).

% equals0D
tff(fact_1889_emptyE,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),bot_bo1488462491386950373nt_int) ).

% emptyE
tff(fact_1890_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_1891_emptyE,axiom,
    ! [A: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),bot_bo2099793752762293965at_nat) ).

% emptyE
tff(fact_1892_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_1893_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_1894_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_1895_Set_Oempty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_ej) ).

% Set.empty_def
tff(fact_1896_Set_Oempty__def,axiom,
    bot_bo1488462491386950373nt_int = collec5210948495886036740nt_int(aTP_Lamm_ek) ).

% Set.empty_def
tff(fact_1897_Set_Oempty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_el) ).

% Set.empty_def
tff(fact_1898_Set_Oempty__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(aTP_Lamm_em) ).

% Set.empty_def
tff(fact_1899_Set_Oempty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_en) ).

% Set.empty_def
tff(fact_1900_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_eo) ).

% Set.empty_def
tff(fact_1901_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_ep) ).

% Set.empty_def
tff(fact_1902_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A) ).

% bot.extremum
tff(fact_1903_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_1904_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_1905_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A) ).

% bot.extremum
tff(fact_1906_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_1907_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_1908_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_1909_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1910_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1911_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1912_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1913_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_1914_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1915_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1916_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1917_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),bot_bo2099793752762293965at_nat) ).

% bot.extremum_strict
tff(fact_1918_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_1919_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1920_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1921_bot_Oextremum__strict,axiom,
    ! [A: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),bot_bot_assn) ).

% bot.extremum_strict
tff(fact_1922_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1923_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,bot_bo2099793752762293965at_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1924_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_1925_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1926_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_1927_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,bot_bot_assn),A) ) ).

% bot.not_eq_extremum
tff(fact_1928_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1929_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),bot_bo8422036546324065075at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1930_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1931_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1932_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),bot_bot_set_o) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1933_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1934_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1935_IdE,axiom,
    ! [P2: produc859450856879609959at_nat] :
      ( member8206827879206165904at_nat(P2,id_Pro2258643101195443293at_nat)
     => ~ ! [X3: product_prod_nat_nat] : P2 != produc6161850002892822231at_nat(X3,X3) ) ).

% IdE
tff(fact_1936_IdE,axiom,
    ! [P2: produc4166570645942440679at_nat] :
      ( member6689249552917799696at_nat(P2,id_mul2649389997224486051at_nat)
     => ~ ! [X3: multis2468970476368604999at_nat] : P2 != produc4348348721325984599at_nat(X3,X3) ) ).

% IdE
tff(fact_1937_IdE,axiom,
    ! [P2: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(P2),id_nat2)
     => ~ ! [X3: nat] : P2 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3) ) ).

% IdE
tff(fact_1938_IdE,axiom,
    ! [P2: product_prod_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(P2),id_int2)
     => ~ ! [X3: int] : P2 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),X3) ) ).

% IdE
tff(fact_1939_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(A,B),id_Pro2258643101195443293at_nat)
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_1940_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( member6689249552917799696at_nat(produc4348348721325984599at_nat(A,B),id_mul2649389997224486051at_nat)
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_1941_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),id_nat2)
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_1942_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: int,B: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),id_int2)
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_1943_subset__emptyI,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] : ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
     => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),bot_bo1488462491386950373nt_int) ) ).

% subset_emptyI
tff(fact_1944_subset__emptyI,axiom,
    ! [A4: set_set_nat] :
      ( ! [X3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X3),A4)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_1945_subset__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_1946_subset__emptyI,axiom,
    ! [A4: set_o] :
      ( ! [X3: $o] : ~ aa_set_o_o(member_o((X3)),A4)
     => aa_set_o_o(ord_less_eq_set_o(A4),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_1947_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),A4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1948_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X3: int] : ~ aa_set_int_o(member_int(X3),A4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1949_Un__empty__left,axiom,
    ! [B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1950_Un__empty__left,axiom,
    ! [B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1951_Un__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1952_Un__empty__left,axiom,
    ! [B4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),B4) = B4 ).

% Un_empty_left
tff(fact_1953_Un__empty__left,axiom,
    ! [B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B4) = B4 ).

% Un_empty_left
tff(fact_1954_Un__empty__left,axiom,
    ! [B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B4) = B4 ).

% Un_empty_left
tff(fact_1955_Un__empty__right,axiom,
    ! [A4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),bot_bo8422036546324065075at_nat) = A4 ).

% Un_empty_right
tff(fact_1956_Un__empty__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),bot_bo228742789529271731at_nat) = A4 ).

% Un_empty_right
tff(fact_1957_Un__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),bot_bo2099793752762293965at_nat) = A4 ).

% Un_empty_right
tff(fact_1958_Un__empty__right,axiom,
    ! [A4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),bot_bot_set_o) = A4 ).

% Un_empty_right
tff(fact_1959_Un__empty__right,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),bot_bot_set_nat) = A4 ).

% Un_empty_right
tff(fact_1960_Un__empty__right,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),bot_bot_set_int) = A4 ).

% Un_empty_right
tff(fact_1961_not__psubset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),bot_bo2099793752762293965at_nat) ).

% not_psubset_empty
tff(fact_1962_not__psubset__empty,axiom,
    ! [A4: set_o] : ~ aa_set_o_o(ord_less_set_o(A4),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1963_not__psubset__empty,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1964_not__psubset__empty,axiom,
    ! [A4: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1965_mod__h__bot__indep,axiom,
    ! [Pa: assn,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(H_a,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_1966_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila5207306265035627125at_nat(sup_su3035147773818789531at_nat,bot_bo8422036546324065075at_nat,aTP_Lamm_bl,aTP_Lamm_bm) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1967_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6534579987270727413at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat,aTP_Lamm_bn,aTP_Lamm_bo) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1968_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila4918476307565957903at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat,aTP_Lamm_eq,aTP_Lamm_er) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1969_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila2554085542299052326_set_o(sup_sup_set_o,bot_bot_set_o,aTP_Lamm_es,aTP_Lamm_et) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1970_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat(sup_sup_set_nat,bot_bot_set_nat,aTP_Lamm_bp,aTP_Lamm_bq) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1971_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila8603258263270017530r_assn(sup_sup_assn,bot_bot_assn,aTP_Lamm_br,aTP_Lamm_bs) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1972_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    semila6712789903965657268et_int(sup_sup_set_int,bot_bot_set_int,aTP_Lamm_bt,aTP_Lamm_bu) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_1973_execute__assert_I1_J,axiom,
    ! [Pa: fun_Product_unit_o,X: product_unit,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Product_unit_o(Pa,X)
     => ( aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T4208721593536448476t_unit(Pa),X)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(X,produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_1974_execute__assert_I1_J,axiom,
    ! [Pa: fun_b_o,X: b,Ha: heap_e7401611519738050253t_unit] :
      ( aa_b_o(Pa,X)
     => ( aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(heap_Time_assert_b(Pa),X)),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_1975_execute__assert_I1_J,axiom,
    ! [Pa: fun_a_o,X: a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_a_o(Pa,X)
     => ( aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_assert_a(Pa),X)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc584006145561248582it_nat(Ha,one_one_nat))) ) ) ).

% execute_assert(1)
tff(fact_1976_the__dflt__None__empty,axiom,
    dflt_N6592383573632408824at_nat(bot_bo2099793752762293965at_nat) = none_s625347054029921090at_nat ).

% the_dflt_None_empty
tff(fact_1977_the__dflt__None__empty,axiom,
    dflt_None_set_o(bot_bot_set_o) = none_set_o ).

% the_dflt_None_empty
tff(fact_1978_the__dflt__None__empty,axiom,
    dflt_None_set_nat(bot_bot_set_nat) = none_set_nat ).

% the_dflt_None_empty
tff(fact_1979_the__dflt__None__empty,axiom,
    dflt_None_set_int(bot_bot_set_int) = none_set_int ).

% the_dflt_None_empty
tff(fact_1980_the__dflt__None__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ( dflt_N6592383573632408824at_nat(S) = some_s147305329494351046at_nat(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_1981_the__dflt__None__nonempty,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ( dflt_None_set_o(S) = some_set_o(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_1982_the__dflt__None__nonempty,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ( dflt_None_set_nat(S) = some_set_nat(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_1983_the__dflt__None__nonempty,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ( dflt_None_set_int(S) = some_set_int(S) ) ) ).

% the_dflt_None_nonempty
tff(fact_1984_the__dflt__None__set,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : the_de3668382347305825784at_nat(bot_bo2099793752762293965at_nat,dflt_N6592383573632408824at_nat(X)) = X ).

% the_dflt_None_set
tff(fact_1985_the__dflt__None__set,axiom,
    ! [X: set_o] : the_default_set_o(bot_bot_set_o,dflt_None_set_o(X)) = X ).

% the_dflt_None_set
tff(fact_1986_the__dflt__None__set,axiom,
    ! [X: set_nat] : the_default_set_nat(bot_bot_set_nat,dflt_None_set_nat(X)) = X ).

% the_dflt_None_set
tff(fact_1987_the__dflt__None__set,axiom,
    ! [X: set_int] : the_default_set_int(bot_bot_set_int,dflt_None_set_int(X)) = X ).

% the_dflt_None_set
tff(fact_1988_bot_Oordering__top__axioms,axiom,
    aa_set3702741264012733747_nat_o(orderi2172309028950807442at_nat(aTP_Lamm_eq,aTP_Lamm_er),bot_bo2099793752762293965at_nat) ).

% bot.ordering_top_axioms
tff(fact_1989_bot_Oordering__top__axioms,axiom,
    aa_set_o_o(ordering_top_set_o(aTP_Lamm_es,aTP_Lamm_et),bot_bot_set_o) ).

% bot.ordering_top_axioms
tff(fact_1990_bot_Oordering__top__axioms,axiom,
    aa_set_nat_o(ordering_top_set_nat(aTP_Lamm_bp,aTP_Lamm_bq),bot_bot_set_nat) ).

% bot.ordering_top_axioms
tff(fact_1991_bot_Oordering__top__axioms,axiom,
    ordering_top_assn(aTP_Lamm_br,aTP_Lamm_bs,bot_bot_assn) ).

% bot.ordering_top_axioms
tff(fact_1992_bot_Oordering__top__axioms,axiom,
    aa_set_int_o(ordering_top_set_int(aTP_Lamm_bt,aTP_Lamm_bu),bot_bot_set_int) ).

% bot.ordering_top_axioms
tff(fact_1993_bot_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_bx,aTP_Lamm_by),bot_bot_nat) ).

% bot.ordering_top_axioms
tff(fact_1994_false__rule,axiom,
    ! [C2: heap_Time_Heap_a,Qa: fun_a_assn] : hoare_hoare_triple_a(bot_bot_assn,C2,Qa) ).

% false_rule
tff(fact_1995_false__rule,axiom,
    ! [C2: heap_Time_Heap_b,Qa: fun_b_assn] : hoare_hoare_triple_b(bot_bot_assn,C2,Qa) ).

% false_rule
tff(fact_1996_false__rule,axiom,
    ! [C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] : hoare_8945653483474564448t_unit(bot_bot_assn,C2,Qa) ).

% false_rule
tff(fact_1997_bijective__Empty,axiom,
    aa_set3702741264012733747_nat_o(bijective_nat_nat,bot_bo2099793752762293965at_nat) ).

% bijective_Empty
tff(fact_1998_set__to__map__empty,axiom,
    ! [X5: nat] : aa_nat_option_nat(set_to_map_nat_nat(bot_bo2099793752762293965at_nat),X5) = none_nat ).

% set_to_map_empty
tff(fact_1999_ordering__top_Oextremum,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => aa_nat_o(aa_nat_fun_nat_o(Less_eq,A),Top) ) ).

% ordering_top.extremum
tff(fact_2000_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ~ aa_nat_o(aa_nat_fun_nat_o(Less,Top),A) ) ).

% ordering_top.extremum_strict
tff(fact_2001_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ( aa_nat_o(aa_nat_fun_nat_o(Less_eq,Top),A)
      <=> ( A = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_2002_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ( ( A != Top )
      <=> aa_nat_o(aa_nat_fun_nat_o(Less,A),Top) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_2003_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: fun_nat_fun_nat_o,Less: fun_nat_fun_nat_o,Top: nat,A: nat] :
      ( aa_nat_o(ordering_top_nat(Less_eq,Less),Top)
     => ( aa_nat_o(aa_nat_fun_nat_o(Less_eq,Top),A)
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_2004_bot__empty__eq2,axiom,
    ! [X5: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X5),Xa2)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa2)),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_2005_bot__empty__eq2,axiom,
    ! [X5: fun_Pr6460944699146269935_nat_o,Xa2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(bot_bo3509460380234399255_nat_o,X5),Xa2)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X5),Xa2),bot_bo1493351289776401898et_nat) ) ).

% bot_empty_eq2
tff(fact_2006_bot__empty__eq2,axiom,
    ! [X5: fun_Pr6460944699146269935_nat_o,Xa2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(bot_bo5031374944739956857_nat_o,X5),Xa2)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X5),Xa2),bot_bo6121529163330120712et_nat) ) ).

% bot_empty_eq2
tff(fact_2007_bot__empty__eq2,axiom,
    ! [X5: b,Xa2: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(bot_bo364399882760469181_nat_o,X5),Xa2)
    <=> member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X5),Xa2),bot_bo8983147464310796868it_nat) ) ).

% bot_empty_eq2
tff(fact_2008_bot__empty__eq2,axiom,
    ! [X5: a,Xa2: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(bot_bo2291057277214747902_nat_o,X5),Xa2)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X5),Xa2),bot_bo8480459777671986371it_nat) ) ).

% bot_empty_eq2
tff(fact_2009_bot__empty__eq2,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X5),Xa2)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_2010_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_2011_bot__set__def,axiom,
    bot_bo1488462491386950373nt_int = collec5210948495886036740nt_int(bot_bo4864556513171353750_int_o) ).

% bot_set_def
tff(fact_2012_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_2013_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo5074786513356115630_nat_o) ).

% bot_set_def
tff(fact_2014_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_2015_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_2016_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_2017_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
tff(fact_2018_mod__false,axiom,
    ! [Ha: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(bot_bot_assn),Ha) ).

% mod_false
tff(fact_2019_less__by__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ).

% less_by_empty
tff(fact_2020_raise__rule,axiom,
    ! [S2: list_char,Qa: fun_a_assn] : hoare_hoare_triple_a(bot_bot_assn,heap_Time_raise_a(S2),Qa) ).

% raise_rule
tff(fact_2021_raise__rule,axiom,
    ! [S2: list_char,Qa: fun_b_assn] : hoare_hoare_triple_b(bot_bot_assn,heap_Time_raise_b(S2),Qa) ).

% raise_rule
tff(fact_2022_raise__rule,axiom,
    ! [S2: list_char,Qa: fun_Pr6626665806141890376t_assn] : hoare_8945653483474564448t_unit(bot_bot_assn,heap_T2927564422264180874t_unit(S2),Qa) ).

% raise_rule
tff(fact_2023_raise__iff,axiom,
    ! [Pa: assn,S2: list_char,Qa: fun_a_assn] :
      ( hoare_hoare_triple_a(Pa,heap_Time_raise_a(S2),Qa)
    <=> ( Pa = bot_bot_assn ) ) ).

% raise_iff
tff(fact_2024_raise__iff,axiom,
    ! [Pa: assn,S2: list_char,Qa: fun_b_assn] :
      ( hoare_hoare_triple_b(Pa,heap_Time_raise_b(S2),Qa)
    <=> ( Pa = bot_bot_assn ) ) ).

% raise_iff
tff(fact_2025_raise__iff,axiom,
    ! [Pa: assn,S2: list_char,Qa: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(Pa,heap_T2927564422264180874t_unit(S2),Qa)
    <=> ( Pa = bot_bot_assn ) ) ).

% raise_iff
tff(fact_2026_set__to__map__empty__iff_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat] : aa_nat_option_nat(set_to_map_nat_nat(S),X4) = none_nat
    <=> ( S = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(1)
tff(fact_2027_set__to__map__empty__iff_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( aTP_Lamm_eu = set_to_map_nat_nat(S) )
    <=> ( S = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(2)
tff(fact_2028_dflt__None__set__def,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      dflt_N6592383573632408824at_nat(S) = $ite(S = bot_bo2099793752762293965at_nat,none_s625347054029921090at_nat,some_s147305329494351046at_nat(S)) ).

% dflt_None_set_def
tff(fact_2029_dflt__None__set__def,axiom,
    ! [S: set_o] :
      dflt_None_set_o(S) = $ite(S = bot_bot_set_o,none_set_o,some_set_o(S)) ).

% dflt_None_set_def
tff(fact_2030_dflt__None__set__def,axiom,
    ! [S: set_nat] :
      dflt_None_set_nat(S) = $ite(S = bot_bot_set_nat,none_set_nat,some_set_nat(S)) ).

% dflt_None_set_def
tff(fact_2031_dflt__None__set__def,axiom,
    ! [S: set_int] :
      dflt_None_set_int(S) = $ite(S = bot_bot_set_int,none_set_int,some_set_int(S)) ).

% dflt_None_set_def
tff(fact_2032_rel__of__empty,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] : rel_of_nat_nat(aTP_Lamm_eu,Pa) = bot_bo2099793752762293965at_nat ).

% rel_of_empty
tff(fact_2033_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_2034_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_2035_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_2036_tap__def,axiom,
    ! [Fa: fun_he6342020312717187381unit_b] : heap_Time_tap_b(Fa) = heap_Time_Heap_b2(aTP_Lamm_ev(Fa)) ).

% tap_def
tff(fact_2037_tap__def,axiom,
    ! [Fa: fun_he6342020308413958580unit_a] : heap_Time_tap_a(Fa) = heap_Time_Heap_a2(aTP_Lamm_ew(Fa)) ).

% tap_def
tff(fact_2038_tap__def,axiom,
    ! [Fa: fun_he5123621388056232515t_unit] : heap_T560649228465745139t_unit(Fa) = heap_T6183433275982383450t_unit(aTP_Lamm_ex(Fa)) ).

% tap_def
tff(fact_2039_execute__tap,axiom,
    ! [Fa: fun_he5123621388056232515t_unit,Ha: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(heap_T560649228465745139t_unit(Fa)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_hea4146273921685161788t_unit(Fa,Ha),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_2040_execute__tap,axiom,
    ! [Fa: fun_he6342020312717187381unit_b,Ha: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_tap_b(Fa)),Ha) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,aa_hea5100639427662539566unit_b(Fa,Ha)),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_2041_execute__tap,axiom,
    ! [Fa: fun_he6342020308413958580unit_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_tap_a(Fa)),Ha) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,aa_hea5100639427662539565unit_a(Fa,Ha)),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_tap
tff(fact_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_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_2050_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_2051_return__wp__rule,axiom,
    ! [Qa: fun_a_assn,X: a] : hoare_hoare_triple_a(aa_a_assn(Qa,X),aa_a_H8829438773920479562Heap_a(heap_Time_return_a,X),Qa) ).

% return_wp_rule
tff(fact_2052_return__wp__rule,axiom,
    ! [Qa: fun_b_assn,X: b] : hoare_hoare_triple_b(aa_b_assn(Qa,X),aa_b_H1867786022829710796Heap_b(heap_Time_return_b,X),Qa) ).

% return_wp_rule
tff(fact_2053_return__wp__rule,axiom,
    ! [Qa: fun_Pr6626665806141890376t_assn,X: product_unit] : hoare_8945653483474564448t_unit(aa_Product_unit_assn(Qa,X),aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X),Qa) ).

% return_wp_rule
tff(fact_2054_wait__rule,axiom,
    ! [N2: nat] : hoare_8945653483474564448t_unit(one_one_assn,heap_Time_wait(N2),aTP_Lamm_ey) ).

% wait_rule
tff(fact_2055_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_2056_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_2057_bounded__Max__nat,axiom,
    ! [Pa: fun_nat_o,X: nat,M6: nat] :
      ( aa_nat_o(Pa,X)
     => ( ! [X3: nat] :
            ( aa_nat_o(Pa,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M6) )
       => ~ ! [M5: nat] :
              ( aa_nat_o(Pa,M5)
             => ~ ! [X5: nat] :
                    ( aa_nat_o(Pa,X5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M5) ) ) ) ) ).

% bounded_Max_nat
tff(fact_2058_mod__emp__simp,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_2059_return__def,axiom,
    ! [X: b] : aa_b_H1867786022829710796Heap_b(heap_Time_return_b,X) = heap_Time_heap_b(aTP_Lamm_ez(X)) ).

% return_def
tff(fact_2060_return__def,axiom,
    ! [X: a] : aa_a_H8829438773920479562Heap_a(heap_Time_return_a,X) = heap_Time_heap_a(aTP_Lamm_fa(X)) ).

% return_def
tff(fact_2061_execute__lookup,axiom,
    ! [R: ref_Product_unit,Ha: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(ref_lo7930775386976318366t_unit(R)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(ref_get_Product_unit(Ha,R),produc584006145561248582it_nat(Ha,one_one_nat))) ).

% execute_lookup
tff(fact_2062_execute__return,axiom,
    ! [X: product_unit] : heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X)) = comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,aTP_Lamm_fb(X)) ).

% execute_return
tff(fact_2063_execute__return,axiom,
    ! [X: b] : heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(heap_Time_return_b,X)) = comp_P6401406225962756324t_unit(some_P2818173045054083285it_nat,aTP_Lamm_ez(X)) ).

% execute_return
tff(fact_2064_execute__return,axiom,
    ! [X: a] : heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_return_a,X)) = comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,aTP_Lamm_fa(X)) ).

% execute_return
tff(fact_2065_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_2066_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_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_map__to__set__empty,axiom,
    map_to_set_nat_nat(aTP_Lamm_eu) = bot_bo2099793752762293965at_nat ).

% map_to_set_empty
tff(fact_2074_type__copy__map__cong0,axiom,
    ! [M6: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger,X: code_integer,N7: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger,Fa: fun_Co1561183069634336733nteger] :
      ( ( aa_Cod1783974652533396728nteger(M6,aa_Cod1783974652533396728nteger(Ga,X)) = aa_Cod1783974652533396728nteger(N7,aa_Cod1783974652533396728nteger(Ha,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Fa,M6),Ga),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2075_type__copy__map__cong0,axiom,
    ! [M6: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger,X: code_integer,N7: fun_fu5689865433563736081nteger,Ha: fun_Co1981900195452921617nteger,Fa: fun_Co1561183069634336733nteger] :
      ( ( aa_Cod1783974652533396728nteger(M6,aa_Cod1783974652533396728nteger(Ga,X)) = aa_fun2937960011630702546nteger(N7,aa_Cod4161879620478718930nteger(Ha,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Fa,M6),Ga),X) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(comp_C7827331287018003580nteger(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2076_type__copy__map__cong0,axiom,
    ! [M6: fun_Co1981900195452921617nteger,Ga: fun_Co8499238673658658103nteger,X: code_integer,N7: fun_Co1981900195452921617nteger,Ha: fun_Co8499238673658658103nteger,Fa: fun_fu4800041728187187127nteger] :
      ( ( aa_Cod4161879620478718930nteger(M6,aa_Cod1783974652533396728nteger(Ga,X)) = aa_Cod4161879620478718930nteger(N7,aa_Cod1783974652533396728nteger(Ha,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(Fa,M6),Ga),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2077_type__copy__map__cong0,axiom,
    ! [M6: fun_Co1981900195452921617nteger,Ga: fun_Co8499238673658658103nteger,X: code_integer,N7: fun_fu1763652423599918571nteger,Ha: fun_Co1981900195452921617nteger,Fa: fun_fu4800041728187187127nteger] :
      ( ( aa_Cod4161879620478718930nteger(M6,aa_Cod1783974652533396728nteger(Ga,X)) = aa_fun4435975922079263404nteger(N7,aa_Cod4161879620478718930nteger(Ha,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(Fa,M6),Ga),X) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2078_type__copy__map__cong0,axiom,
    ! [M6: fun_fu5689865433563736081nteger,Ga: fun_Co1981900195452921617nteger,X: code_integer,N7: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger,Fa: fun_Co1561183069634336733nteger] :
      ( ( aa_fun2937960011630702546nteger(M6,aa_Cod4161879620478718930nteger(Ga,X)) = aa_Cod1783974652533396728nteger(N7,aa_Cod1783974652533396728nteger(Ha,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(comp_C7827331287018003580nteger(Fa,M6),Ga),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2079_type__copy__map__cong0,axiom,
    ! [M6: fun_fu1763652423599918571nteger,Ga: fun_Co1981900195452921617nteger,X: code_integer,N7: fun_Co1981900195452921617nteger,Ha: fun_Co8499238673658658103nteger,Fa: fun_fu4800041728187187127nteger] :
      ( ( aa_fun4435975922079263404nteger(M6,aa_Cod4161879620478718930nteger(Ga,X)) = aa_Cod4161879620478718930nteger(N7,aa_Cod1783974652533396728nteger(Ha,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(Fa,M6),Ga),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2080_type__copy__map__cong0,axiom,
    ! [M6: fun_int_int,Ga: fun_int_int,X: int,N7: fun_int_int,Ha: fun_int_int,Fa: fun_int_nat] :
      ( ( aa_int_int(M6,aa_int_int(Ga,X)) = aa_int_int(N7,aa_int_int(Ha,X)) )
     => ( aa_int_nat(comp_int_nat_int(comp_int_nat_int(Fa,M6),Ga),X) = aa_int_nat(comp_int_nat_int(comp_int_nat_int(Fa,N7),Ha),X) ) ) ).

% type_copy_map_cong0
tff(fact_2081_comp__cong__left,axiom,
    ! [X: fun_Co1561183069634336733nteger,Y: fun_Co1561183069634336733nteger,Fa: fun_Co8499238673658658103nteger] :
      ( ( X = Y )
     => ( comp_C7848664941760151970nteger(X,Fa) = comp_C7848664941760151970nteger(Y,Fa) ) ) ).

% comp_cong_left
tff(fact_2082_comp__cong__left,axiom,
    ! [X: fun_fu4800041728187187127nteger,Y: fun_fu4800041728187187127nteger,Fa: fun_Co1981900195452921617nteger] :
      ( ( X = Y )
     => ( comp_f6739083541385432444nteger(X,Fa) = comp_f6739083541385432444nteger(Y,Fa) ) ) ).

% comp_cong_left
tff(fact_2083_comp__cong__left,axiom,
    ! [X: fun_int_nat,Y: fun_int_nat,Fa: fun_int_int] :
      ( ( X = Y )
     => ( comp_int_nat_int(X,Fa) = comp_int_nat_int(Y,Fa) ) ) ).

% comp_cong_left
tff(fact_2084_comp__cong__right,axiom,
    ! [X: fun_Co8499238673658658103nteger,Y: fun_Co8499238673658658103nteger,Fa: fun_Co1561183069634336733nteger] :
      ( ( X = Y )
     => ( comp_C7848664941760151970nteger(Fa,X) = comp_C7848664941760151970nteger(Fa,Y) ) ) ).

% comp_cong_right
tff(fact_2085_comp__cong__right,axiom,
    ! [X: fun_Co1981900195452921617nteger,Y: fun_Co1981900195452921617nteger,Fa: fun_fu4800041728187187127nteger] :
      ( ( X = Y )
     => ( comp_f6739083541385432444nteger(Fa,X) = comp_f6739083541385432444nteger(Fa,Y) ) ) ).

% comp_cong_right
tff(fact_2086_comp__cong__right,axiom,
    ! [X: fun_int_int,Y: fun_int_int,Fa: fun_int_nat] :
      ( ( X = Y )
     => ( comp_int_nat_int(Fa,X) = comp_int_nat_int(Fa,Y) ) ) ).

% comp_cong_right
tff(fact_2087_fun__comp__eq__conv,axiom,
    ! [Fa: fun_Co1561183069634336733nteger,Ga: fun_Co8499238673658658103nteger,Fg: fun_Co1561183069634336733nteger] :
      ( ( comp_C7848664941760151970nteger(Fa,Ga) = Fg )
    <=> ! [X4: code_integer] : aa_Cod1978467581589928094nteger(Fa,aa_Cod1783974652533396728nteger(Ga,X4)) = aa_Cod1978467581589928094nteger(Fg,X4) ) ).

% fun_comp_eq_conv
tff(fact_2088_fun__comp__eq__conv,axiom,
    ! [Fa: fun_fu4800041728187187127nteger,Ga: fun_Co1981900195452921617nteger,Fg: fun_Co1561183069634336733nteger] :
      ( ( comp_f6739083541385432444nteger(Fa,Ga) = Fg )
    <=> ! [X4: code_integer] : aa_fun2509385170507305080nteger(Fa,aa_Cod4161879620478718930nteger(Ga,X4)) = aa_Cod1978467581589928094nteger(Fg,X4) ) ).

% fun_comp_eq_conv
tff(fact_2089_fun__comp__eq__conv,axiom,
    ! [Fa: fun_int_nat,Ga: fun_int_int,Fg: fun_int_nat] :
      ( ( comp_int_nat_int(Fa,Ga) = Fg )
    <=> ! [X4: int] : aa_int_nat(Fa,aa_int_int(Ga,X4)) = aa_int_nat(Fg,X4) ) ).

% fun_comp_eq_conv
tff(fact_2090_comp__cong,axiom,
    ! [Fa: fun_Co1561183069634336733nteger,Ga: fun_Co8499238673658658103nteger,X: code_integer,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger,X8: code_integer] :
      ( ( aa_Cod1978467581589928094nteger(Fa,aa_Cod1783974652533396728nteger(Ga,X)) = aa_Cod1978467581589928094nteger(F,aa_Cod1783974652533396728nteger(G,X8)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(Fa,Ga),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(F,G),X8) ) ) ).

% comp_cong
tff(fact_2091_comp__cong,axiom,
    ! [Fa: fun_Co1561183069634336733nteger,Ga: fun_Co8499238673658658103nteger,X: code_integer,F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,X8: code_integer] :
      ( ( aa_Cod1978467581589928094nteger(Fa,aa_Cod1783974652533396728nteger(Ga,X)) = aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X8)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(Fa,Ga),X) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),X8) ) ) ).

% comp_cong
tff(fact_2092_comp__cong,axiom,
    ! [Fa: fun_fu4800041728187187127nteger,Ga: fun_Co1981900195452921617nteger,X: code_integer,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger,X8: code_integer] :
      ( ( aa_fun2509385170507305080nteger(Fa,aa_Cod4161879620478718930nteger(Ga,X)) = aa_Cod1978467581589928094nteger(F,aa_Cod1783974652533396728nteger(G,X8)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(Fa,Ga),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(F,G),X8) ) ) ).

% comp_cong
tff(fact_2093_comp__cong,axiom,
    ! [Fa: fun_fu4800041728187187127nteger,Ga: fun_Co1981900195452921617nteger,X: code_integer,F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,X8: code_integer] :
      ( ( aa_fun2509385170507305080nteger(Fa,aa_Cod4161879620478718930nteger(Ga,X)) = aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X8)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(Fa,Ga),X) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),X8) ) ) ).

% comp_cong
tff(fact_2094_comp__cong,axiom,
    ! [Fa: fun_int_nat,Ga: fun_int_int,X: int,F: fun_int_nat,G: fun_int_int,X8: int] :
      ( ( aa_int_nat(Fa,aa_int_int(Ga,X)) = aa_int_nat(F,aa_int_int(G,X8)) )
     => ( aa_int_nat(comp_int_nat_int(Fa,Ga),X) = aa_int_nat(comp_int_nat_int(F,G),X8) ) ) ).

% comp_cong
tff(fact_2095_execute__heap,axiom,
    ! [Fa: fun_he4944555133437423336it_nat] : heap_T875086893843062177t_unit(heap_T6927113302350381334t_unit(Fa)) = comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,Fa) ).

% execute_heap
tff(fact_2096_execute__heap,axiom,
    ! [Fa: fun_he168995381318928090it_nat] : heap_Time_execute_b(heap_Time_heap_b(Fa)) = comp_P6401406225962756324t_unit(some_P2818173045054083285it_nat,Fa) ).

% execute_heap
tff(fact_2097_execute__heap,axiom,
    ! [Fa: fun_he8889679731534893401it_nat] : heap_Time_execute_a(heap_Time_heap_a(Fa)) = comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,Fa) ).

% execute_heap
tff(fact_2098_heap__def,axiom,
    ! [Fa: fun_he168995381318928090it_nat] : heap_Time_heap_b(Fa) = heap_Time_Heap_b2(comp_P6401406225962756324t_unit(some_P2818173045054083285it_nat,Fa)) ).

% heap_def
tff(fact_2099_heap__def,axiom,
    ! [Fa: fun_he8889679731534893401it_nat] : heap_Time_heap_a(Fa) = heap_Time_Heap_a2(comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,Fa)) ).

% heap_def
tff(fact_2100_heap__def,axiom,
    ! [Fa: fun_he4944555133437423336it_nat] : heap_T6927113302350381334t_unit(Fa) = heap_T6183433275982383450t_unit(comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,Fa)) ).

% heap_def
tff(fact_2101_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% is_num_normalize(1)
tff(fact_2102_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% is_num_normalize(1)
tff(fact_2103_map__to__set__empty__iff_I2_J,axiom,
    ! [M: fun_nat_option_nat] :
      ( ( bot_bo2099793752762293965at_nat = map_to_set_nat_nat(M) )
    <=> ! [X4: nat] : aa_nat_option_nat(M,X4) = none_nat ) ).

% map_to_set_empty_iff(2)
tff(fact_2104_map__to__set__empty__iff_I1_J,axiom,
    ! [M: fun_nat_option_nat] :
      ( ( map_to_set_nat_nat(M) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: nat] : aa_nat_option_nat(M,X4) = none_nat ) ).

% map_to_set_empty_iff(1)
tff(fact_2105_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [Fa: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Fa,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Fa,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(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(Fa,A),Acc))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_2106_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc))))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_2107_Set_Ois__empty__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_emp1662574758705540307at_nat,A4)
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
tff(fact_2108_Set_Ois__empty__def,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_empty_o,A4)
    <=> ( A4 = bot_bot_set_o ) ) ).

% Set.is_empty_def
tff(fact_2109_Set_Ois__empty__def,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_empty_nat,A4)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_2110_Set_Ois__empty__def,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_empty_int,A4)
    <=> ( A4 = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_2111_Heap__lub__empty,axiom,
    heap_T4436805183663264329t_unit(bot_bo8105976889876132193t_unit) = heap_T6183433275982383450t_unit(aTP_Lamm_dv) ).

% Heap_lub_empty
tff(fact_2112_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [Fa: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Fa,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(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(Fa,A),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2113_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2114_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_2115_pure__assn__eq__conv,axiom,
    ! [Pa: $o,Qa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = aa_o_assn(pure_assn,(Qa)) )
    <=> ( (Pa)
      <=> (Qa) ) ) ).

% pure_assn_eq_conv
tff(fact_2116_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        | (B) )) ).

% merge_pure_or
tff(fact_2117_pure__assn__eq__emp__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = one_one_assn )
    <=> (Pa) ) ).

% pure_assn_eq_emp_iff
tff(fact_2118_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_2119_pure__assn__eq__false__iff,axiom,
    ! [Pa: $o] :
      ( ( aa_o_assn(pure_assn,(Pa)) = bot_bot_assn )
    <=> ~ (Pa) ) ).

% pure_assn_eq_false_iff
tff(fact_2120_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_2121_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,Fa: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( hoare_hoare_triple_a(aa_o_assn(pure_assn,(B)),Fa,Qa)
    <=> ( (B)
       => hoare_hoare_triple_a(one_one_assn,Fa,Qa) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_2122_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,Fa: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( hoare_hoare_triple_b(aa_o_assn(pure_assn,(B)),Fa,Qa)
    <=> ( (B)
       => hoare_hoare_triple_b(one_one_assn,Fa,Qa) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_2123_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,Fa: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(aa_o_assn(pure_assn,(B)),Fa,Qa)
    <=> ( (B)
       => hoare_8945653483474564448t_unit(one_one_assn,Fa,Qa) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_2124_norm__pre__pure__rule2,axiom,
    ! [B: $o,Fa: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( ( (B)
       => hoare_hoare_triple_a(one_one_assn,Fa,Qa) )
     => hoare_hoare_triple_a(aa_o_assn(pure_assn,(B)),Fa,Qa) ) ).

% norm_pre_pure_rule2
tff(fact_2125_norm__pre__pure__rule2,axiom,
    ! [B: $o,Fa: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( ( (B)
       => hoare_hoare_triple_b(one_one_assn,Fa,Qa) )
     => hoare_hoare_triple_b(aa_o_assn(pure_assn,(B)),Fa,Qa) ) ).

% norm_pre_pure_rule2
tff(fact_2126_norm__pre__pure__rule2,axiom,
    ! [B: $o,Fa: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( ( (B)
       => hoare_8945653483474564448t_unit(one_one_assn,Fa,Qa) )
     => hoare_8945653483474564448t_unit(aa_o_assn(pure_assn,(B)),Fa,Qa) ) ).

% norm_pre_pure_rule2
tff(fact_2127_lookup__rule,axiom,
    ! [P2: ref_Product_unit,X: product_unit] : hoare_8945653483474564448t_unit(aa_Product_unit_assn(sngr_a5825115052027484668t_unit(P2),X),ref_lo7930775386976318366t_unit(P2),aa_Pro6018021088111102263t_assn(aTP_Lamm_fc(P2),X)) ).

% lookup_rule
tff(fact_2128_K__record__comp,axiom,
    ! [C2: fun_Pr5876161306168176259nteger,Fa: fun_Co8499238673658658103nteger,X5: code_integer] : aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(aTP_Lamm_fd(C2),Fa),X5) = C2 ).

% K_record_comp
tff(fact_2129_K__record__comp,axiom,
    ! [C2: fun_Pr5876161306168176259nteger,Fa: fun_Co1981900195452921617nteger,X5: code_integer] : aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(aTP_Lamm_fe(C2),Fa),X5) = C2 ).

% K_record_comp
tff(fact_2130_K__record__comp,axiom,
    ! [C2: nat,Fa: fun_int_int,X5: int] : aa_int_nat(comp_int_nat_int(aa_nat_fun_int_nat(aTP_Lamm_ff,C2),Fa),X5) = C2 ).

% K_record_comp
tff(fact_2131_execute__ureturn,axiom,
    ! [X: product_unit] : heap_T875086893843062177t_unit(aa_Pro1108561329362217960t_unit(heap_T4284346855313245393t_unit,X)) = comp_P3118722334806803912t_unit(some_P1914260805536162275it_nat,aTP_Lamm_fg(X)) ).

% execute_ureturn
tff(fact_2132_execute__ureturn,axiom,
    ! [X: b] : heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(heap_Time_ureturn_b,X)) = comp_P6401406225962756324t_unit(some_P2818173045054083285it_nat,aTP_Lamm_fh(X)) ).

% execute_ureturn
tff(fact_2133_execute__ureturn,axiom,
    ! [X: a] : heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,X)) = comp_P8552900652980440422t_unit(some_P7913643980934408916it_nat,aTP_Lamm_fi(X)) ).

% execute_ureturn
tff(fact_2134_mlex__leq,axiom,
    ! [Fa: fun_int_nat,X: int,Y: int,Ra: set_Pr958786334691620121nt_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X)),aa_int_nat(Fa,Y))
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),Ra)
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),mlex_prod_int(Fa,Ra)) ) ) ).

% mlex_leq
tff(fact_2135_mlex__iff,axiom,
    ! [X: int,Y: int,Fa: fun_int_nat,Ra: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),mlex_prod_int(Fa,Ra))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Fa,X)),aa_int_nat(Fa,Y))
        | ( ( aa_int_nat(Fa,X) = aa_int_nat(Fa,Y) )
          & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),Ra) ) ) ) ).

% mlex_iff
tff(fact_2136_mlex__less,axiom,
    ! [Fa: fun_int_nat,X: int,Y: int,Ra: set_Pr958786334691620121nt_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Fa,X)),aa_int_nat(Fa,Y))
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),mlex_prod_int(Fa,Ra)) ) ).

% mlex_less
tff(fact_2137_ent__pure__post__iff__sng,axiom,
    ! [Pa: assn,B: $o] :
      ( entails(Pa,aa_o_assn(pure_assn,(B)))
    <=> ( ! [H4: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H4)
           => (B) )
        & entails(Pa,one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_2138_pure__assn__raw_Osimps,axiom,
    ! [B: $o,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((B)),produc7507926704131184380et_nat(Ha,Asa))
    <=> ( ( Asa = bot_bot_set_nat )
        & (B) ) ) ).

% pure_assn_raw.simps
tff(fact_2139_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H,As) )
           => ( (Y)
            <=> ~ ( ( As = bot_bot_set_nat )
                  & (X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_2140_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H,As) )
           => ~ ( ( As = bot_bot_set_nat )
                & (X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_2141_le__zero__eq,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),zero_zero_nat)
    <=> ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2142_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2143_add__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% add_0
tff(fact_2144_add__0,axiom,
    ! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(zero_z1048942125864253310at_nat),A) = A ).

% add_0
tff(fact_2145_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_2146_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_2147_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_2148_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2149_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2150_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_2151_add__cancel__right__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B) )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
tff(fact_2152_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_2153_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_2154_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_2155_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_2156_add__cancel__right__left,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),A) )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
tff(fact_2157_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_2158_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_2159_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_2160_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_2161_add__cancel__left__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B) = A )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
tff(fact_2162_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_2163_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_2164_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_2165_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_2166_add__cancel__left__left,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),A) = A )
    <=> ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
tff(fact_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_add_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% add.right_neutral
tff(fact_2174_add_Oright__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),zero_z1048942125864253310at_nat) = A ).

% add.right_neutral
tff(fact_2175_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_2176_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_2177_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_2178_less__nat__zero__code,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_2179_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% neq0_conv
tff(fact_2180_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_2181_le0,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N2) ).

% le0
tff(fact_2182_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_2183_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2184_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_2185_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        & (B) )) ).

% merge_pure_star
tff(fact_2186_star__false__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),Pa) = bot_bot_assn ).

% star_false_left
tff(fact_2187_star__false__right,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_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_2201_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_2202_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_2203_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_2204_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_2205_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_2206_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_2207_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_2208_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_2209_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_2210_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_2211_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_2212_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_2213_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_2214_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_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_add__gr__0,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ( 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),N2) ) ) ).

% add_gr_0
tff(fact_2233_less__one,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),one_one_nat)
    <=> ( N2 = zero_zero_nat ) ) ).

% less_one
tff(fact_2234_mod__pure__star__dist,axiom,
    ! [Pa: assn,B: $o,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B)))),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_2235_ent__pure__pre__iff,axiom,
    ! [Pa: assn,B: $o,Qa: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Qa)
    <=> ( (B)
       => entails(Pa,Qa) ) ) ).

% ent_pure_pre_iff
tff(fact_2236_norm__pre__pure__iff,axiom,
    ! [Pa: assn,B: $o,Fa: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( hoare_hoare_triple_a(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Fa,Qa)
    <=> ( (B)
       => hoare_hoare_triple_a(Pa,Fa,Qa) ) ) ).

% norm_pre_pure_iff
tff(fact_2237_norm__pre__pure__iff,axiom,
    ! [Pa: assn,B: $o,Fa: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( hoare_hoare_triple_b(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Fa,Qa)
    <=> ( (B)
       => hoare_hoare_triple_b(Pa,Fa,Qa) ) ) ).

% norm_pre_pure_iff
tff(fact_2238_norm__pre__pure__iff,axiom,
    ! [Pa: assn,B: $o,Fa: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Fa,Qa)
    <=> ( (B)
       => hoare_8945653483474564448t_unit(Pa,Fa,Qa) ) ) ).

% norm_pre_pure_iff
tff(fact_2239_ent__false__iff,axiom,
    ! [Pa: assn] :
      ( entails(Pa,bot_bot_assn)
    <=> ! [H4: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),H4) ) ).

% ent_false_iff
tff(fact_2240_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Qa: assn] :
      ( entails(aa_o_assn(pure_assn,(B)),Qa)
    <=> ( (B)
       => entails(one_one_assn,Qa) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_2241_mod__h__bot__iff_I5_J,axiom,
    ! [Pa: assn,Qa: assn,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_2242_ent__pure__post__iff,axiom,
    ! [Pa: assn,Qa: assn,B: $o] :
      ( entails(Pa,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),aa_o_assn(pure_assn,(B))))
    <=> ( ! [H4: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H4)
           => (B) )
        & entails(Pa,Qa) ) ) ).

% ent_pure_post_iff
tff(fact_2243_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_2244_lambda__zero,axiom,
    aTP_Lamm_fj = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger) ).

% lambda_zero
tff(fact_2245_lambda__zero,axiom,
    aTP_Lamm_fk = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_2246_lambda__zero,axiom,
    aTP_Lamm_fl = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_2247_lambda__zero,axiom,
    aTP_Lamm_fm = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_2248_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_2249_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_2250_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_2251_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_2252_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_2253_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_2254_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_2255_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_2256_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_2257_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_2258_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_2259_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_2260_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_2261_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_2262_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2263_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2264_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2265_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_2266_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_2267_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_2268_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2269_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2270_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2271_mult__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono
tff(fact_2272_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_2273_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_2274_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_2275_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2276_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2277_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2278_mult__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono
tff(fact_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_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_2292_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_2293_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_2294_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_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2306_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2307_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2308_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2309_ent__iffI,axiom,
    ! [A4: assn,B4: assn] :
      ( entails(A4,B4)
     => ( entails(B4,A4)
       => ( A4 = B4 ) ) ) ).

% ent_iffI
tff(fact_2310_ent__refl,axiom,
    ! [Pa: assn] : entails(Pa,Pa) ).

% ent_refl
tff(fact_2311_ent__trans,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] :
      ( entails(Pa,Qa)
     => ( entails(Qa,Ra)
       => entails(Pa,Ra) ) ) ).

% ent_trans
tff(fact_2312_ent__star__mono,axiom,
    ! [Pa: assn,P3: assn,Qa: assn,Q2: assn] :
      ( entails(Pa,P3)
     => ( entails(Qa,Q2)
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),Q2)) ) ) ).

% ent_star_mono
tff(fact_2313_assn__times__comm,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Pa) ).

% assn_times_comm
tff(fact_2314_assn__times__assoc,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),Ra) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Ra)) ).

% assn_times_assoc
tff(fact_2315_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_2316_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_2317_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_2318_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_2319_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_2320_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_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_mult__less__cancel__left__neg,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2352_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2353_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2354_mult__less__cancel__left__pos,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2355_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2356_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2357_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2358_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2359_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2360_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2361_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2362_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2363_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2364_mult__less__cancel__left__disj,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2365_mult__less__cancel__left__disj,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2366_mult__less__cancel__left__disj,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2367_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2368_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2369_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: 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,C2),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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2370_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2371_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2372_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2373_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_2374_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2375_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2376_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2377_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2378_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2379_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2380_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2381_mult__le__cancel__left,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2382_mult__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2383_mult__le__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2384_mult__le__cancel__right,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2385_mult__le__cancel__right,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2386_mult__le__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2387_mult__left__less__imp__less,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2388_mult__left__less__imp__less,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2389_mult__left__less__imp__less,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2390_mult__left__less__imp__less,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2391_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2392_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2393_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2394_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2395_mult__less__cancel__left,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2396_mult__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2397_mult__less__cancel__left,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2398_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2399_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2400_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2401_mult__right__less__imp__less,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2402_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2403_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2404_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2405_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2406_mult__less__cancel__right,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2407_mult__less__cancel__right,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2408_mult__less__cancel__right,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2409_mult__le__cancel__left__neg,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2410_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2411_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2412_mult__le__cancel__left__pos,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2413_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2414_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2415_mult__left__le__imp__le,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2416_mult__left__le__imp__le,axiom,
    ! [C2: 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,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2417_mult__left__le__imp__le,axiom,
    ! [C2: 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,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2418_mult__left__le__imp__le,axiom,
    ! [C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2419_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2420_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: 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),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2421_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: 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),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2422_mult__right__le__imp__le,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2423_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2424_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2425_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2426_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2427_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2428_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2429_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2430_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( 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),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2431_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: 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,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2432_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: 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,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2433_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2434_mult__left__le,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),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),C2)),A) ) ) ).

% mult_left_le
tff(fact_2435_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),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),C2)),A) ) ) ).

% mult_left_le
tff(fact_2436_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),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),C2)),A) ) ) ).

% mult_left_le
tff(fact_2437_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),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),C2)),A) ) ) ).

% mult_left_le
tff(fact_2438_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_2439_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_2440_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_2441_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_2442_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2443_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2444_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2445_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2446_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2447_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2448_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger) ).

% not_sum_squares_lt_zero
tff(fact_2449_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: 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,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2450_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2451_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C2: 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)),C2)) = 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)),C2) ).

% combine_common_factor
tff(fact_2452_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),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)),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C2) ).

% combine_common_factor
tff(fact_2453_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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)),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C2) ).

% combine_common_factor
tff(fact_2454_distrib__right,axiom,
    ! [A: rat,B: rat,C2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% distrib_right
tff(fact_2455_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% distrib_right
tff(fact_2456_distrib__right,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% distrib_right
tff(fact_2457_distrib__left,axiom,
    ! [A: rat,B: rat,C2: 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),C2)) = 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),C2)) ).

% distrib_left
tff(fact_2458_distrib__left,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) ).

% distrib_left
tff(fact_2459_distrib__left,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ).

% distrib_left
tff(fact_2460_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2461_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2462_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% comm_semiring_class.distrib
tff(fact_2463_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C2: 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),C2)) = 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),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2464_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_2465_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2466_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_2467_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_2468_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_2469_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_2470_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_2471_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_2472_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_2473_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_2474_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_2475_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_2476_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_2477_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2478_not__less__zero,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% not_less_zero
tff(fact_2479_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% gr_zeroI
tff(fact_2480_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% add.group_left_neutral
tff(fact_2481_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_2482_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_2483_add_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% add.comm_neutral
tff(fact_2484_add_Ocomm__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),zero_z1048942125864253310at_nat) = A ).

% add.comm_neutral
tff(fact_2485_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_2486_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_2487_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_2488_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_2489_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(zero_z1048942125864253310at_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2490_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_2491_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_2492_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_2493_verit__sum__simplify,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% verit_sum_simplify
tff(fact_2494_verit__sum__simplify,axiom,
    ! [A: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),zero_z1048942125864253310at_nat) = A ).

% verit_sum_simplify
tff(fact_2495_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_2496_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_2497_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_2498_ent__fwd,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat,Qa: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
     => ( entails(Pa,Qa)
       => aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% ent_fwd
tff(fact_2499_entailsD,axiom,
    ! [Pa: assn,Qa: assn,Ha: produc3658429121746597890et_nat] :
      ( entails(Pa,Qa)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
       => aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% entailsD
tff(fact_2500_entailsI,axiom,
    ! [Pa: assn,Qa: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H)
         => aa_Pro653082453696054616_nat_o(rep_assn(Qa),H) )
     => entails(Pa,Qa) ) ).

% entailsI
tff(fact_2501_entails__def,axiom,
    ! [Pa: assn,Qa: assn] :
      ( entails(Pa,Qa)
    <=> ! [H4: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),H4)
         => aa_Pro653082453696054616_nat_o(rep_assn(Qa),H4) ) ) ).

% entails_def
tff(fact_2502_mod__starD,axiom,
    ! [A4: assn,B4: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),Ha)
     => ? [H12: produc3658429121746597890et_nat,H23: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(A4),H12)
          & aa_Pro653082453696054616_nat_o(rep_assn(B4),H23) ) ) ).

% mod_starD
tff(fact_2503_mod__starE,axiom,
    ! [A: assn,B: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),Ha)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(rep_assn(A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(B),H_2) ) ) ).

% mod_starE
tff(fact_2504_infinite__descent0,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
           => ( ~ aa_nat_o(Pa,N5)
             => ? [M4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N5)
                  & ~ aa_nat_o(Pa,M4) ) ) )
       => aa_nat_o(Pa,N2) ) ) ).

% infinite_descent0
tff(fact_2505_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_2506_less__zeroE,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% less_zeroE
tff(fact_2507_not__less0,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),zero_zero_nat) ).

% not_less0
tff(fact_2508_not__gr0,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ( N2 = zero_zero_nat ) ) ).

% not_gr0
tff(fact_2509_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% gr0I
tff(fact_2510_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_2511_le__0__eq,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),zero_zero_nat)
    <=> ( N2 = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_2512_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_2513_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_2514_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N2) ).

% less_eq_nat.simps(1)
tff(fact_2515_plus__nat_Oadd__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N2) = N2 ).

% plus_nat.add_0
tff(fact_2516_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_2517_cons__post__rule,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn,Q2: fun_a_assn] :
      ( hoare_hoare_triple_a(Pa,C2,Qa)
     => ( ! [X3: a] : entails(aa_a_assn(Qa,X3),aa_a_assn(Q2,X3))
       => hoare_hoare_triple_a(Pa,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_2518_cons__post__rule,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn,Q2: fun_b_assn] :
      ( hoare_hoare_triple_b(Pa,C2,Qa)
     => ( ! [X3: b] : entails(aa_b_assn(Qa,X3),aa_b_assn(Q2,X3))
       => hoare_hoare_triple_b(Pa,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_2519_cons__post__rule,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,Q2: fun_Pr6626665806141890376t_assn] :
      ( hoare_8945653483474564448t_unit(Pa,C2,Qa)
     => ( ! [X3: product_unit] : entails(aa_Product_unit_assn(Qa,X3),aa_Product_unit_assn(Q2,X3))
       => hoare_8945653483474564448t_unit(Pa,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_2520_cons__rule,axiom,
    ! [Pa: assn,P3: assn,Qa: fun_a_assn,Q2: fun_a_assn,C2: heap_Time_Heap_a] :
      ( entails(Pa,P3)
     => ( ! [X3: a] : entails(aa_a_assn(Qa,X3),aa_a_assn(Q2,X3))
       => ( hoare_hoare_triple_a(P3,C2,Qa)
         => hoare_hoare_triple_a(Pa,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_2521_cons__rule,axiom,
    ! [Pa: assn,P3: assn,Qa: fun_b_assn,Q2: fun_b_assn,C2: heap_Time_Heap_b] :
      ( entails(Pa,P3)
     => ( ! [X3: b] : entails(aa_b_assn(Qa,X3),aa_b_assn(Q2,X3))
       => ( hoare_hoare_triple_b(P3,C2,Qa)
         => hoare_hoare_triple_b(Pa,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_2522_cons__rule,axiom,
    ! [Pa: assn,P3: assn,Qa: fun_Pr6626665806141890376t_assn,Q2: fun_Pr6626665806141890376t_assn,C2: heap_T5738788834812785303t_unit] :
      ( entails(Pa,P3)
     => ( ! [X3: product_unit] : entails(aa_Product_unit_assn(Qa,X3),aa_Product_unit_assn(Q2,X3))
       => ( hoare_8945653483474564448t_unit(P3,C2,Qa)
         => hoare_8945653483474564448t_unit(Pa,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_2523_ent__disjE,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( entails(A4,C3)
     => ( entails(B4,C3)
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A4),B4),C3) ) ) ).

% ent_disjE
tff(fact_2524_ent__disjI1,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa),Ra)
     => entails(Pa,Ra) ) ).

% ent_disjI1
tff(fact_2525_ent__disjI2,axiom,
    ! [Pa: assn,Qa: assn,Ra: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa),Ra)
     => entails(Qa,Ra) ) ).

% ent_disjI2
tff(fact_2526_ent__disjI1_H,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( entails(A4,B4)
     => entails(A4,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B4),C3)) ) ).

% ent_disjI1'
tff(fact_2527_ent__disjI2_H,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( entails(A4,C3)
     => entails(A4,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B4),C3)) ) ).

% ent_disjI2'
tff(fact_2528_ent__disjI1__direct,axiom,
    ! [A4: assn,B4: assn] : entails(A4,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A4),B4)) ).

% ent_disjI1_direct
tff(fact_2529_ent__disjI2__direct,axiom,
    ! [B4: assn,A4: assn] : entails(B4,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A4),B4)) ).

% ent_disjI2_direct
tff(fact_2530_star__or__dist1,axiom,
    ! [A4: assn,B4: assn,C3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A4),B4)),C3) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),C3)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B4),C3)) ).

% star_or_dist1
tff(fact_2531_star__or__dist2,axiom,
    ! [C3: assn,A4: assn,B4: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),A4)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),B4)) ).

% star_or_dist2
tff(fact_2532_assn__one__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),Pa) = Pa ).

% assn_one_left
tff(fact_2533_ent__false,axiom,
    ! [Pa: assn] : entails(bot_bot_assn,Pa) ).

% ent_false
tff(fact_2534_mult__le__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2535_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),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_2536_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => 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,C2),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_2537_mult__le__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),C2)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2538_mult__le__cancel__left2,axiom,
    ! [C2: 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,C2),A)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),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_2539_mult__le__cancel__left2,axiom,
    ! [C2: 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,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => 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,C2),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_2540_mult__le__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2541_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),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_2542_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => 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,C2),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_2543_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),C2)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2544_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: 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),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),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_2545_mult__le__cancel__right2,axiom,
    ! [A: int,C2: 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),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => 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,C2),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_2546_mult__less__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2547_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2548_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2549_mult__less__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),A)),C2)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2550_mult__less__cancel__left2,axiom,
    ! [C2: 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,C2),A)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2551_mult__less__cancel__left2,axiom,
    ! [C2: 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,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2552_mult__less__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2553_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2554_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2555_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2)),C2)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2556_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2557_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => 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,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2558_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),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),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2559_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(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),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2560_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2561_lambda__one,axiom,
    aTP_Lamm_fn = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer) ).

% lambda_one
tff(fact_2562_lambda__one,axiom,
    aTP_Lamm_fo = aa_ass954072293579886259n_assn(times_times_assn,one_one_assn) ).

% lambda_one
tff(fact_2563_lambda__one,axiom,
    aTP_Lamm_fp = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_2564_lambda__one,axiom,
    aTP_Lamm_fq = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_2565_lambda__one,axiom,
    aTP_Lamm_fr = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_2566_cons__pre__rule,axiom,
    ! [Pa: assn,P3: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( entails(Pa,P3)
     => ( hoare_hoare_triple_a(P3,C2,Qa)
       => hoare_hoare_triple_a(Pa,C2,Qa) ) ) ).

% cons_pre_rule
tff(fact_2567_cons__pre__rule,axiom,
    ! [Pa: assn,P3: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( entails(Pa,P3)
     => ( hoare_hoare_triple_b(P3,C2,Qa)
       => hoare_hoare_triple_b(Pa,C2,Qa) ) ) ).

% cons_pre_rule
tff(fact_2568_cons__pre__rule,axiom,
    ! [Pa: assn,P3: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( entails(Pa,P3)
     => ( hoare_8945653483474564448t_unit(P3,C2,Qa)
       => hoare_8945653483474564448t_unit(Pa,C2,Qa) ) ) ).

% cons_pre_rule
tff(fact_2569_frame__rule,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_a,Qa: fun_a_assn,Ra: assn] :
      ( hoare_hoare_triple_a(Pa,C2,Qa)
     => hoare_hoare_triple_a(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Ra),C2,aa_assn_fun_a_assn(aTP_Lamm_fs(Qa),Ra)) ) ).

% frame_rule
tff(fact_2570_frame__rule,axiom,
    ! [Pa: assn,C2: heap_Time_Heap_b,Qa: fun_b_assn,Ra: assn] :
      ( hoare_hoare_triple_b(Pa,C2,Qa)
     => hoare_hoare_triple_b(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Ra),C2,aa_assn_fun_b_assn(aTP_Lamm_ft(Qa),Ra)) ) ).

% frame_rule
tff(fact_2571_frame__rule,axiom,
    ! [Pa: assn,C2: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn,Ra: assn] :
      ( hoare_8945653483474564448t_unit(Pa,C2,Qa)
     => hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Ra),C2,aa_ass3177796706412234594t_assn(aTP_Lamm_fu(Qa),Ra)) ) ).

% frame_rule
tff(fact_2572_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),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),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2573_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(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),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2574_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2575_less__1__mult,axiom,
    ! [M: code_integer,N2: 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),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M),N2)) ) ) ).

% less_1_mult
tff(fact_2576_less__1__mult,axiom,
    ! [M: rat,N2: 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),N2)
       => 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),N2)) ) ) ).

% less_1_mult
tff(fact_2577_less__1__mult,axiom,
    ! [M: nat,N2: 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),N2)
       => 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),N2)) ) ) ).

% less_1_mult
tff(fact_2578_less__1__mult,axiom,
    ! [M: int,N2: 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),N2)
       => 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),N2)) ) ) ).

% less_1_mult
tff(fact_2579_add__decreasing,axiom,
    ! [A: code_integer,C2: 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,C2),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_2580_add__decreasing,axiom,
    ! [A: rat,C2: 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,C2),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),C2)),B) ) ) ).

% add_decreasing
tff(fact_2581_add__decreasing,axiom,
    ! [A: nat,C2: 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,C2),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),C2)),B) ) ) ).

% add_decreasing
tff(fact_2582_add__decreasing,axiom,
    ! [A: int,C2: 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,C2),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),C2)),B) ) ) ).

% add_decreasing
tff(fact_2583_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C2)) ) ) ).

% add_increasing
tff(fact_2584_add__increasing,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => 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),C2)) ) ) ).

% add_increasing
tff(fact_2585_add__increasing,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => 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),C2)) ) ) ).

% add_increasing
tff(fact_2586_add__increasing,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => 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),C2)) ) ) ).

% add_increasing
tff(fact_2587_add__decreasing2,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),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),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2588_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),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),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2589_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),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),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2590_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),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),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2591_add__increasing2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
     => ( 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),C2)) ) ) ).

% add_increasing2
tff(fact_2592_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
     => ( 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),C2)) ) ) ).

% add_increasing2
tff(fact_2593_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( 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),C2)) ) ) ).

% add_increasing2
tff(fact_2594_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( 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),C2)) ) ) ).

% add_increasing2
tff(fact_2595_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_2596_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_2597_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_2598_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_2599_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_2600_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_2601_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_2602_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_2603_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2604_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2605_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2606_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2607_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),zero_z3403309356797280102nteger)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2608_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2609_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2610_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2611_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_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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_2618_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_2619_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_2620_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_2621_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_2622_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_2623_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2624_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2625_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2626_pos__add__strict,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2627_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),zero_z3403309356797280102nteger) ) ) ).

% add_less_zeroD
tff(fact_2637_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_2638_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_2639_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_2640_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_2641_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_2642_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_2643_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_2644_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_2645_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_2646_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_2647_zero__less__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_2648_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_2649_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_2650_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_2651_ex__least__nat__le,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,N2)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
               => ~ aa_nat_o(Pa,I4) )
            & aa_nat_o(Pa,K) ) ) ) ).

% ex_least_nat_le
tff(fact_2652_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ? [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_2653_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),X)
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_2654_norm__pre__pure__rule1,axiom,
    ! [B: $o,Pa: assn,Fa: heap_Time_Heap_a,Qa: fun_a_assn] :
      ( ( (B)
       => hoare_hoare_triple_a(Pa,Fa,Qa) )
     => hoare_hoare_triple_a(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Fa,Qa) ) ).

% norm_pre_pure_rule1
tff(fact_2655_norm__pre__pure__rule1,axiom,
    ! [B: $o,Pa: assn,Fa: heap_Time_Heap_b,Qa: fun_b_assn] :
      ( ( (B)
       => hoare_hoare_triple_b(Pa,Fa,Qa) )
     => hoare_hoare_triple_b(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Fa,Qa) ) ).

% norm_pre_pure_rule1
tff(fact_2656_norm__pre__pure__rule1,axiom,
    ! [B: $o,Pa: assn,Fa: heap_T5738788834812785303t_unit,Qa: fun_Pr6626665806141890376t_assn] :
      ( ( (B)
       => hoare_8945653483474564448t_unit(Pa,Fa,Qa) )
     => hoare_8945653483474564448t_unit(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_o_assn(pure_assn,(B))),Fa,Qa) ) ).

% norm_pre_pure_rule1
tff(fact_2657_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_bx,aTP_Lamm_by),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_2658_ureturn__def,axiom,
    ! [X: b] : aa_b_H1867786022829710796Heap_b(heap_Time_ureturn_b,X) = heap_Time_heap_b(aTP_Lamm_fh(X)) ).

% ureturn_def
tff(fact_2659_ureturn__def,axiom,
    ! [X: a] : aa_a_H8829438773920479562Heap_a(heap_Time_ureturn_a,X) = heap_Time_heap_a(aTP_Lamm_fi(X)) ).

% ureturn_def
tff(fact_2660_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_2661_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_2662_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_2663_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_2664_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_2665_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_2666_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_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_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_2675_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_2676_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2677_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2678_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2679_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2680_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2681_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2682_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2683_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2684_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_2685_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_2686_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_2687_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_2688_return__sp__rule,axiom,
    ! [Pa: assn,X: a] : hoare_hoare_triple_a(Pa,aa_a_H8829438773920479562Heap_a(heap_Time_return_a,X),aa_a_fun_a_assn(aTP_Lamm_fv(Pa),X)) ).

% return_sp_rule
tff(fact_2689_return__sp__rule,axiom,
    ! [Pa: assn,X: b] : hoare_hoare_triple_b(Pa,aa_b_H1867786022829710796Heap_b(heap_Time_return_b,X),aa_b_fun_b_assn(aTP_Lamm_fw(Pa),X)) ).

% return_sp_rule
tff(fact_2690_return__sp__rule,axiom,
    ! [Pa: assn,X: product_unit] : hoare_8945653483474564448t_unit(Pa,aa_Pro1108561329362217960t_unit(heap_T7507251653302230130t_unit,X),aa_Pro6018021088111102263t_assn(aTP_Lamm_fx(Pa),X)) ).

% return_sp_rule
tff(fact_2691_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H,As) )
           => ( ( As = bot_bot_set_nat )
              & (X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_2692_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),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,Z2),X)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_2693_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_2694_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_2695_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_2696_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: 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,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2697_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: 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,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2698_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2699_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger)
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2700_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: 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,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2701_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2702_mult__le__cancel__iff1,axiom,
    ! [Z3: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2703_mult__le__cancel__iff1,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2704_mult__le__cancel__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2705_mult__le__cancel__iff2,axiom,
    ! [Z3: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z3),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z3),Y))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2706_mult__le__cancel__iff2,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2707_mult__le__cancel__iff2,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z3),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2708_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_epsilon
tff(fact_2709_divides__aux__eq,axiom,
    ! [Q4: code_integer,R: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q4,R))
    <=> ( R = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_2710_divides__aux__eq,axiom,
    ! [Q4: nat,R: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R))
    <=> ( R = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2711_divides__aux__eq,axiom,
    ! [Q4: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R))
    <=> ( R = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2712_pure__assn__raw_Opelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
     => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xa = produc7507926704131184380et_nat(H,As) )
             => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H,As)))
               => ( ( As = bot_bot_set_nat )
                  & (X) ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
tff(fact_2713_pure__assn__raw_Opelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
     => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xa = produc7507926704131184380et_nat(H,As) )
             => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H,As)))
               => ~ ( ( As = bot_bot_set_nat )
                    & (X) ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
tff(fact_2714_mult__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2) )
    <=> ( ( M = N2 )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2715_mult__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2) )
    <=> ( ( M = N2 )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2716_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_2717_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2718_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2719_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
    <=> ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2720_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ( 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),N2) ) ) ).

% nat_0_less_mult_iff
tff(fact_2721_mult__less__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% mult_less_cancel2
tff(fact_2722_mult__le__cancel2,axiom,
    ! [M: nat,K2: nat,N2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% mult_le_cancel2
tff(fact_2723_mult__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N2) = zero_zero_nat ).

% mult_0
tff(fact_2724_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_2725_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_2726_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),L)) ) ) ).

% mult_le_mono
tff(fact_2727_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K2)) ) ).

% mult_le_mono1
tff(fact_2728_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J2)) ) ).

% mult_le_mono2
tff(fact_2729_add__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2)) ).

% add_mult_distrib2
tff(fact_2730_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2)) ).

% add_mult_distrib
tff(fact_2731_nat__mult__1__right,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),one_one_nat) = N2 ).

% nat_mult_1_right
tff(fact_2732_nat__mult__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N2) = N2 ).

% nat_mult_1
tff(fact_2733_mult__less__mono2,axiom,
    ! [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,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J2)) ) ) ).

% mult_less_mono2
tff(fact_2734_mult__less__mono1,axiom,
    ! [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,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K2)) ) ) ).

% mult_less_mono1
tff(fact_2735_mlex__snd__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N7: nat] :
      ( ( A = A2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B2)
       => 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,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)) ) ) ).

% mlex_snd_decrI
tff(fact_2736_mlex__fst__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,N7: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N7)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N7)
         => 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,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)) ) ) ) ).

% mlex_fst_decrI
tff(fact_2737_mlex__bound,axiom,
    ! [A: nat,A4: nat,B: nat,N7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N7)
       => 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,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N7)) ) ) ).

% mlex_bound
tff(fact_2738_mlex__leI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B2)
       => 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,A),N7)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)) ) ) ).

% mlex_leI
tff(fact_2739_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2740_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X5) ).

% linordered_field_no_lb
tff(fact_2741_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),X_1) ).

% linordered_field_no_ub
tff(fact_2742_mult__less__iff1,axiom,
    ! [Z3: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2743_mult__less__iff1,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2744_mult__less__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z3)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2745_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = zero_z3403309356797280102nteger )
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2746_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2747_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2748_pure__assn__raw_Opelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
      <=> (Y) )
     => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),Xa))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xa = produc7507926704131184380et_nat(H,As) )
             => ( ( (Y)
                <=> ( ( As = bot_bot_set_nat )
                    & (X) ) )
               => ~ accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),produc7507926704131184380et_nat(H,As))) ) ) ) ) ).

% pure_assn_raw.pelims(1)
tff(fact_2749_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2750_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2751_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% nat_mult_le_cancel1
tff(fact_2752_add__scale__eq__noteq,axiom,
    ! [R: code_integer,A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( R != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),C2)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2753_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C2: rat,D2: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),C2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2754_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C2: nat,D2: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2755_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C2: int,D2: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2756_Euclid__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(Pa,B3),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,A3),zero_zero_nat)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,A3),B3)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,A),B) ) ) ) ).

% Euclid_induct
tff(fact_2757_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2) )
      <=> ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2758_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),U)),K2) ).

% left_add_mult_distrib
tff(fact_2759_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_2760_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_2761_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_2762_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_2763_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2764_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2765_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% crossproduct_noteq
tff(fact_2766_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z3: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
tff(fact_2767_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Z3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
tff(fact_2768_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z3)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
tff(fact_2769_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% nat_mult_less_cancel1
tff(fact_2770_ent__mp,axiom,
    ! [Pa: assn,Qa: assn] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Qa)),Qa) ).

% ent_mp
tff(fact_2771_ent__wandI,axiom,
    ! [Qa: assn,Pa: assn,Ra: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Pa),Ra)
     => entails(Pa,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Qa),Ra)) ) ).

% ent_wandI
tff(fact_2772_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2773_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2774_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2775_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2776_mod__star__conv,axiom,
    ! [A4: assn,B4: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),Ha)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
          ( ( Ha = produc7507926704131184380et_nat(Hr,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22)) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(rep_assn(A4),produc7507926704131184380et_nat(Hr,As1))
          & aa_Pro653082453696054616_nat_o(rep_assn(B4),produc7507926704131184380et_nat(Hr,As22)) ) ) ).

% mod_star_conv
tff(fact_2777_star__assnI,axiom,
    ! [Pa: assn,Ha: heap_e7401611519738050253t_unit,Asa: set_nat,Qa: assn,As4: set_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,Asa))
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,As4))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Asa),As4) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As4))) ) ) ) ).

% star_assnI
tff(fact_2778_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N2: 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),N2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_2779_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: 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),N2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_2780_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: 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),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_2781_power__strict__mono,axiom,
    ! [A: int,B: int,N2: 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),N2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ) ).

% power_strict_mono
tff(fact_2782_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: 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),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2783_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2784_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2785_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2786_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2787_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2788_Int__iff,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),inf_inf_set_o(A4,B4))
    <=> ( aa_set_o_o(member_o((C2)),A4)
        & aa_set_o_o(member_o((C2)),B4) ) ) ).

% Int_iff
tff(fact_2789_Int__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),inf_in8396524679539076455nt_int(A4,B4))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
        & aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% Int_iff
tff(fact_2790_Int__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A4,B4))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A4)
        & aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% Int_iff
tff(fact_2791_Int__iff,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
    <=> ( aa_set_int_o(member_int(C2),A4)
        & aa_set_int_o(member_int(C2),B4) ) ) ).

% Int_iff
tff(fact_2792_Int__iff,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4) ) ) ).

% Int_iff
tff(fact_2793_Int__iff,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
    <=> ( aa_set_nat_o(member_nat(C2),A4)
        & aa_set_nat_o(member_nat(C2),B4) ) ) ).

% Int_iff
tff(fact_2794_IntI,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),A4)
     => ( aa_set_o_o(member_o((C2)),B4)
       => aa_set_o_o(member_o((C2)),inf_inf_set_o(A4,B4)) ) ) ).

% IntI
tff(fact_2795_IntI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),inf_in8396524679539076455nt_int(A4,B4)) ) ) ).

% IntI
tff(fact_2796_IntI,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A4)
     => ( aa_set_set_nat_o(member_set_nat(C2),B4)
       => aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A4,B4)) ) ) ).

% IntI
tff(fact_2797_IntI,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),A4)
     => ( aa_set_int_o(member_int(C2),B4)
       => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) ) ) ).

% IntI
tff(fact_2798_IntI,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ) ) ).

% IntI
tff(fact_2799_IntI,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A4)
     => ( aa_set_nat_o(member_nat(C2),B4)
       => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ) ) ).

% IntI
tff(fact_2800_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2801_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2802_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2803_le__inf__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3))
    <=> ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
        & aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2804_le__inf__iff,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z3))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2805_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z3))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2806_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z3))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2807_le__inf__iff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z3))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3) ) ) ).

% le_inf_iff
tff(fact_2808_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2809_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2810_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2811_inf_Obounded__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C2))
    <=> ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
        & aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2812_inf_Obounded__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C2))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2813_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2))
    <=> ( 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,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2814_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))
    <=> ( 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),C2) ) ) ).

% inf.bounded_iff
tff(fact_2815_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))
    <=> ( 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,A),C2) ) ) ).

% inf.bounded_iff
tff(fact_2816_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: 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,X),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2817_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2818_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2819_inf__sup__absorb,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2820_inf__sup__absorb,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2821_inf__sup__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2822_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2823_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2824_inf__sup__absorb,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2825_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2826_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2827_sup__inf__absorb,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2828_sup__inf__absorb,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2829_sup__inf__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2830_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2831_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2832_sup__inf__absorb,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2833_Int__subset__iff,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),A4)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),B4) ) ) ).

% Int_subset_iff
tff(fact_2834_Int__subset__iff,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),A4)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),B4) ) ) ).

% Int_subset_iff
tff(fact_2835_Int__subset__iff,axiom,
    ! [C3: set_int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),A4)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),B4) ) ) ).

% Int_subset_iff
tff(fact_2836_inf__Some,axiom,
    ! [X: assn,Y: assn] : inf_inf_option_assn(some_assn(X),some_assn(Y)) = some_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) ).

% inf_Some
tff(fact_2837_inf__Some,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : inf_in777885744494410645at_nat(some_s147305329494351046at_nat(X),some_s147305329494351046at_nat(Y)) = some_s147305329494351046at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) ).

% inf_Some
tff(fact_2838_inf__Some,axiom,
    ! [X: set_nat,Y: set_nat] : inf_in7812914138253463912et_nat(some_set_nat(X),some_set_nat(Y)) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) ).

% inf_Some
tff(fact_2839_inf__Some,axiom,
    ! [X: nat,Y: nat] : inf_inf_option_nat(aa_nat_option_nat(some_nat,X),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) ).

% inf_Some
tff(fact_2840_inf__Some,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : inf_in3196829186289650628_int_o(some_f3270274345126892989_int_o(X),some_f3270274345126892989_int_o(Y)) = some_f3270274345126892989_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) ).

% inf_Some
tff(fact_2841_Int__Un__eq_I4_J,axiom,
    ! [T7: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T7),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T7)) = T7 ).

% Int_Un_eq(4)
tff(fact_2842_Int__Un__eq_I4_J,axiom,
    ! [T7: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,T7),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T7)) = T7 ).

% Int_Un_eq(4)
tff(fact_2843_Int__Un__eq_I4_J,axiom,
    ! [T7: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,T7),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T7)) = T7 ).

% Int_Un_eq(4)
tff(fact_2844_Int__Un__eq_I4_J,axiom,
    ! [T7: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T7),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T7)) = T7 ).

% Int_Un_eq(4)
tff(fact_2845_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T7)) = S ).

% Int_Un_eq(3)
tff(fact_2846_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T7)) = S ).

% Int_Un_eq(3)
tff(fact_2847_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T7)) = S ).

% Int_Un_eq(3)
tff(fact_2848_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T7: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T7)) = S ).

% Int_Un_eq(3)
tff(fact_2849_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T7)),T7) = T7 ).

% Int_Un_eq(2)
tff(fact_2850_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T7)),T7) = T7 ).

% Int_Un_eq(2)
tff(fact_2851_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T7)),T7) = T7 ).

% Int_Un_eq(2)
tff(fact_2852_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T7: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T7)),T7) = T7 ).

% Int_Un_eq(2)
tff(fact_2853_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T7)),S) = S ).

% Int_Un_eq(1)
tff(fact_2854_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),T7)),S) = S ).

% Int_Un_eq(1)
tff(fact_2855_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T7)),S) = S ).

% Int_Un_eq(1)
tff(fact_2856_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T7: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T7)),S) = S ).

% Int_Un_eq(1)
tff(fact_2857_Un__Int__eq_I4_J,axiom,
    ! [T7: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T7),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T7)) = T7 ).

% Un_Int_eq(4)
tff(fact_2858_Un__Int__eq_I4_J,axiom,
    ! [T7: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,T7),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T7)) = T7 ).

% Un_Int_eq(4)
tff(fact_2859_Un__Int__eq_I4_J,axiom,
    ! [T7: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,T7),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T7)) = T7 ).

% Un_Int_eq(4)
tff(fact_2860_Un__Int__eq_I4_J,axiom,
    ! [T7: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T7),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T7)) = T7 ).

% Un_Int_eq(4)
tff(fact_2861_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T7)) = S ).

% Un_Int_eq(3)
tff(fact_2862_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,S),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T7)) = S ).

% Un_Int_eq(3)
tff(fact_2863_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T7)) = S ).

% Un_Int_eq(3)
tff(fact_2864_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T7: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T7)) = S ).

% Un_Int_eq(3)
tff(fact_2865_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T7)),T7) = T7 ).

% Un_Int_eq(2)
tff(fact_2866_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T7)),T7) = T7 ).

% Un_Int_eq(2)
tff(fact_2867_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T7)),T7) = T7 ).

% Un_Int_eq(2)
tff(fact_2868_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T7: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T7)),T7) = T7 ).

% Un_Int_eq(2)
tff(fact_2869_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T7)),S) = S ).

% Un_Int_eq(1)
tff(fact_2870_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T7)),S) = S ).

% Un_Int_eq(1)
tff(fact_2871_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T7)),S) = S ).

% Un_Int_eq(1)
tff(fact_2872_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T7: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T7)),S) = S ).

% Un_Int_eq(1)
tff(fact_2873_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2874_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2875_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2876_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: 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),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2877_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2878_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2879_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2880_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2881_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2882_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2883_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2884_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_2885_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: 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),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2886_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2887_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2888_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2889_power__eq__0__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N2) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_2890_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_2891_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_2892_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% power_eq_0_iff
tff(fact_2893_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2894_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2895_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2896_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2897_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2898_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: 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),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2899_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N2: 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),N2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2900_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: 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),N2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2901_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: 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),N2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2902_power__mono__iff,axiom,
    ! [A: int,B: int,N2: 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),N2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2903_Int__left__commute,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C3)) ).

% Int_left_commute
tff(fact_2904_Int__left__commute,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C3)) ).

% Int_left_commute
tff(fact_2905_Int__left__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ).

% Int_left_absorb
tff(fact_2906_Int__left__absorb,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ).

% Int_left_absorb
tff(fact_2907_Int__commute,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),A4) ).

% Int_commute
tff(fact_2908_Int__commute,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A4) ).

% Int_commute
tff(fact_2909_Int__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),A4) = A4 ).

% Int_absorb
tff(fact_2910_Int__absorb,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),A4) = A4 ).

% Int_absorb
tff(fact_2911_Int__assoc,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) ).

% Int_assoc
tff(fact_2912_Int__assoc,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) ).

% Int_assoc
tff(fact_2913_IntD2,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),inf_inf_set_o(A4,B4))
     => aa_set_o_o(member_o((C2)),B4) ) ).

% IntD2
tff(fact_2914_IntD2,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),inf_in8396524679539076455nt_int(A4,B4))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ).

% IntD2
tff(fact_2915_IntD2,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A4,B4))
     => aa_set_set_nat_o(member_set_nat(C2),B4) ) ).

% IntD2
tff(fact_2916_IntD2,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
     => aa_set_int_o(member_int(C2),B4) ) ).

% IntD2
tff(fact_2917_IntD2,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4) ) ).

% IntD2
tff(fact_2918_IntD2,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
     => aa_set_nat_o(member_nat(C2),B4) ) ).

% IntD2
tff(fact_2919_IntD1,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),inf_inf_set_o(A4,B4))
     => aa_set_o_o(member_o((C2)),A4) ) ).

% IntD1
tff(fact_2920_IntD1,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),inf_in8396524679539076455nt_int(A4,B4))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4) ) ).

% IntD1
tff(fact_2921_IntD1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A4,B4))
     => aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% IntD1
tff(fact_2922_IntD1,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
     => aa_set_int_o(member_int(C2),A4) ) ).

% IntD1
tff(fact_2923_IntD1,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4) ) ).

% IntD1
tff(fact_2924_IntD1,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
     => aa_set_nat_o(member_nat(C2),A4) ) ).

% IntD1
tff(fact_2925_IntE,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),inf_inf_set_o(A4,B4))
     => ~ ( aa_set_o_o(member_o((C2)),A4)
         => ~ aa_set_o_o(member_o((C2)),B4) ) ) ).

% IntE
tff(fact_2926_IntE,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),inf_in8396524679539076455nt_int(A4,B4))
     => ~ ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% IntE
tff(fact_2927_IntE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),inf_inf_set_set_nat(A4,B4))
     => ~ ( aa_set_set_nat_o(member_set_nat(C2),A4)
         => ~ aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% IntE
tff(fact_2928_IntE,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
     => ~ ( aa_set_int_o(member_int(C2),A4)
         => ~ aa_set_int_o(member_int(C2),B4) ) ) ).

% IntE
tff(fact_2929_IntE,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4) ) ) ).

% IntE
tff(fact_2930_IntE,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
     => ~ ( aa_set_nat_o(member_nat(C2),A4)
         => ~ aa_set_nat_o(member_nat(C2),B4) ) ) ).

% IntE
tff(fact_2931_Int__def,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(A4,B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_fy(A4),B4)) ).

% Int_def
tff(fact_2932_Int__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : inf_inf_set_list_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_fz(A4),B4)) ).

% Int_def
tff(fact_2933_Int__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] : inf_in8396524679539076455nt_int(A4,B4) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_ga(A4),B4)) ).

% Int_def
tff(fact_2934_Int__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : inf_inf_set_set_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_gb(A4),B4)) ).

% Int_def
tff(fact_2935_Int__def,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_gc(A4),B4)) ).

% Int_def
tff(fact_2936_Int__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_gd(A4),B4)) ).

% Int_def
tff(fact_2937_Int__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ge(A4),B4)) ).

% Int_def
tff(fact_2938_Int__Collect,axiom,
    ! [X: $o,A4: set_o,Pa: fun_o_o] :
      ( aa_set_o_o(member_o((X)),inf_inf_set_o(A4,collect_o(Pa)))
    <=> ( aa_set_o_o(member_o((X)),A4)
        & aa_o_o(Pa,(X)) ) ) ).

% Int_Collect
tff(fact_2939_Int__Collect,axiom,
    ! [X: list_nat,A4: set_list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(X,inf_inf_set_list_nat(A4,collect_list_nat(Pa)))
    <=> ( member_list_nat(X,A4)
        & aa_list_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_2940_Int__Collect,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),inf_in8396524679539076455nt_int(A4,collec5210948495886036740nt_int(Pa)))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
        & aa_set6170482117830564987_int_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_2941_Int__Collect,axiom,
    ! [X: set_nat,A4: set_set_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(X),inf_inf_set_set_nat(A4,collect_set_nat(Pa)))
    <=> ( aa_set_set_nat_o(member_set_nat(X),A4)
        & aa_set_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_2942_Int__Collect,axiom,
    ! [X: int,A4: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))
    <=> ( aa_set_int_o(member_int(X),A4)
        & aa_int_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_2943_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),collec3392354462482085612at_nat(Pa)))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
        & aa_Pro6150637604445964947_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_2944_Int__Collect,axiom,
    ! [X: nat,A4: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))
    <=> ( aa_set_nat_o(member_nat(X),A4)
        & aa_nat_o(Pa,X) ) ) ).

% Int_Collect
tff(fact_2945_Collect__conj__eq,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ap(Pa),Qa)) = inf_inf_set_list_nat(collect_list_nat(Pa),collect_list_nat(Qa)) ).

% Collect_conj_eq
tff(fact_2946_Collect__conj__eq,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] : collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_aq(Pa),Qa)) = inf_in8396524679539076455nt_int(collec5210948495886036740nt_int(Pa),collec5210948495886036740nt_int(Qa)) ).

% Collect_conj_eq
tff(fact_2947_Collect__conj__eq,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ar(Pa),Qa)) = inf_inf_set_set_nat(collect_set_nat(Pa),collect_set_nat(Qa)) ).

% Collect_conj_eq
tff(fact_2948_Collect__conj__eq,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_at(Pa),Qa)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,collect_int(Pa)),collect_int(Qa)) ).

% Collect_conj_eq
tff(fact_2949_Collect__conj__eq,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] : collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_gf(Pa),Qa)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,collec3392354462482085612at_nat(Pa)),collec3392354462482085612at_nat(Qa)) ).

% Collect_conj_eq
tff(fact_2950_Collect__conj__eq,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_as(Pa),Qa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,collect_nat(Pa)),collect_nat(Qa)) ).

% Collect_conj_eq
tff(fact_2951_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2952_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2953_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2954_inf__sup__ord_I2_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2955_inf__sup__ord_I2_J,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2956_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2957_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2958_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_2959_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2960_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2961_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2962_inf__sup__ord_I1_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2963_inf__sup__ord_I1_J,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2964_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2965_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2966_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_2967_inf__le1,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_le1
tff(fact_2968_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_le1
tff(fact_2969_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_le1
tff(fact_2970_inf__le1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),X) ).

% inf_le1
tff(fact_2971_inf__le1,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),X) ).

% inf_le1
tff(fact_2972_inf__le1,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X) ).

% inf_le1
tff(fact_2973_inf__le1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_le1
tff(fact_2974_inf__le1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_le1
tff(fact_2975_inf__le2,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_le2
tff(fact_2976_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_le2
tff(fact_2977_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_le2
tff(fact_2978_inf__le2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) ).

% inf_le2
tff(fact_2979_inf__le2,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),Y) ).

% inf_le2
tff(fact_2980_inf__le2,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y) ).

% inf_le2
tff(fact_2981_inf__le2,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_le2
tff(fact_2982_inf__le2,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_le2
tff(fact_2983_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B) ) ) ).

% le_infE
tff(fact_2984_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B) ) ) ).

% le_infE
tff(fact_2985_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B) ) ) ).

% le_infE
tff(fact_2986_le__infE,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),A)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),B) ) ) ).

% le_infE
tff(fact_2987_le__infE,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B))
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),A)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),B) ) ) ).

% le_infE
tff(fact_2988_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B))
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),B) ) ) ).

% le_infE
tff(fact_2989_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B) ) ) ).

% le_infE
tff(fact_2990_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B) ) ) ).

% le_infE
tff(fact_2991_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% le_infI
tff(fact_2992_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% le_infI
tff(fact_2993_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% le_infI
tff(fact_2994_le__infI,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),A)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),B)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% le_infI
tff(fact_2995_le__infI,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),B)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)) ) ) ).

% le_infI
tff(fact_2996_le__infI,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)) ) ) ).

% le_infI
tff(fact_2997_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% le_infI
tff(fact_2998_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% le_infI
tff(fact_2999_inf__mono,axiom,
    ! [A: assn,C2: assn,B: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,C2),D2)) ) ) ).

% inf_mono
tff(fact_3000_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),D2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D2)) ) ) ).

% inf_mono
tff(fact_3001_inf__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D2)) ) ) ).

% inf_mono
tff(fact_3002_inf__mono,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,D2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C2)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),D2)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,C2),D2)) ) ) ).

% inf_mono
tff(fact_3003_inf__mono,axiom,
    ! [A: set_int,C2: set_int,B: set_int,D2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C2),D2)) ) ) ).

% inf_mono
tff(fact_3004_inf__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C2),D2)) ) ) ).

% inf_mono
tff(fact_3005_inf__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C2),D2)) ) ) ).

% inf_mono
tff(fact_3006_inf__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C2),D2)) ) ) ).

% inf_mono
tff(fact_3007_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI1
tff(fact_3008_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3009_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3010_le__infI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% le_infI1
tff(fact_3011_le__infI1,axiom,
    ! [A: set_int,X: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),X) ) ).

% le_infI1
tff(fact_3012_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% le_infI1
tff(fact_3013_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI1
tff(fact_3014_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI1
tff(fact_3015_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI2
tff(fact_3016_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3017_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3018_le__infI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% le_infI2
tff(fact_3019_le__infI2,axiom,
    ! [B: set_int,X: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),X)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),X) ) ).

% le_infI2
tff(fact_3020_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% le_infI2
tff(fact_3021_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI2
tff(fact_3022_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI2
tff(fact_3023_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.orderE
tff(fact_3024_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3025_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3026_inf_OorderE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ) ).

% inf.orderE
tff(fact_3027_inf_OorderE,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) ) ) ).

% inf.orderE
tff(fact_3028_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.orderE
tff(fact_3029_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_3030_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_3031_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).

% inf.orderI
tff(fact_3032_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B) ) ).

% inf.orderI
tff(fact_3033_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% inf.orderI
tff(fact_3034_inf_OorderI,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B) ) ).

% inf.orderI
tff(fact_3035_inf_OorderI,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% inf.orderI
tff(fact_3036_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% inf.orderI
tff(fact_3037_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% inf.orderI
tff(fact_3038_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% inf.orderI
tff(fact_3039_inf__unique,axiom,
    ! [Fa: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(Fa,X3),Y3)),X3)
     => ( ! [X3: assn,Y3: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(Fa,X3),Y3)),Y3)
       => ( ! [X3: assn,Y3: assn,Z2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),Y3)
             => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),Z2)
               => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(Fa,Y3),Z2)) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3040_inf__unique,axiom,
    ! [Fa: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(Fa,X3),Y3)),X3)
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(Fa,X3),Y3)),Y3)
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y3)
             => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Z2)
               => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(Fa,Y3),Z2)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3041_inf__unique,axiom,
    ! [Fa: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,X3),Y3)),X3)
     => ( ! [X3: set_nat,Y3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,X3),Y3)),Y3)
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,Y3),Z2)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3042_inf__unique,axiom,
    ! [Fa: fun_fu959379683396315552_int_o,X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: fun_Pr3238374608701176162_int_o,Y3: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(Fa,X3),Y3)),X3)
     => ( ! [X3: fun_Pr3238374608701176162_int_o,Y3: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(Fa,X3),Y3)),Y3)
       => ( ! [X3: fun_Pr3238374608701176162_int_o,Y3: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] :
              ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X3),Y3)
             => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X3),Z2)
               => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X3),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(Fa,Y3),Z2)) ) )
         => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3043_inf__unique,axiom,
    ! [Fa: fun_se6573205548028396576et_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,X3),Y3)),X3)
     => ( ! [X3: set_int,Y3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,X3),Y3)),Y3)
       => ( ! [X3: set_int,Y3: set_int,Z2: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Y3)
             => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Z2)
               => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,Y3),Z2)) ) )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = aa_set_int_set_int(aa_set4234024024286713313et_int(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3044_inf__unique,axiom,
    ! [Fa: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(Fa,X3),Y3)),X3)
     => ( ! [X3: rat,Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(Fa,X3),Y3)),Y3)
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y3)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z2)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(Fa,Y3),Z2)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3045_inf__unique,axiom,
    ! [Fa: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(Fa,X3),Y3)),X3)
     => ( ! [X3: nat,Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(Fa,X3),Y3)),Y3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(Fa,Y3),Z2)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3046_inf__unique,axiom,
    ! [Fa: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(Fa,X3),Y3)),X3)
     => ( ! [X3: int,Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(Fa,X3),Y3)),Y3)
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),aa_int_int(aa_int_fun_int_int(Fa,Y3),Z2)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(Fa,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_3047_le__iff__inf,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3048_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3049_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3050_le__iff__inf,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3051_le__iff__inf,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3052_le__iff__inf,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3053_le__iff__inf,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3054_le__iff__inf,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_3055_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3056_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3057_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3058_inf_Oabsorb1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3059_inf_Oabsorb1,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3060_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3061_inf_Oabsorb1,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3062_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_3063_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3064_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3065_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3066_inf_Oabsorb2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),A)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3067_inf_Oabsorb2,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3068_inf_Oabsorb2,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(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3069_inf_Oabsorb2,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(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3070_inf_Oabsorb2,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(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_3071_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3072_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3073_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3074_inf__absorb1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3075_inf__absorb1,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3076_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3077_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3078_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_3079_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3080_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Y),X)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3081_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3082_inf__absorb2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,Y),X)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3083_inf__absorb2,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3084_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3085_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3086_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_3087_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2) ) ) ).

% inf.boundedE
tff(fact_3088_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2) ) ) ).

% inf.boundedE
tff(fact_3089_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2) ) ) ).

% inf.boundedE
tff(fact_3090_inf_OboundedE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C2))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C2) ) ) ).

% inf.boundedE
tff(fact_3091_inf_OboundedE,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C2))
     => ~ ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
         => ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2) ) ) ).

% inf.boundedE
tff(fact_3092_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2))
     => ~ ( 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,A),C2) ) ) ).

% inf.boundedE
tff(fact_3093_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))
     => ~ ( 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),C2) ) ) ).

% inf.boundedE
tff(fact_3094_inf_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))
     => ~ ( 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,A),C2) ) ) ).

% inf.boundedE
tff(fact_3095_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3096_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3097_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3098_inf_OboundedI,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C2)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3099_inf_OboundedI,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3100_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C2: 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,A),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3101_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3102_inf_OboundedI,axiom,
    ! [A: int,B: int,C2: 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,A),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2)) ) ) ).

% inf.boundedI
tff(fact_3103_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z3)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3104_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z3)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3105_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3106_inf__greatest,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Z3)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3107_inf__greatest,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Z3)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3108_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3109_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3110_inf__greatest,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z3)) ) ) ).

% inf_greatest
tff(fact_3111_inf_Oorder__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.order_iff
tff(fact_3112_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3113_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3114_inf_Oorder__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
    <=> ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ) ).

% inf.order_iff
tff(fact_3115_inf_Oorder__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( A = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) ) ) ).

% inf.order_iff
tff(fact_3116_inf_Oorder__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.order_iff
tff(fact_3117_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_3118_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_3119_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),A) ).

% inf.cobounded1
tff(fact_3120_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3121_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3122_inf_Ocobounded1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),A) ).

% inf.cobounded1
tff(fact_3123_inf_Ocobounded1,axiom,
    ! [A: set_int,B: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),A) ).

% inf.cobounded1
tff(fact_3124_inf_Ocobounded1,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(inf_inf_rat,A),B)),A) ).

% inf.cobounded1
tff(fact_3125_inf_Ocobounded1,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(inf_inf_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_3126_inf_Ocobounded1,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(inf_inf_int,A),B)),A) ).

% inf.cobounded1
tff(fact_3127_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) ).

% inf.cobounded2
tff(fact_3128_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3129_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3130_inf_Ocobounded2,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),B) ).

% inf.cobounded2
tff(fact_3131_inf_Ocobounded2,axiom,
    ! [A: set_int,B: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),B) ).

% inf.cobounded2
tff(fact_3132_inf_Ocobounded2,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(inf_inf_rat,A),B)),B) ).

% inf.cobounded2
tff(fact_3133_inf_Ocobounded2,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(inf_inf_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_3134_inf_Ocobounded2,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(inf_inf_int,A),B)),B) ).

% inf.cobounded2
tff(fact_3135_inf_Oabsorb__iff1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3136_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3137_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3138_inf_Oabsorb__iff1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3139_inf_Oabsorb__iff1,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3140_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3141_inf_Oabsorb__iff1,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3142_inf_Oabsorb__iff1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_3143_inf_Oabsorb__iff2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3144_inf_Oabsorb__iff2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3145_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3146_inf_Oabsorb__iff2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),A)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3147_inf_Oabsorb__iff2,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3148_inf_Oabsorb__iff2,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(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3149_inf_Oabsorb__iff2,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(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3150_inf_Oabsorb__iff2,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(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_3151_inf_OcoboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3152_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3153_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3154_inf_OcoboundedI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C2)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3155_inf_OcoboundedI1,axiom,
    ! [A: set_int,C2: set_int,B: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C2)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3156_inf_OcoboundedI1,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3157_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3158_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.coboundedI1
tff(fact_3159_inf_OcoboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3160_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3161_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3162_inf_OcoboundedI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),C2)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3163_inf_OcoboundedI2,axiom,
    ! [B: set_int,C2: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),C2)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3164_inf_OcoboundedI2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3165_inf_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3166_inf_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.coboundedI2
tff(fact_3167_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI1
tff(fact_3168_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI1
tff(fact_3169_less__infI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% less_infI1
tff(fact_3170_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI1
tff(fact_3171_less__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% less_infI1
tff(fact_3172_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI1
tff(fact_3173_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI1
tff(fact_3174_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI2
tff(fact_3175_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI2
tff(fact_3176_less__infI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% less_infI2
tff(fact_3177_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI2
tff(fact_3178_less__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% less_infI2
tff(fact_3179_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI2
tff(fact_3180_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI2
tff(fact_3181_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3182_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3183_inf_Oabsorb3,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3184_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3185_inf_Oabsorb3,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(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3186_inf_Oabsorb3,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(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3187_inf_Oabsorb3,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(inf_inf_int,A),B) = A ) ) ).

% inf.absorb3
tff(fact_3188_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3189_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3190_inf_Oabsorb4,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),A)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3191_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3192_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3193_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3194_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb4
tff(fact_3195_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3196_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3197_inf_Ostrict__boundedE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C2))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3198_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3199_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C2))
     => ~ ( 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),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3200_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C2))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3201_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C2))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% inf.strict_boundedE
tff(fact_3202_inf_Ostrict__order__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
    <=> ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3203_inf_Ostrict__order__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3204_inf_Ostrict__order__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
    <=> ( ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3205_inf_Ostrict__order__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3206_inf_Ostrict__order__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3207_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3208_inf_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_3209_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3210_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3211_inf_Ostrict__coboundedI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),C2)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3212_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3213_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3214_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3215_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.strict_coboundedI1
tff(fact_3216_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),C2)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3217_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3218_inf_Ostrict__coboundedI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,C2: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),C2)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3219_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3220_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3221_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3222_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C2) ) ).

% inf.strict_coboundedI2
tff(fact_3223_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y3),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y3)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Z2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3224_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y3),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Y3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X3),Z2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3225_distrib__imp1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: fun_Pr3238374608701176162_int_o,Y3: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X3),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y3),Z2)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X3),Y3)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X3),Z2))
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3226_distrib__imp1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y3),Z2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X3),Y3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X3),Z2))
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3227_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y3),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),Y3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X3),Z2))
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3228_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y3),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Z2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3229_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y3),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Z2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3230_distrib__imp1,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ! [X3: int,Y3: int,Z2: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y3),Z2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y3)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Z2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z3)) ) ) ).

% distrib_imp1
tff(fact_3231_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y3),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y3)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Z2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3232_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y3),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Y3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X3),Z2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3233_distrib__imp2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: fun_Pr3238374608701176162_int_o,Y3: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X3),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y3),Z2)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X3),Y3)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X3),Z2))
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3234_distrib__imp2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ! [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X3),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y3),Z2)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X3),Y3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X3),Z2))
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3235_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y3),Z2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Y3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X3),Z2))
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3236_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y3),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Z2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3237_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y3),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Z2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3238_distrib__imp2,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ! [X3: int,Y3: int,Z2: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y3),Z2)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y3)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Z2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z3)) ) ) ).

% distrib_imp2
tff(fact_3239_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3240_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3241_inf__sup__distrib1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3242_inf__sup__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3243_inf__sup__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3244_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3245_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3246_inf__sup__distrib1,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z3)) ).

% inf_sup_distrib1
tff(fact_3247_inf__sup__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3248_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3249_inf__sup__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3250_inf__sup__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3251_inf__sup__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3252_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3253_inf__sup__distrib2,axiom,
    ! [Y: nat,Z3: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3)),X) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3254_inf__sup__distrib2,axiom,
    ! [Y: int,Z3: int,X: int] : aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3)),X) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),X)),aa_int_int(aa_int_fun_int_int(inf_inf_int,Z3),X)) ).

% inf_sup_distrib2
tff(fact_3255_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3256_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3257_sup__inf__distrib1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3258_sup__inf__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3259_sup__inf__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3260_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3261_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3262_sup__inf__distrib1,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z3)) ).

% sup_inf_distrib1
tff(fact_3263_sup__inf__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3264_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3265_sup__inf__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3266_sup__inf__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z3)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3267_sup__inf__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z3)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3268_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3269_sup__inf__distrib2,axiom,
    ! [Y: nat,Z3: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z3)),X) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3270_sup__inf__distrib2,axiom,
    ! [Y: int,Z3: int,X: int] : aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z3)),X) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),X)),aa_int_int(aa_int_fun_int_int(sup_sup_int,Z3),X)) ).

% sup_inf_distrib2
tff(fact_3271_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z3)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3272_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z3)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3273_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z3)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3274_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z3)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3275_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z3)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3276_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z3)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_3277_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z3)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3278_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z3)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3279_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z3)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3280_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z3)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3281_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z3)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3282_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_3283_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z3),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3284_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z3),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3285_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Z3),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3286_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Z3),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3287_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z3),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3288_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z3),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_3289_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z3),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3290_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z3),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3291_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Z3),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3292_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z3)),X) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),X)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Z3),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3293_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z3)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z3),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3294_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z3),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_3295_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) ).

% add_divide_distrib
tff(fact_3296_disjointI,axiom,
    ! [A: set_se6260736226359567993nt_int,B: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),B) )
     => ( inf_in8396524679539076455nt_int(A,B) = bot_bo1488462491386950373nt_int ) ) ).

% disjointI
tff(fact_3297_disjointI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A)
         => ~ aa_set_set_nat_o(member_set_nat(X3),B) )
     => ( inf_inf_set_set_nat(A,B) = bot_bot_set_set_nat ) ) ).

% disjointI
tff(fact_3298_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
tff(fact_3299_disjointI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => ~ aa_set_o_o(member_o((X3)),B) )
     => ( inf_inf_set_o(A,B) = bot_bot_set_o ) ) ).

% disjointI
tff(fact_3300_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ~ aa_set_nat_o(member_nat(X3),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_3301_disjointI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ~ aa_set_int_o(member_int(X3),B) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_3302_disjoint__iff__not__equal,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
         => ! [Xa3: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),B4)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3303_disjoint__iff__not__equal,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A4)
         => ! [Xa3: $o] :
              ( aa_set_o_o(member_o((Xa3)),B4)
             => ( (X4)
              <=> ~ (Xa3) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3304_disjoint__iff__not__equal,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),B4)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3305_disjoint__iff__not__equal,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),B4)
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3306_Int__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_3307_Int__empty__right,axiom,
    ! [A4: set_o] : inf_inf_set_o(A4,bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_3308_Int__empty__right,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_3309_Int__empty__right,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_3310_Int__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B4) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_3311_Int__empty__left,axiom,
    ! [B4: set_o] : inf_inf_set_o(bot_bot_set_o,B4) = bot_bot_set_o ).

% Int_empty_left
tff(fact_3312_Int__empty__left,axiom,
    ! [B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B4) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_3313_Int__empty__left,axiom,
    ! [B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),B4) = bot_bot_set_int ).

% Int_empty_left
tff(fact_3314_disjoint__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ( inf_in8396524679539076455nt_int(A4,B4) = bot_bo1488462491386950373nt_int )
    <=> ! [X4: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),A4)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X4),B4) ) ) ).

% disjoint_iff
tff(fact_3315_disjoint__iff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ( inf_inf_set_set_nat(A4,B4) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X4),A4)
         => ~ aa_set_set_nat_o(member_set_nat(X4),B4) ) ) ).

% disjoint_iff
tff(fact_3316_disjoint__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B4) ) ) ).

% disjoint_iff
tff(fact_3317_disjoint__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
    <=> ! [X4: $o] :
          ( aa_set_o_o(member_o((X4)),A4)
         => ~ aa_set_o_o(member_o((X4)),B4) ) ) ).

% disjoint_iff
tff(fact_3318_disjoint__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ~ aa_set_nat_o(member_nat(X4),B4) ) ) ).

% disjoint_iff
tff(fact_3319_disjoint__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => ~ aa_set_int_o(member_int(X4),B4) ) ) ).

% disjoint_iff
tff(fact_3320_Int__emptyI,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),B4) )
     => ( inf_in8396524679539076455nt_int(A4,B4) = bot_bo1488462491386950373nt_int ) ) ).

% Int_emptyI
tff(fact_3321_Int__emptyI,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A4)
         => ~ aa_set_set_nat_o(member_set_nat(X3),B4) )
     => ( inf_inf_set_set_nat(A4,B4) = bot_bot_set_set_nat ) ) ).

% Int_emptyI
tff(fact_3322_Int__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),B4) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_3323_Int__emptyI,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => ~ aa_set_o_o(member_o((X3)),B4) )
     => ( inf_inf_set_o(A4,B4) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_3324_Int__emptyI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ~ aa_set_nat_o(member_nat(X3),B4) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_3325_Int__emptyI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ~ aa_set_int_o(member_int(X3),B4) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_3326_Int__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),C3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),D4)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),D4)) ) ) ).

% Int_mono
tff(fact_3327_Int__mono,axiom,
    ! [A4: set_nat,C3: set_nat,B4: set_nat,D4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),D4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),D4)) ) ) ).

% Int_mono
tff(fact_3328_Int__mono,axiom,
    ! [A4: set_int,C3: set_int,B4: set_int,D4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),C3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),D4)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,C3),D4)) ) ) ).

% Int_mono
tff(fact_3329_Int__lower1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),A4) ).

% Int_lower1
tff(fact_3330_Int__lower1,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),A4) ).

% Int_lower1
tff(fact_3331_Int__lower1,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)),A4) ).

% Int_lower1
tff(fact_3332_Int__lower2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),B4) ).

% Int_lower2
tff(fact_3333_Int__lower2,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),B4) ).

% Int_lower2
tff(fact_3334_Int__lower2,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)),B4) ).

% Int_lower2
tff(fact_3335_Int__absorb1,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_3336_Int__absorb1,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_3337_Int__absorb1,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_3338_Int__absorb2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = A4 ) ) ).

% Int_absorb2
tff(fact_3339_Int__absorb2,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = A4 ) ) ).

% Int_absorb2
tff(fact_3340_Int__absorb2,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = A4 ) ) ).

% Int_absorb2
tff(fact_3341_Int__greatest,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),B4)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ) ) ).

% Int_greatest
tff(fact_3342_Int__greatest,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),B4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ) ) ).

% Int_greatest
tff(fact_3343_Int__greatest,axiom,
    ! [C3: set_int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),B4)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) ) ) ).

% Int_greatest
tff(fact_3344_Int__Collect__mono,axiom,
    ! [A4: set_o,B4: set_o,Pa: fun_o_o,Qa: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
           => ( aa_o_o(Pa,(X3))
             => aa_o_o(Qa,(X3)) ) )
       => aa_set_o_o(ord_less_eq_set_o(inf_inf_set_o(A4,collect_o(Pa))),inf_inf_set_o(B4,collect_o(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3345_Int__Collect__mono,axiom,
    ! [A4: set_list_nat,B4: set_list_nat,Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A4,B4)
     => ( ! [X3: list_nat] :
            ( member_list_nat(X3,A4)
           => ( aa_list_nat_o(Pa,X3)
             => aa_list_nat_o(Qa,X3) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A4,collect_list_nat(Pa)),inf_inf_set_list_nat(B4,collect_list_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3346_Int__Collect__mono,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int,Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)
     => ( ! [X3: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
           => ( aa_set6170482117830564987_int_o(Pa,X3)
             => aa_set6170482117830564987_int_o(Qa,X3) ) )
       => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(inf_in8396524679539076455nt_int(A4,collec5210948495886036740nt_int(Pa))),inf_in8396524679539076455nt_int(B4,collec5210948495886036740nt_int(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3347_Int__Collect__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
     => ( ! [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A4)
           => ( aa_set_nat_o(Pa,X3)
             => aa_set_nat_o(Qa,X3) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(inf_inf_set_set_nat(A4,collect_set_nat(Pa))),inf_inf_set_set_nat(B4,collect_set_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3348_Int__Collect__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
           => ( aa_Pro6150637604445964947_nat_o(Pa,X3)
             => aa_Pro6150637604445964947_nat_o(Qa,X3) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),collec3392354462482085612at_nat(Pa))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),collec3392354462482085612at_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3349_Int__Collect__mono,axiom,
    ! [A4: set_nat,B4: set_nat,Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => ( aa_nat_o(Pa,X3)
             => aa_nat_o(Qa,X3) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),collect_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3350_Int__Collect__mono,axiom,
    ! [A4: set_int,B4: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => ( aa_int_o(Pa,X3)
             => aa_int_o(Qa,X3) ) )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),collect_int(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_3351_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),S4)
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),S4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),S4) )
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),S) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_3352_inter__eq__subsetI,axiom,
    ! [S: set_nat,S4: set_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),S4)
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),S4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),S4) )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_3353_inter__eq__subsetI,axiom,
    ! [S: set_int,S4: set_int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),S4)
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),S4) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),S4) )
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),S) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_3354_Un__Int__crazy,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),A4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C3),A4)) ).

% Un_Int_crazy
tff(fact_3355_Un__Int__crazy,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B4),C3))),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,C3),A4)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3))),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C3),A4)) ).

% Un_Int_crazy
tff(fact_3356_Un__Int__crazy,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C3),A4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C3),A4)) ).

% Un_Int_crazy
tff(fact_3357_Un__Int__crazy,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),A4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C3),A4)) ).

% Un_Int_crazy
tff(fact_3358_Int__Un__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C3)) ).

% Int_Un_distrib
tff(fact_3359_Int__Un__distrib,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),C3)) ).

% Int_Un_distrib
tff(fact_3360_Int__Un__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),C3)) ).

% Int_Un_distrib
tff(fact_3361_Int__Un__distrib,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C3)) ).

% Int_Un_distrib
tff(fact_3362_Un__Int__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),C3)) ).

% Un_Int_distrib
tff(fact_3363_Un__Int__distrib,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B4),C3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),C3)) ).

% Un_Int_distrib
tff(fact_3364_Un__Int__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),C3)) ).

% Un_Int_distrib
tff(fact_3365_Un__Int__distrib,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),C3)) ).

% Un_Int_distrib
tff(fact_3366_Int__Un__distrib2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)),A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),A4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),A4)) ).

% Int_Un_distrib2
tff(fact_3367_Int__Un__distrib2,axiom,
    ! [B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)),A4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B4),A4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,C3),A4)) ).

% Int_Un_distrib2
tff(fact_3368_Int__Un__distrib2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)),A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),A4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C3),A4)) ).

% Int_Un_distrib2
tff(fact_3369_Int__Un__distrib2,axiom,
    ! [B4: set_nat,C3: set_nat,A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)),A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),A4)) ).

% Int_Un_distrib2
tff(fact_3370_Un__Int__distrib2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)),A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),A4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C3),A4)) ).

% Un_Int_distrib2
tff(fact_3371_Un__Int__distrib2,axiom,
    ! [B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B4),C3)),A4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),A4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,C3),A4)) ).

% Un_Int_distrib2
tff(fact_3372_Un__Int__distrib2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3)),A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),A4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C3),A4)) ).

% Un_Int_distrib2
tff(fact_3373_Un__Int__distrib2,axiom,
    ! [B4: set_nat,C3: set_nat,A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)),A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C3),A4)) ).

% Un_Int_distrib2
tff(fact_3374_inf__option__def,axiom,
    ! [X: option_assn,Y: option_assn] : inf_inf_option_assn(X,Y) = case_o4484465799723439917n_assn(none_assn,aTP_Lamm_gh(Y),X) ).

% inf_option_def
tff(fact_3375_inf__option__def,axiom,
    ! [X: option8963830502488799655at_nat,Y: option8963830502488799655at_nat] : inf_in777885744494410645at_nat(X,Y) = case_o311359030874850053at_nat(none_s625347054029921090at_nat,aTP_Lamm_gj(Y),X) ).

% inf_option_def
tff(fact_3376_inf__option__def,axiom,
    ! [X: option_set_nat,Y: option_set_nat] : inf_in7812914138253463912et_nat(X,Y) = case_o4054078431260844265et_nat(none_set_nat,aTP_Lamm_gl(Y),X) ).

% inf_option_def
tff(fact_3377_inf__option__def,axiom,
    ! [X: option_nat,Y: option_nat] : inf_inf_option_nat(X,Y) = case_o7429725398727453821at_nat(none_nat,aTP_Lamm_gn(Y),X) ).

% inf_option_def
tff(fact_3378_inf__option__def,axiom,
    ! [X: option2108551362046372146_int_o,Y: option2108551362046372146_int_o] : inf_in3196829186289650628_int_o(X,Y) = case_o7864645398209642145_int_o(none_f6270875889056561217_int_o,aTP_Lamm_gp(Y),X) ).

% inf_option_def
tff(fact_3379_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z3: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z3))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z3))) ).

% distrib_sup_le
tff(fact_3380_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z3))) ).

% distrib_sup_le
tff(fact_3381_distrib__sup__le,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z3))),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z3))) ).

% distrib_sup_le
tff(fact_3382_distrib__sup__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,Y),Z3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Z3))) ).

% distrib_sup_le
tff(fact_3383_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z3))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z3))) ).

% distrib_sup_le
tff(fact_3384_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z3))) ).

% distrib_sup_le
tff(fact_3385_distrib__sup__le,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),Z3))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Z3))) ).

% distrib_sup_le
tff(fact_3386_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z3))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Z3))) ).

% distrib_sup_le
tff(fact_3387_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z3))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z3))) ).

% distrib_sup_le
tff(fact_3388_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z3))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z3))) ).

% distrib_sup_le
tff(fact_3389_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z3: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z3))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3))) ).

% distrib_inf_le
tff(fact_3390_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z3))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3))) ).

% distrib_inf_le
tff(fact_3391_distrib__inf__le,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z3))),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3))) ).

% distrib_inf_le
tff(fact_3392_distrib__inf__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Z3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3))) ).

% distrib_inf_le
tff(fact_3393_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z3))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3))) ).

% distrib_inf_le
tff(fact_3394_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z3))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3))) ).

% distrib_inf_le
tff(fact_3395_distrib__inf__le,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Z3))),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z3))) ).

% distrib_inf_le
tff(fact_3396_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Z3))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z3))) ).

% distrib_inf_le
tff(fact_3397_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z3))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z3))) ).

% distrib_inf_le
tff(fact_3398_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z3))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z3))) ).

% distrib_inf_le
tff(fact_3399_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N2: 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),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ) ) ).

% power_mono
tff(fact_3400_power__mono,axiom,
    ! [A: rat,B: rat,N2: 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),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ).

% power_mono
tff(fact_3401_power__mono,axiom,
    ! [A: nat,B: nat,N2: 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)) ) ) ).

% power_mono
tff(fact_3402_power__mono,axiom,
    ! [A: int,B: int,N2: 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ).

% power_mono
tff(fact_3403_zero__le__power,axiom,
    ! [A: code_integer,N2: 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),N2)) ) ).

% zero_le_power
tff(fact_3404_zero__le__power,axiom,
    ! [A: rat,N2: 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),N2)) ) ).

% zero_le_power
tff(fact_3405_zero__le__power,axiom,
    ! [A: nat,N2: 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),N2)) ) ).

% zero_le_power
tff(fact_3406_zero__le__power,axiom,
    ! [A: int,N2: 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),N2)) ) ).

% zero_le_power
tff(fact_3407_zero__less__power,axiom,
    ! [A: code_integer,N2: 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),N2)) ) ).

% zero_less_power
tff(fact_3408_zero__less__power,axiom,
    ! [A: rat,N2: 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),N2)) ) ).

% zero_less_power
tff(fact_3409_zero__less__power,axiom,
    ! [A: nat,N2: 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),N2)) ) ).

% zero_less_power
tff(fact_3410_zero__less__power,axiom,
    ! [A: int,N2: 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),N2)) ) ).

% zero_less_power
tff(fact_3411_one__le__power,axiom,
    ! [A: code_integer,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3412_one__le__power,axiom,
    ! [A: rat,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3413_one__le__power,axiom,
    ! [A: nat,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3414_one__le__power,axiom,
    ! [A: int,N2: 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),N2)) ) ).

% one_le_power
tff(fact_3415_divide__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(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_3416_divide__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) ) ) ).

% divide_right_mono
tff(fact_3417_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3418_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3419_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3420_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3421_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3422_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C2: 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,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_3423_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_neg_neg
tff(fact_3424_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_3425_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_3426_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_pos_pos
tff(fact_3427_divide__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(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_3428_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3429_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_3430_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_3431_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: 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,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3432_power__add,axiom,
    ! [A: assn,M: nat,N2: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M)),aa_nat_assn(power_power_assn(A),N2)) ).

% power_add
tff(fact_3433_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = 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),N2)) ).

% power_add
tff(fact_3434_power__add,axiom,
    ! [A: nat,M: nat,N2: 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),N2)) = 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),N2)) ).

% power_add
tff(fact_3435_power__add,axiom,
    ! [A: int,M: nat,N2: 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),N2)) = 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),N2)) ).

% power_add
tff(fact_3436_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% nat_power_less_imp_less
tff(fact_3437_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),A2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),B2)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_3438_disjoint__mono,axiom,
    ! [A: set_o,A2: set_o,B: set_o,B2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),A2)
     => ( aa_set_o_o(ord_less_eq_set_o(B),B2)
       => ( ( inf_inf_set_o(A2,B2) = bot_bot_set_o )
         => ( inf_inf_set_o(A,B) = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
tff(fact_3439_disjoint__mono,axiom,
    ! [A: set_nat,A2: set_nat,B: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_3440_disjoint__mono,axiom,
    ! [A: set_int,A2: set_int,B: set_int,B2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),A2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),B2)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
tff(fact_3441_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)) )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),A4) ) ).

% Un_Int_assoc_eq
tff(fact_3442_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4)),C3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) )
    <=> ord_le8081472938463900775at_nat(C3,A4) ) ).

% Un_Int_assoc_eq
tff(fact_3443_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),C3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) )
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C3),A4) ) ).

% Un_Int_assoc_eq
tff(fact_3444_Un__Int__assoc__eq,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),A4) ) ).

% Un_Int_assoc_eq
tff(fact_3445_Un__Int__assoc__eq,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)),C3) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B4),C3)) )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),A4) ) ).

% Un_Int_assoc_eq
tff(fact_3446_power__less__imp__less__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))
     => ( 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_3447_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))
     => ( 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_3448_power__less__imp__less__base,axiom,
    ! [A: nat,N2: 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))
     => ( 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_3449_power__less__imp__less__base,axiom,
    ! [A: int,N2: 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2))
     => ( 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_3450_power__le__one,axiom,
    ! [A: code_integer,N2: 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),N2)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_3451_power__le__one,axiom,
    ! [A: rat,N2: 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),N2)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3452_power__le__one,axiom,
    ! [A: nat,N2: 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),N2)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3453_power__le__one,axiom,
    ! [A: int,N2: 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),N2)),one_one_int) ) ) ).

% power_le_one
tff(fact_3454_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3455_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3456_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3457_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3458_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => 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,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3459_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Z3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_3460_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_3461_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_3462_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_3463_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_3464_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_3465_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_3466_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_3467_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_3468_power__gt1__lemma,axiom,
    ! [A: code_integer,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3469_power__gt1__lemma,axiom,
    ! [A: rat,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3470_power__gt1__lemma,axiom,
    ! [A: nat,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3471_power__gt1__lemma,axiom,
    ! [A: int,N2: 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),N2))) ) ).

% power_gt1_lemma
tff(fact_3472_power__less__power__Suc,axiom,
    ! [A: code_integer,N2: 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),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))) ) ).

% power_less_power_Suc
tff(fact_3473_power__less__power__Suc,axiom,
    ! [A: rat,N2: 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),N2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),N2))) ) ).

% power_less_power_Suc
tff(fact_3474_power__less__power__Suc,axiom,
    ! [A: nat,N2: 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ).

% power_less_power_Suc
tff(fact_3475_power__less__power__Suc,axiom,
    ! [A: int,N2: 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),N2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2))) ) ).

% power_less_power_Suc
tff(fact_3476_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3477_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3478_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3479_divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3480_less__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3481_neg__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3482_neg__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_3483_pos__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),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),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_3484_pos__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3485_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z3) ) ) ).

% mult_imp_div_pos_less
tff(fact_3486_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3487_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: 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),C2)
       => ( 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,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3488_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3489_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3490_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3491_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z3: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z3)),B) = $ite(Z3 = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(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),Z3))),Z3)) ).

% add_divide_eq_if_simps(2)
tff(fact_3492_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z3: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z3)) = $ite(Z3 = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(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),Z3)),B)),Z3)) ).

% add_divide_eq_if_simps(1)
tff(fact_3493_add__frac__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z3)) = aa_rat_rat(aa_rat_fun_rat_rat(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,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3)) ) ) ) ).

% add_frac_eq
tff(fact_3494_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z3) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3495_add__num__frac,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3496_add__divide__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z3)) = aa_rat_rat(aa_rat_fun_rat_rat(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,X),Z3)),Y)),Z3) ) ) ).

% add_divide_eq_iff
tff(fact_3497_divide__add__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z3)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3))),Z3) ) ) ).

% divide_add_eq_iff
tff(fact_3498_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3499_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3500_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3501_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_3502_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3503_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_rat(power_power_rat(A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3504_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3505_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N7)) ) ) ).

% power_strict_increasing
tff(fact_3506_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N7)) ) ) ).

% power_increasing
tff(fact_3507_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_rat(power_power_rat(A),N7)) ) ) ).

% power_increasing
tff(fact_3508_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N7)) ) ) ).

% power_increasing
tff(fact_3509_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N7)) ) ) ).

% power_increasing
tff(fact_3510_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N2) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_3511_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N2) = zero_zero_rat ) ) ).

% zero_power
tff(fact_3512_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N2) = zero_zero_nat ) ) ).

% zero_power
tff(fact_3513_zero__power,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),N2) = zero_zero_int ) ) ).

% zero_power
tff(fact_3514_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_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_3515_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_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))) ) ).

% less_half_sum
tff(fact_3516_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) ) ) ).

% nat_mult_div_cancel1
tff(fact_3517_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6529551498584149819at_nat(inf_in2572325071724192079at_nat,ord_le3146513528884898305at_nat,ord_le7866589430770878221at_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3518_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat(inf_inf_set_nat,ord_less_eq_set_nat,ord_less_set_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3519_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8142664068130256392_int_o(inf_in1782598092406300404_int_o,ord_le14110778710936002_int_o,ord_le7515035663206590902_int_o) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3520_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn(inf_inf_assn,ord_less_eq_assn,ord_less_assn) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3521_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8154731777541915528et_int(inf_inf_set_int,ord_less_eq_set_int,ord_less_set_int) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3522_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5374021519246970238et_rat(inf_inf_rat,ord_less_eq_rat,ord_less_rat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3523_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat(inf_inf_nat,ord_less_eq_nat,ord_less_nat) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3524_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int(inf_inf_int,ord_less_eq_int,ord_less_int) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_3525_power__Suc__less,axiom,
    ! [A: code_integer,N2: 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),N2))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% power_Suc_less
tff(fact_3526_power__Suc__less,axiom,
    ! [A: rat,N2: 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),N2))),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_Suc_less
tff(fact_3527_power__Suc__less,axiom,
    ! [A: nat,N2: 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),N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_Suc_less
tff(fact_3528_power__Suc__less,axiom,
    ! [A: int,N2: 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),N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ).

% power_Suc_less
tff(fact_3529_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: 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,C2),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,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3530_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3531_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z3) ) ) ).

% mult_imp_div_pos_le
tff(fact_3532_pos__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3533_pos__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),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),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_3534_neg__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
      <=> 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),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_3535_neg__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),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),C2)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3536_divide__left__mono,axiom,
    ! [B: rat,A: rat,C2: 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),C2)
       => ( 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,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C2),B)) ) ) ) ).

% divide_left_mono
tff(fact_3537_le__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3538_divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3539_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3540_divide__le__eq__1,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(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_3541_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3542_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3543_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3544_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N7)
     => ( 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),N7)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_3545_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3546_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3547_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3548_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N7)
     => ( 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),N7)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ) ).

% power_decreasing
tff(fact_3549_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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),N2) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3550_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),N2) = aa_nat_rat(power_power_rat(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3551_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3552_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3553_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),N2) = aa_nat_Code_integer(power_8256067586552552935nteger(B),N2) )
     => ( 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),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3554_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N2) = aa_nat_rat(power_power_rat(B),N2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3555_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3556_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3557_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3558_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3559_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3560_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: 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),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_3561_self__le__power,axiom,
    ! [A: code_integer,N2: 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),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% self_le_power
tff(fact_3562_self__le__power,axiom,
    ! [A: rat,N2: 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),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% self_le_power
tff(fact_3563_self__le__power,axiom,
    ! [A: nat,N2: 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),N2)
       => 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),N2)) ) ) ).

% self_le_power
tff(fact_3564_self__le__power,axiom,
    ! [A: int,N2: 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),N2)
       => 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),N2)) ) ) ).

% self_le_power
tff(fact_3565_one__less__power,axiom,
    ! [A: code_integer,N2: 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),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ).

% one_less_power
tff(fact_3566_one__less__power,axiom,
    ! [A: rat,N2: 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),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% one_less_power
tff(fact_3567_one__less__power,axiom,
    ! [A: nat,N2: 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),N2)
       => 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),N2)) ) ) ).

% one_less_power
tff(fact_3568_one__less__power,axiom,
    ! [A: int,N2: 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),N2)
       => 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),N2)) ) ) ).

% one_less_power
tff(fact_3569_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C2),B))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self1
tff(fact_3570_div__mult__self1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_3571_div__mult__self1,axiom,
    ! [B: int,A: int,C2: 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,C2),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_3572_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(C2,B)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C2),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self1
tff(fact_3573_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2))),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self2
tff(fact_3574_div__mult__self2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_3575_div__mult__self2,axiom,
    ! [B: int,A: int,C2: 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),C2))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_3576_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(B,C2)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C2),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self2
tff(fact_3577_div__mult__self3,axiom,
    ! [B: code_integer,C2: 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,C2),B)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self3
tff(fact_3578_div__mult__self3,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_3579_div__mult__self3,axiom,
    ! [B: int,C2: 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,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_3580_div__mult__self3,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(C2,B)),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C2),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self3
tff(fact_3581_div__mult__self4,axiom,
    ! [B: code_integer,C2: 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),C2)),A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),B)) ) ) ).

% div_mult_self4
tff(fact_3582_div__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_3583_div__mult__self4,axiom,
    ! [B: int,C2: 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),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_3584_div__mult__self4,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,C2)),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C2),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self4
tff(fact_3585_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),N2) = M ) ) ).

% div_mult_self_is_m
tff(fact_3586_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M)),N2) = M ) ) ).

% div_mult_self1_is_m
tff(fact_3587_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat ) ) ).

% div_less
tff(fact_3588_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)))) ) ).

% dividend_less_times_div
tff(fact_3589_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),N2))) ) ).

% dividend_less_div_times
tff(fact_3590_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A)
        & (B) )) ).

% merge_pure_and
tff(fact_3591_mod__h__bot__iff_I6_J,axiom,
    ! [Pa: assn,Qa: assn,Ha: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(6)
tff(fact_3592_inf__Int__eq2,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,X5: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(inf_in8908015913983109984_int_o(aTP_Lamm_cl(Ra),aTP_Lamm_cl(S)),X5),Xa2)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa2)),inf_in2269163501485487111nt_int(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_3593_inf__Int__eq2,axiom,
    ! [Ra: set_Pr1312406793078187350et_nat,S: set_Pr1312406793078187350et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(inf_in3476382694143140117_nat_o(aTP_Lamm_cm(Ra),aTP_Lamm_cm(S)),X5),Xa2)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X5),Xa2),inf_in4791168559842171624et_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_3594_inf__Int__eq2,axiom,
    ! [Ra: set_Pr4040394263813967220et_nat,S: set_Pr4040394263813967220et_nat,X5: fun_Pr6460944699146269935_nat_o,Xa2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(inf_in6333540121576287351_nat_o(aTP_Lamm_cn(Ra),aTP_Lamm_cn(S)),X5),Xa2)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X5),Xa2),inf_in6821596156192049670et_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_3595_inf__Int__eq2,axiom,
    ! [Ra: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat,X5: b,Xa2: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(inf_in692675228589143739_nat_o(aTP_Lamm_co(Ra),aTP_Lamm_co(S)),X5),Xa2)
    <=> member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X5),Xa2),inf_in3995795194662975554it_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_3596_inf__Int__eq2,axiom,
    ! [Ra: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat,X5: a,Xa2: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(inf_in2619332623043422460_nat_o(aTP_Lamm_cp(Ra),aTP_Lamm_cp(S)),X5),Xa2)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X5),Xa2),inf_in3493107508024165057it_nat(Ra,S)) ) ).

% inf_Int_eq2
tff(fact_3597_inf__Int__eq2,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aTP_Lamm_ef(Ra),aTP_Lamm_ef(S)),X5),Xa2)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ra),S)) ) ).

% inf_Int_eq2
tff(fact_3598_mod__and__dist,axiom,
    ! [Pa: assn,Qa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
        & aa_Pro653082453696054616_nat_o(rep_assn(Qa),Ha) ) ) ).

% mod_and_dist
tff(fact_3599_ent__conjI,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( entails(A4,B4)
     => ( entails(A4,C3)
       => entails(A4,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B4),C3)) ) ) ).

% ent_conjI
tff(fact_3600_ent__conjE1,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( entails(A4,C3)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A4),B4),C3) ) ).

% ent_conjE1
tff(fact_3601_ent__conjE2,axiom,
    ! [B4: assn,C3: assn,A4: assn] :
      ( entails(B4,C3)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A4),B4),C3) ) ).

% ent_conjE2
tff(fact_3602_inf__Int__eq,axiom,
    ! [Ra: set_o,S: set_o,X5: $o] :
      ( aa_o_o(inf_inf_fun_o_o(aTP_Lamm_a(Ra),aTP_Lamm_a(S)),(X5))
    <=> aa_set_o_o(member_o((X5)),inf_inf_set_o(Ra,S)) ) ).

% inf_Int_eq
tff(fact_3603_inf__Int__eq,axiom,
    ! [Ra: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int,X5: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(inf_in9107477634626221460_int_o(aTP_Lamm_ab(Ra),aTP_Lamm_ab(S)),X5)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X5),inf_in8396524679539076455nt_int(Ra,S)) ) ).

% inf_Int_eq
tff(fact_3604_inf__Int__eq,axiom,
    ! [Ra: set_set_nat,S: set_set_nat,X5: set_nat] :
      ( aa_set_nat_o(inf_in7602907407060649817_nat_o(aTP_Lamm_ac(Ra),aTP_Lamm_ac(S)),X5)
    <=> aa_set_set_nat_o(member_set_nat(X5),inf_inf_set_set_nat(Ra,S)) ) ).

% inf_Int_eq
tff(fact_3605_inf__Int__eq,axiom,
    ! [Ra: set_int,S: set_int,X5: int] :
      ( aa_int_o(inf_inf_fun_int_o(aTP_Lamm_ae(Ra),aTP_Lamm_ae(S)),X5)
    <=> aa_set_int_o(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Ra),S)) ) ).

% inf_Int_eq
tff(fact_3606_inf__Int__eq,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_gq(Ra)),aTP_Lamm_gq(S)),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Ra),S)) ) ).

% inf_Int_eq
tff(fact_3607_inf__Int__eq,axiom,
    ! [Ra: set_nat,S: set_nat,X5: nat] :
      ( aa_nat_o(inf_inf_fun_nat_o(aTP_Lamm_ad(Ra),aTP_Lamm_ad(S)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Ra),S)) ) ).

% inf_Int_eq
tff(fact_3608_inf__Int__eq,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aTP_Lamm_gr(Ra)),aTP_Lamm_gr(S)),X5)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X5),inf_in2269163501485487111nt_int(Ra,S)) ) ).

% inf_Int_eq
tff(fact_3609_inf__set__def,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(A4,B4) = collect_o(inf_inf_fun_o_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% inf_set_def
tff(fact_3610_inf__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : inf_inf_set_list_nat(A4,B4) = collect_list_nat(inf_in1107944134336960767_nat_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ).

% inf_set_def
tff(fact_3611_inf__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] : inf_in8396524679539076455nt_int(A4,B4) = collec5210948495886036740nt_int(inf_in9107477634626221460_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% inf_set_def
tff(fact_3612_inf__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : inf_inf_set_set_nat(A4,B4) = collect_set_nat(inf_in7602907407060649817_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% inf_set_def
tff(fact_3613_inf__set__def,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = collect_int(inf_inf_fun_int_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% inf_set_def
tff(fact_3614_inf__set__def,axiom,
    ! [A4: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : inf_in2269163501485487111nt_int(A4,B4) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aTP_Lamm_gr(A4)),aTP_Lamm_gr(B4))) ).

% inf_set_def
tff(fact_3615_inf__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(inf_in3341167027036195756_nat_o(aTP_Lamm_gq(A4)),aTP_Lamm_gq(B4))) ).

% inf_set_def
tff(fact_3616_inf__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = collect_nat(inf_inf_fun_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% inf_set_def
tff(fact_3617_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),M) ).

% div_le_dividend
tff(fact_3618_div__le__mono,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),K2)) ) ).

% div_le_mono
tff(fact_3619_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3620_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),I2) ) ).

% less_mult_imp_div_less
tff(fact_3621_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),N2)),M) ).

% div_times_less_eq_dividend
tff(fact_3622_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))),M) ).

% times_div_less_eq_dividend
tff(fact_3623_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K2: 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),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),M)) ) ) ).

% div_le_mono2
tff(fact_3624_div__greater__zero__iff,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% div_greater_zero_iff
tff(fact_3625_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),N2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ) ) ).

% div_less_iff_less_mult
tff(fact_3626_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: 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),N2) = M )
      <=> ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3627_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N2)
     => ( 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),N2)),M) ) ) ).

% div_less_dividend
tff(fact_3628_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_3629_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_3630_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_3631_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_3632_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_3633_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_3634_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_3635_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_3636_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),Q4))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),N2) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3637_split__div,axiom,
    ! [Pa: fun_nat_o,M: nat,N2: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))
    <=> ( ( ( N2 = zero_zero_nat )
         => aa_nat_o(Pa,zero_zero_nat) )
        & ( ( N2 != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),N2)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I)),J) )
               => aa_nat_o(Pa,I) ) ) ) ) ) ).

% split_div
tff(fact_3638_wand__assnI,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat,Qa: assn,Ra: assn] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
     => ( ! [H3: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Asa),As5) = bot_bot_set_nat )
           => ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H3)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,Asa))
               => ( aa_Pro653082453696054616_nat_o(rep_assn(Qa),produc7507926704131184380et_nat(H3,As5))
                 => aa_Pro653082453696054616_nat_o(rep_assn(Ra),produc7507926704131184380et_nat(H3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As5))) ) ) ) )
       => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Qa),Ra)),produc7507926704131184380et_nat(Ha,Asa)) ) ) ).

% wand_assnI
tff(fact_3639_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ? [As12: set_nat,As23: set_nat] :
                ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As12))
                & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_3640_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ~ ? [As13: set_nat,As24: set_nat] :
                  ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                  & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                  & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As13))
                  & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As24)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_3641_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( (Y)
            <=> ~ ? [As1: set_nat,As22: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As1))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As22)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_3642_times__assn__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Pa),Qa),produc7507926704131184380et_nat(Ha,Asa))
    <=> ? [As1: set_nat,As22: set_nat] :
          ( ( Asa = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(Ha,As1))
          & aa_Pro653082453696054616_nat_o(Qa,produc7507926704131184380et_nat(Ha,As22)) ) ) ).

% times_assn_raw.simps
tff(fact_3643_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As))))
               => ? [As12: set_nat,As23: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As12))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_3644_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As))))
               => ~ ? [As13: set_nat,As24: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As13))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As24)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_3645_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(X),Xa),Xb)
      <=> (Y) )
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( ( (Y)
                <=> ? [As1: set_nat,As22: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H,As1))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As22)) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_3646_in__range__empty,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ).

% in_range_empty
tff(fact_3647_in__range__dist__union,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat,As4: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As4)))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,As4)) ) ) ).

% in_range_dist_union
tff(fact_3648_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_3649_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% less_eq_assn_def
tff(fact_3650_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
tff(fact_3651_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(Uu),Uv) = product_Unity ).

% sup_unit_def
tff(fact_3652_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
tff(fact_3653_models__in__range,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
     => aa_Pro653082453696054616_nat_o(in_range,Ha) ) ).

% models_in_range
tff(fact_3654_relH__in__rangeI_I2_J,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H_a,Asa)) ) ).

% relH_in_rangeI(2)
tff(fact_3655_relH__in__rangeI_I1_J,axiom,
    ! [Asa: set_nat,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(relH(Asa,Ha),H_a)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa)) ) ).

% relH_in_rangeI(1)
tff(fact_3656_relH__refl,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
     => aa_hea7994211679605476167unit_o(relH(Asa,Ha),Ha) ) ).

% relH_refl
tff(fact_3657_in__range__subset,axiom,
    ! [Asa: set_nat,As4: set_nat,Ha: heap_e7401611519738050253t_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Asa),As4)
     => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,As4))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa)) ) ) ).

% in_range_subset
tff(fact_3658_wait__def,axiom,
    ! [N2: nat] : heap_Time_wait(N2) = heap_T6183433275982383450t_unit(aTP_Lamm_gs(N2)) ).

% wait_def
tff(fact_3659_in__range_Osimps,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),Asa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(Ha)) ) ) ).

% in_range.simps
tff(fact_3660_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( (Y)
            <=> ~ ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),As)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_3661_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ~ ! [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),As)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),lim_Product_unit(H)) ) ) ) ).

% in_range.elims(2)
tff(fact_3662_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),As)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H)) ) ) ) ).

% in_range.elims(3)
tff(fact_3663_wand__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(Pa),Qa),produc7507926704131184380et_nat(Ha,Asa))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(Ha,Asa))
        & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Asa),As6) = bot_bot_set_nat )
              & aa_hea7994211679605476167unit_o(relH(Asa,Ha),H5)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,Asa))
              & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H5,As6)) )
           => aa_Pro653082453696054616_nat_o(Qa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Asa),As6))) ) ) ) ).

% wand_raw.simps
tff(fact_3664_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( (Y)
            <=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As))
                  & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                        & aa_hea7994211679605476167unit_o(relH(As,H),H5)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H5,As6)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_3665_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As))
                & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As7) = bot_bot_set_nat )
                      & aa_hea7994211679605476167unit_o(relH(As,H),H6)
                      & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As7)) )
                   => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As7))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_3666_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As))
              & ! [H3: heap_e7401611519738050253t_unit,As5: set_nat] :
                  ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                    & aa_hea7994211679605476167unit_o(relH(As,H),H3)
                    & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As5)) )
                 => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_3667_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
      <=> (Y) )
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( ( (Y)
                <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As))
                    & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                          & aa_hea7994211679605476167unit_o(relH(As,H),H5)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H5,As6)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_3668_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As))))
               => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As))
                    & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As7) = bot_bot_set_nat )
                          & aa_hea7994211679605476167unit_o(relH(As,H),H6)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As7)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As7))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_3669_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(X),Xa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xa),produc7507926704131184380et_nat(H,As))))
               => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As))
                  & ! [H3: heap_e7401611519738050253t_unit,As5: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                        & aa_hea7994211679605476167unit_o(relH(As,H),H3)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As5)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_3670_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As) )
             => ( ( (Y)
                <=> ! [X4: nat] :
                      ( aa_set_nat_o(member_nat(X4),As)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H)) ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H,As)) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_3671_less__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ( A != B ) ) ) ).

% less_assn_def
tff(fact_3672_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H,As))
               => ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),As)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H)) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_3673_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H,As))
               => ~ ! [X5: nat] :
                      ( aa_set_nat_o(member_nat(X5),As)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),lim_Product_unit(H)) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_3674_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
tff(fact_3675_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),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(aa_nat_fun_nat_nat(divide_divide_nat,A4),N2)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),N2) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),N2)) ) ) ).

% verit_le_mono_div
tff(fact_3676_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: 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,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3677_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: 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,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3678_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: 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,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3679_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,X)),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3680_wand__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Qa) = abs_assn(aa_fun2638059832795814268_nat_o(wand_raw(rep_assn(Pa)),rep_assn(Qa))) ).

% wand_assn_def
tff(fact_3681_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C2: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),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),C2)) = 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),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3682_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3683_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C2: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),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),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3684_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C2: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),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),C2)) = 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),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3685_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3686_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),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),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3687_distrib__left__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: rat,B: rat,C2: rat] :
      ( aa_rat_o(nO_MAT3836730678284324717al_rat(divide5121882707175180666atural(X,Y)),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),C2)) = 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),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3688_distrib__left__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: nat,B: nat,C2: nat] :
      ( aa_nat_o(nO_MAT4471860738370820453al_nat(divide5121882707175180666atural(X,Y)),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),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3689_distrib__left__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: int,B: int,C2: int] :
      ( aa_int_o(nO_MAT4469370267861770177al_int(divide5121882707175180666atural(X,Y)),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),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3690_distrib__left__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(nO_MATCH_rat_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),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),C2)) = 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),C2)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_3691_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_3692_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_3693_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_3694_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_3695_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_3696_mod__add__self1,axiom,
    ! [B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_add_self1
tff(fact_3697_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_3698_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_3699_mod__add__self2,axiom,
    ! [A: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),B) = modulo8411746178871703098atural(A,B) ).

% mod_add_self2
tff(fact_3700_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = M ) ) ).

% mod_less
tff(fact_3701_Rep__assn__inverse,axiom,
    ! [X: assn] : abs_assn(rep_assn(X)) = X ).

% Rep_assn_inverse
tff(fact_3702_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_3703_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_3704_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_3705_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_3706_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_3707_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_3708_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_3709_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_3710_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3711_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3712_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3713_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3714_mod__mult__self1,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_3715_mod__mult__self1,axiom,
    ! [A: int,C2: 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,C2),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_3716_mod__mult__self1,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(C2,B)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self1
tff(fact_3717_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_3718_mod__mult__self2,axiom,
    ! [A: int,B: int,C2: 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),C2))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_3719_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(B,C2)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self2
tff(fact_3720_mod__mult__self3,axiom,
    ! [C2: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_3721_mod__mult__self3,axiom,
    ! [C2: 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,C2),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_3722_mod__mult__self3,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(C2,B)),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self3
tff(fact_3723_mod__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_3724_mod__mult__self4,axiom,
    ! [B: int,C2: 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),C2)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_3725_mod__mult__self4,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,C2)),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self4
tff(fact_3726_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3727_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3728_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3729_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_iff
tff(fact_3730_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3731_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3732_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3733_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% of_nat_le_iff
tff(fact_3734_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_add
tff(fact_3735_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_add
tff(fact_3736_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_add
tff(fact_3737_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_add
tff(fact_3738_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_mult
tff(fact_3739_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_mult
tff(fact_3740_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_mult
tff(fact_3741_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_mult
tff(fact_3742_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N2) = one_one_rat )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3743_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N2) = one_one_int )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3744_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N2) = one_one_nat )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3745_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N2) = one_one_Code_integer )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3746_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3747_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3748_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3749_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3750_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3751_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3752_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3753_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_3754_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_3755_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_3756_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_3757_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_3758_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_3759_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_3760_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_3761_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% of_nat_0_less_iff
tff(fact_3762_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3763_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3764_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3765_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3766_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3767_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3768_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3769_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3770_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3771_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3772_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3773_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3774_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3775_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3776_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3777_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3778_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_right_eq
tff(fact_3779_mod__add__right__eq,axiom,
    ! [A: int,B: int,C2: 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),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_right_eq
tff(fact_3780_mod__add__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),modulo8411746178871703098atural(B,C2)),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) ).

% mod_add_right_eq
tff(fact_3781_mod__add__left__eq,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_left_eq
tff(fact_3782_mod__add__left__eq,axiom,
    ! [A: int,C2: 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),C2)),B)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_left_eq
tff(fact_3783_mod__add__left__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C2)),B),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) ).

% mod_add_left_eq
tff(fact_3784_mod__add__cong,axiom,
    ! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ) ) ) ).

% mod_add_cong
tff(fact_3785_mod__add__cong,axiom,
    ! [A: int,C2: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C2) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C2) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C2) ) ) ) ).

% mod_add_cong
tff(fact_3786_mod__add__cong,axiom,
    ! [A: code_natural,C2: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural(A,C2) = modulo8411746178871703098atural(A2,C2) )
     => ( ( modulo8411746178871703098atural(B,C2) = modulo8411746178871703098atural(B2,C2) )
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A2),B2),C2) ) ) ) ).

% mod_add_cong
tff(fact_3787_mod__add__eq,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) ).

% mod_add_eq
tff(fact_3788_mod__add__eq,axiom,
    ! [A: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% mod_add_eq
tff(fact_3789_mod__add__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C2)),modulo8411746178871703098atural(B,C2)),C2) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) ).

% mod_add_eq
tff(fact_3790_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_3791_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_3792_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_3793_mult__of__nat__commute,axiom,
    ! [X: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,X)) ).

% mult_of_nat_commute
tff(fact_3794_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),M) ).

% mod_less_eq_dividend
tff(fact_3795_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_3796_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_3797_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% zle_int
tff(fact_3798_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_3799_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z3: 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,N2)),Z3)) = 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),N2))),Z3) ).

% zadd_int_left
tff(fact_3800_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_3801_int__plus,axiom,
    ! [N2: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_3802_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_3803_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_3804_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M: nat,N2: nat] : modulo8411746178871703098atural(A,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),aa_nat_Code_natural(semiri3763490453095760265atural,N2))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,M),modulo8411746178871703098atural(divide5121882707175180666atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M)),aa_nat_Code_natural(semiri3763490453095760265atural,N2)))),modulo8411746178871703098atural(A,aa_nat_Code_natural(semiri3763490453095760265atural,M))) ).

% mod_mult2_eq'
tff(fact_3805_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: 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,N2))) = 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,N2)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_3806_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: 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,N2))) = 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,N2)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_3807_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))) = 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,N2)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_3808_nat__less__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_less_as_int
tff(fact_3809_nat__leq__as__int,axiom,
    ! [X5: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_leq_as_int
tff(fact_3810_Abs__assn__eqI_I1_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H)
        <=> aa_Pro653082453696054616_nat_o(rep_assn(Pr),H) )
     => ( abs_assn(Pa) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_3811_Abs__assn__eqI_I2_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H)
        <=> aa_Pro653082453696054616_nat_o(rep_assn(Pr),H) )
     => ( Pr = abs_assn(Pa) ) ) ).

% Abs_assn_eqI(2)
tff(fact_3812_pure__assn__def,axiom,
    ! [B: $o] : aa_o_assn(pure_assn,(B)) = abs_assn(pure_a825153325127701367it_nat((B))) ).

% pure_assn_def
tff(fact_3813_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_3814_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_3815_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_3816_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_3817_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_3818_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_3819_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_0_le_iff
tff(fact_3820_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_0_le_iff
tff(fact_3821_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_0_le_iff
tff(fact_3822_of__nat__0__le__iff,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_0_le_iff
tff(fact_3823_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_3824_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_3825_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_3826_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_3827_mod__eqE,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2) )
     => ~ ! [D: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D)) ) ).

% mod_eqE
tff(fact_3828_div__add1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2)) ).

% div_add1_eq
tff(fact_3829_div__add1__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))),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),C2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C2))),C2)) ).

% div_add1_eq
tff(fact_3830_div__add1__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C2)),divide5121882707175180666atural(B,C2))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,C2)),modulo8411746178871703098atural(B,C2)),C2)) ).

% div_add1_eq
tff(fact_3831_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ) ).

% less_imp_of_nat_less
tff(fact_3832_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% less_imp_of_nat_less
tff(fact_3833_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ) ).

% less_imp_of_nat_less
tff(fact_3834_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ) ).

% less_imp_of_nat_less
tff(fact_3835_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_3836_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_3837_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_3838_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% of_nat_less_imp_less
tff(fact_3839_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I2)),aa_nat_Code_integer(semiri4939895301339042750nteger,J2)) ) ).

% of_nat_mono
tff(fact_3840_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I2)),aa_nat_rat(semiri681578069525770553at_rat,J2)) ) ).

% of_nat_mono
tff(fact_3841_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I2)),aa_nat_nat(semiri1316708129612266289at_nat,J2)) ) ).

% of_nat_mono
tff(fact_3842_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J2)) ) ).

% of_nat_mono
tff(fact_3843_gcd__nat__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,M: nat,N2: nat] :
      ( ! [M5: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,M5),zero_zero_nat)
     => ( ! [M5: nat,N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
           => ( aa_nat_o(aa_nat_fun_nat_o(Pa,N5),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M5),N5))
             => aa_nat_o(aa_nat_fun_nat_o(Pa,M5),N5) ) )
       => aa_nat_o(aa_nat_fun_nat_o(Pa,M),N2) ) ) ).

% gcd_nat_induct
tff(fact_3844_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),N2) ) ).

% mod_less_divisor
tff(fact_3845_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J2: int,K2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => 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,K2)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),J2)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_3846_pos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ~ ! [N5: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) ) ) ).

% pos_int_cases
tff(fact_3847_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
          & ( K2 = aa_nat_int(semiri1314217659103216013at_int,N5) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_3848_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N2) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3849_bot__assn__def,axiom,
    bot_bot_assn = abs_assn(aTP_Lamm_gt) ).

% bot_assn_def
tff(fact_3850_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_3851_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_3852_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_3853_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_3854_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_3855_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_3856_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_3857_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_3858_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B)) = A ).

% mult_div_mod_eq
tff(fact_3859_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_3860_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_3861_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))) = A ).

% mod_mult_div_eq
tff(fact_3862_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_3863_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_3864_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)) = A ).

% mod_div_mult_eq
tff(fact_3865_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_3866_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_3867_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)),modulo8411746178871703098atural(A,B)) = A ).

% div_mult_mod_eq
tff(fact_3868_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_3869_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_3870_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] : A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)),modulo8411746178871703098atural(A,B)) ).

% mod_div_decomp
tff(fact_3871_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3872_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3873_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(divide5121882707175180666atural(A,B),B)),modulo8411746178871703098atural(A,B))),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C2) ).

% cancel_div_mod_rules(1)
tff(fact_3874_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3875_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3876_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(B,divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B))),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C2) ).

% cancel_div_mod_rules(2)
tff(fact_3877_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C2))),C2)) ).

% div_mult1_eq
tff(fact_3878_div__mult1__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2))),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),C2))),C2)) ).

% div_mult1_eq
tff(fact_3879_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] : divide5121882707175180666atural(times_2397367101498566445atural(A,B),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(A,divide5121882707175180666atural(B,C2))),divide5121882707175180666atural(times_2397367101498566445atural(A,modulo8411746178871703098atural(B,C2)),C2)) ).

% div_mult1_eq
tff(fact_3880_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),N2) ) ).

% mod_le_divisor
tff(fact_3881_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ? [Q3: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3882_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ~ ! [S5: nat] : N2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S5)) ) ) ).

% mod_eq_nat2E
tff(fact_3883_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ~ ! [S5: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S5)) ) ) ).

% mod_eq_nat1E
tff(fact_3884_div__less__mono,axiom,
    ! [A4: nat,B4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),N2) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),N2) = 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,A4),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),N2)) ) ) ) ) ).

% div_less_mono
tff(fact_3885_div__mod__decomp,axiom,
    ! [A4: nat,N2: nat] : A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),N2)),N2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),N2)) ).

% div_mod_decomp
tff(fact_3886_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),Q4))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) ).

% mod_mult2_eq
tff(fact_3887_sup__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa) = abs_assn(aa_ass4243947775541419017_nat_o(aTP_Lamm_gu(Pa),Qa)) ).

% sup_assn_def
tff(fact_3888_inf__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa) = abs_assn(aa_ass4243947775541419017_nat_o(aTP_Lamm_gv(Pa),Qa)) ).

% inf_assn_def
tff(fact_3889_split__mod,axiom,
    ! [Pa: fun_nat_o,M: nat,N2: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2))
    <=> ( ( ( N2 = zero_zero_nat )
         => aa_nat_o(Pa,M) )
        & ( ( N2 != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),N2)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I)),J) )
               => aa_nat_o(Pa,J) ) ) ) ) ) ).

% split_mod
tff(fact_3890_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C2)) = 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)),C2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3891_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3892_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3893_times__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa) = abs_assn(aa_fun2638059832795814268_nat_o(times_assn_raw(rep_assn(Pa)),rep_assn(Qa))) ).

% times_assn_def
tff(fact_3894_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3895_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: 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,X),Y)),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3896_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: 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,X),Y)),C2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3897_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3898_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: 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,X),Y)),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3899_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: 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,X),Y)),C2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3900_distrib__right__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C2: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MAT3836730678284324717al_rat(divide5121882707175180666atural(X,Y)),C2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3901_distrib__right__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C2: nat,A: nat,B: nat] :
      ( aa_nat_o(nO_MAT4471860738370820453al_nat(divide5121882707175180666atural(X,Y)),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3902_distrib__right__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C2: int,A: int,B: int] :
      ( aa_int_o(nO_MAT4469370267861770177al_int(divide5121882707175180666atural(X,Y)),C2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3903_distrib__right__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(nO_MATCH_rat_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),C2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_3904_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N5)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_3905_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,N2) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_gw,N2),zero_zero_rat) ).

% of_nat_code
tff(fact_3906_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,N2) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_gx),N2),zero_zero_int) ).

% of_nat_code
tff(fact_3907_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_gy),N2),zero_zero_nat) ).

% of_nat_code
tff(fact_3908_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,N2) = aa_Cod1783974652533396728nteger(semiri4055485073559036834nteger(aTP_Lamm_gz,N2),zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_3909_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N5)) ).

% reals_Archimedean2
tff(fact_3910_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N5)) ).

% real_arch_simple
tff(fact_3911_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: 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),N2))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3912_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_3913_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_3914_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_less_eq
tff(fact_3915_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) ) ).

% Suc_mono
tff(fact_3916_lessI,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,N2)) ).

% lessI
tff(fact_3917_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M) ) ).

% Suc_le_mono
tff(fact_3918_add__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% add_Suc_right
tff(fact_3919_less__Suc0,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,zero_zero_nat))
    <=> ( N2 = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_3920_zero__less__Suc,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N2)) ).

% zero_less_Suc
tff(fact_3921_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_3922_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_3923_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ).

% mult_Suc_right
tff(fact_3924_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_3925_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_3926_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3927_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3928_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_3929_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_3930_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_3931_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_3932_one__le__mult__iff,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ( 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)),N2) ) ) ).

% one_le_mult_iff
tff(fact_3933_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K2: 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,N2),K2)),M))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self4
tff(fact_3934_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N2: 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,K2),N2)),M))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self3
tff(fact_3935_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self2
tff(fact_3936_Suc__mod__mult__self1,axiom,
    ! [M: nat,K2: nat,N2: 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,K2),N2)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self1
tff(fact_3937_zle__add1__eq__le,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z3) ) ).

% zle_add1_eq_le
tff(fact_3938_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_3939_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_3940_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I2),K2) = I2 )
    <=> ( ( K2 = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2) ) ) ) ).

% zmod_trivial_iff
tff(fact_3941_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K2: 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,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K2)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_3942_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ? [N5: nat] : K2 = aa_nat_int(semiri1314217659103216013at_int,N5) ) ).

% zero_le_imp_eq_int
tff(fact_3943_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ~ ! [N5: nat] : K2 != aa_nat_int(semiri1314217659103216013at_int,N5) ) ).

% nonneg_int_cases
tff(fact_3944_le__imp__0__less,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => 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),Z3)) ) ).

% le_imp_0_less
tff(fact_3945_int__one__le__iff__zero__less,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3) ) ).

% int_one_le_iff_zero_less
tff(fact_3946_zless__imp__add1__zle,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z3) ) ).

% zless_imp_add1_zle
tff(fact_3947_add1__zle__eq,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ).

% add1_zle_eq
tff(fact_3948_zle__iff__zadd,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z3)
    <=> ? [N: nat] : Z3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% zle_iff_zadd
tff(fact_3949_incr__mult__lemma,axiom,
    ! [D2: int,Pa: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X5: int] :
              ( aa_int_o(Pa,X5)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3950_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R5: int,Q4: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R5)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3951_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R5: int,Q4: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R5)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R5),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q4) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3952_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R: int,B2: int,Q5: int,R5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R5) )
     => ( 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,B2),Q5)),R5)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R5)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3953_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R: int,B2: int,Q5: int,R5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R5) )
     => ( 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,B2),Q5)),R5))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R5),B2)
         => ( 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,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3954_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R) )
     => ( 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),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3955_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3956_q__pos__lemma,axiom,
    ! [B2: int,Q5: int,R5: 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,B2),Q5)),R5))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R5),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3957_split__zmod,axiom,
    ! [Pa: fun_int_o,N2: int,K2: int] :
      ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N2),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(Pa,N2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [I: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J),K2)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => aa_int_o(Pa,J) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),zero_zero_int)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => aa_int_o(Pa,J) ) ) ) ) ).

% split_zmod
tff(fact_3958_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_3959_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3960_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3961_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)) ) ) ).

% zdiv_mono1
tff(fact_3962_zdiv__mono2,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ) ).

% zdiv_mono2
tff(fact_3963_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K2) = zero_zero_int )
    <=> ( ( K2 = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3964_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_3965_zdiv__mono2__neg,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3966_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L))
    <=> ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_3967_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L)) ) ) ).

% div_positive_int
tff(fact_3968_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3969_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3970_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I2) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3971_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3972_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3973_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3974_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N2)
       => 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,A4),N2)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),N2) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),N2)) ) ) ).

% verit_le_mono_div_int
tff(fact_3975_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3976_split__pos__lemma,axiom,
    ! [K2: int,Pa: fun_int_fun_int_o,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,N2),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N2),K2))
      <=> ! [I: int,J: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J),K2)
              & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
           => aa_int_o(aa_int_fun_int_o(Pa,I),J) ) ) ) ).

% split_pos_lemma
tff(fact_3977_split__neg__lemma,axiom,
    ! [K2: int,Pa: fun_int_fun_int_o,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,N2),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N2),K2))
      <=> ! [I: int,J: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),zero_zero_int)
              & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
           => aa_int_o(aa_int_fun_int_o(Pa,I),J) ) ) ) ).

% split_neg_lemma
tff(fact_3978_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R) )
     => ( 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),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3979_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3980_split__zdiv,axiom,
    ! [Pa: fun_int_o,N2: int,K2: int] :
      ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,N2),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(Pa,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [I: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J),K2)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => aa_int_o(Pa,I) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),zero_zero_int)
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J) ) )
             => aa_int_o(Pa,I) ) ) ) ) ).

% split_zdiv
tff(fact_3981_div__mod__decomp__int,axiom,
    ! [A4: int,N2: int] : A4 = 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,A4),N2)),N2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),N2)) ).

% div_mod_decomp_int
tff(fact_3982_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K2)),X) ) ) ).

% int_div_less_self
tff(fact_3983_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N2)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),N2) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),N2) = 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,A4),N2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),N2)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3984_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3985_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3986_div__neg__pos__less0,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(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_3987_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_3988_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) ) ).

% neg_mod_bound
tff(fact_3989_zmult__zless__mono2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),J2)) ) ) ).

% zmult_zless_mono2
tff(fact_3990_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: 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),N2) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3991_int__Suc,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int) ).

% int_Suc
tff(fact_3992_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_3993_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_3994_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_3995_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3)
    <=> ? [N: nat] : Z3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% zless_iff_Suc_zadd
tff(fact_3996_int__gr__induct,axiom,
    ! [K2: int,I2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I3)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_gr_induct
tff(fact_3997_zless__add1__eq,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3)
        | ( W2 = Z3 ) ) ) ).

% zless_add1_eq
tff(fact_3998_odd__less__0__iff,axiom,
    ! [Z3: 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),Z3)),Z3)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_3999_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_4000_plus__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),zero_zero_int) = K2 ).

% plus_int_code(1)
tff(fact_4001_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_4002_odd__nonzero,axiom,
    ! [Z3: 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),Z3)),Z3) != zero_zero_int ).

% odd_nonzero
tff(fact_4003_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_4004_n__not__Suc__n,axiom,
    ! [N2: nat] : N2 != aa_nat_nat(suc,N2) ).

% n_not_Suc_n
tff(fact_4005_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(2)
tff(fact_4006_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(1)
tff(fact_4007_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_4008_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
      | ( A = X )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A) ) ).

% verit_la_generic
tff(fact_4009_int__ge__induct,axiom,
    ! [K2: int,I2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I2)
     => ( aa_int_o(Pa,K2)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I3)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_ge_induct
tff(fact_4010_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_4011_imp__le__cong,axiom,
    ! [X: int,X8: int,Pa: $o,P3: $o] :
      ( ( X = X8 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
         => ( (Pa)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => (Pa) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_4012_conj__le__cong,axiom,
    ! [X: int,X8: int,Pa: $o,P3: $o] :
      ( ( X = X8 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
         => ( (Pa)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
            & (Pa) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X8)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_4013_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_4014_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_4015_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_4016_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_4017_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_4018_nat__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,N5)
           => aa_nat_o(Pa,aa_nat_nat(suc,N5)) )
       => aa_nat_o(Pa,N2) ) ) ).

% nat_induct
tff(fact_4019_diff__induct,axiom,
    ! [Pa: fun_nat_fun_nat_o,M: nat,N2: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,X3),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Pa,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,M),N2) ) ) ) ).

% diff_induct
tff(fact_4020_zero__induct,axiom,
    ! [Pa: fun_nat_o,K2: nat] :
      ( aa_nat_o(Pa,K2)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,aa_nat_nat(suc,N5))
           => aa_nat_o(Pa,N5) )
       => aa_nat_o(Pa,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_4021_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_4022_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_4023_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_4024_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] : N2 = aa_nat_nat(suc,M5) ) ).

% not0_implies_Suc
tff(fact_4025_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M))
      <=> ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_4026_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ( ! [I3: nat] :
            ( ( J2 = aa_nat_nat(suc,I3) )
           => aa_nat_o(Pa,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( aa_nat_o(Pa,aa_nat_nat(suc,I3))
               => aa_nat_o(Pa,I3) ) )
         => aa_nat_o(Pa,I2) ) ) ) ).

% strict_inc_induct
tff(fact_4027_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,Pa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),K)
               => ( aa_nat_o(aa_nat_fun_nat_o(Pa,I3),J3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(Pa,J3),K)
                   => aa_nat_o(aa_nat_fun_nat_o(Pa,I3),K) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,I2),J2) ) ) ) ).

% less_Suc_induct
tff(fact_4028_less__trans__Suc,axiom,
    ! [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(ord_less_nat,aa_nat_nat(suc,I2)),K2) ) ) ).

% less_trans_Suc
tff(fact_4029_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_less_SucD
tff(fact_4030_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M))
       => ( M = N2 ) ) ) ).

% less_antisym
tff(fact_4031_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M)
    <=> ? [M7: nat] :
          ( ( M = aa_nat_nat(suc,M7) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M7) ) ) ).

% Suc_less_eq2
tff(fact_4032_All__less__Suc,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,N2))
         => aa_nat_o(Pa,I) )
    <=> ( aa_nat_o(Pa,N2)
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
           => aa_nat_o(Pa,I) ) ) ) ).

% All_less_Suc
tff(fact_4033_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_4034_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( M = N2 ) ) ) ).

% less_Suc_eq
tff(fact_4035_Ex__less__Suc,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,N2))
          & aa_nat_o(Pa,I) )
    <=> ( aa_nat_o(Pa,N2)
        | ? [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
            & aa_nat_o(Pa,I) ) ) ) ).

% Ex_less_Suc
tff(fact_4036_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2)) ) ).

% less_SucI
tff(fact_4037_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => ( M = N2 ) ) ) ).

% less_SucE
tff(fact_4038_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( ( aa_nat_nat(suc,M) != N2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N2) ) ) ).

% Suc_lessI
tff(fact_4039_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K2)
     => ~ ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J3)
           => ( K2 != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_4040_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_lessD
tff(fact_4041_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K2)
     => ( ( K2 != aa_nat_nat(suc,I2) )
       => ~ ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J3)
             => ( K2 != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_4042_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,Ra: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(Ra,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(Ra,Y3),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Z2) ) )
         => ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(Ra,N5),aa_nat_nat(suc,N5))
           => aa_nat_o(aa_nat_fun_nat_o(Ra,M),N2) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_4043_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(Pa,M)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N5)
             => ( aa_nat_o(Pa,N5)
               => aa_nat_o(Pa,aa_nat_nat(suc,N5)) ) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_induct_at_least
tff(fact_4044_full__nat__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M4)),N5)
             => aa_nat_o(Pa,M4) )
         => aa_nat_o(Pa,N5) )
     => aa_nat_o(Pa,N2) ) ).

% full_nat_induct
tff(fact_4045_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),M) ) ).

% not_less_eq_eq
tff(fact_4046_Suc__n__not__le__n,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),N2) ).

% Suc_n_not_le_n
tff(fact_4047_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        | ( M = aa_nat_nat(suc,N2) ) ) ) ).

% le_Suc_eq
tff(fact_4048_Suc__le__D,axiom,
    ! [N2: nat,M8: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),M8)
     => ? [M5: nat] : M8 = aa_nat_nat(suc,M5) ) ).

% Suc_le_D
tff(fact_4049_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2)) ) ).

% le_SucI
tff(fact_4050_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( M = aa_nat_nat(suc,N2) ) ) ) ).

% le_SucE
tff(fact_4051_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% Suc_leD
tff(fact_4052_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_4053_add__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% add_Suc
tff(fact_4054_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N2)) ).

% add_Suc_shift
tff(fact_4055_Suc__mult__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N2) )
    <=> ( M = N2 ) ) ).

% Suc_mult_cancel1
tff(fact_4056_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_assn,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,N5)),aa_nat_assn(Fa,aa_nat_nat(suc,N5)))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,N2)),aa_nat_assn(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4057_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_rat,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N5)),aa_nat_rat(Fa,aa_nat_nat(suc,N5)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N2)),aa_nat_rat(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4058_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_num,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N5)),aa_nat_num(Fa,aa_nat_nat(suc,N5)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N2)),aa_nat_num(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4059_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_nat,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N5)),aa_nat_nat(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N2)),aa_nat_nat(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4060_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_int,N2: nat,M: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N5)),aa_nat_int(Fa,aa_nat_nat(suc,N5)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N2)),aa_nat_int(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_4061_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_assn,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,N5)),aa_nat_assn(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(Fa,N2)),aa_nat_assn(Fa,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4062_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N5)),aa_nat_rat(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N2)),aa_nat_rat(Fa,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4063_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_num,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N5)),aa_nat_num(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N2)),aa_nat_num(Fa,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4064_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N5)),aa_nat_nat(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N2)),aa_nat_nat(Fa,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4065_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N5)),aa_nat_int(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N2)),aa_nat_int(Fa,N6)) ) ) ).

% lift_Suc_mono_less
tff(fact_4066_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_set_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(Fa,aa_nat_nat(suc,N5))),aa_nat_set_int(Fa,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(Fa,N6)),aa_nat_set_int(Fa,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4067_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,aa_nat_nat(suc,N5))),aa_nat_rat(Fa,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,N6)),aa_nat_rat(Fa,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4068_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_num,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,aa_nat_nat(suc,N5))),aa_nat_num(Fa,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,N6)),aa_nat_num(Fa,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4069_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,aa_nat_nat(suc,N5))),aa_nat_nat(Fa,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N6)),aa_nat_nat(Fa,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4070_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,aa_nat_nat(suc,N5))),aa_nat_int(Fa,N5))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N6)),aa_nat_int(Fa,N2)) ) ) ).

% lift_Suc_antimono_le
tff(fact_4071_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_set_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(Fa,N5)),aa_nat_set_int(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_nat_set_int(Fa,N2)),aa_nat_set_int(Fa,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4072_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_rat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,N5)),aa_nat_rat(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,N2)),aa_nat_rat(Fa,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4073_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_num,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,N5)),aa_nat_num(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,N2)),aa_nat_num(Fa,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4074_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_nat,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N5)),aa_nat_nat(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N2)),aa_nat_nat(Fa,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4075_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_int,N2: nat,N6: nat] :
      ( ! [N5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N5)),aa_nat_int(Fa,aa_nat_nat(suc,N5)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N6)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N2)),aa_nat_int(Fa,N6)) ) ) ).

% lift_Suc_mono_le
tff(fact_4076_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_4077_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_4078_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N2)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_4079_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N2)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_4080_Ex__less__Suc2,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,N2))
          & aa_nat_o(Pa,I) )
    <=> ( aa_nat_o(Pa,zero_zero_nat)
        | ? [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
            & aa_nat_o(Pa,aa_nat_nat(suc,I)) ) ) ) ).

% Ex_less_Suc2
tff(fact_4081_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
    <=> ? [M3: nat] : N2 = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_4082_All__less__Suc2,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,N2))
         => aa_nat_o(Pa,I) )
    <=> ( aa_nat_o(Pa,zero_zero_nat)
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N2)
           => aa_nat_o(Pa,aa_nat_nat(suc,I)) ) ) ) ).

% All_less_Suc2
tff(fact_4083_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ? [M5: nat] : N2 = aa_nat_nat(suc,M5) ) ).

% gr0_implies_Suc
tff(fact_4084_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
    <=> ( ( M = zero_zero_nat )
        | ? [J: nat] :
            ( ( M = aa_nat_nat(suc,J) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),N2) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_4085_nat__compl__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N5)
               => aa_nat_o(Pa,Nn) )
           => aa_nat_o(Pa,aa_nat_nat(suc,N5)) )
       => aa_nat_o(Pa,N2) ) ) ).

% nat_compl_induct
tff(fact_4086_nat__compl__induct_H,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N5)
               => aa_nat_o(Pa,Nn) )
           => aa_nat_o(Pa,aa_nat_nat(suc,N5)) )
       => aa_nat_o(Pa,N2) ) ) ).

% nat_compl_induct'
tff(fact_4087_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_4088_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_4089_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2) ) ).

% Suc_leI
tff(fact_4090_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_le_eq
tff(fact_4091_dec__induct,axiom,
    ! [I2: nat,J2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(Pa,I2)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N5)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),J2)
               => ( aa_nat_o(Pa,N5)
                 => aa_nat_o(Pa,aa_nat_nat(suc,N5)) ) ) )
         => aa_nat_o(Pa,J2) ) ) ) ).

% dec_induct
tff(fact_4092_inc__induct,axiom,
    ! [I2: nat,J2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(Pa,J2)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N5)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),J2)
               => ( aa_nat_o(Pa,aa_nat_nat(suc,N5))
                 => aa_nat_o(Pa,N5) ) ) )
         => aa_nat_o(Pa,I2) ) ) ) ).

% inc_induct
tff(fact_4093_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_le_lessD
tff(fact_4094_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(suc,M))
      <=> ( N2 = M ) ) ) ).

% le_less_Suc_eq
tff(fact_4095_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% less_Suc_eq_le
tff(fact_4096_less__eq__Suc__le,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),M) ) ).

% less_eq_Suc_le
tff(fact_4097_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N2)) ) ).

% le_imp_less_Suc
tff(fact_4098_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_4099_nat__in__between__eq_I1_J,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,B),aa_nat_nat(suc,A)) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_4100_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ~ ! [Q3: nat] : N2 != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_4101_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M))) ).

% less_add_Suc1
tff(fact_4102_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I2))) ).

% less_add_Suc2
tff(fact_4103_less__iff__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
    <=> ? [K3: nat] : N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_4104_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ? [K: nat] : N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)) ) ).

% less_imp_Suc_add
tff(fact_4105_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N2: 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,K2)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% Suc_mult_less_cancel1
tff(fact_4106_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_4107_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N2: 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,K2)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% Suc_mult_le_cancel1
tff(fact_4108_mult__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ).

% mult_Suc
tff(fact_4109_Suc__eq__plus1,axiom,
    ! [N2: nat] : aa_nat_nat(suc,N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat) ).

% Suc_eq_plus1
tff(fact_4110_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_4111_Suc__eq__plus1__left,axiom,
    ! [N2: nat] : aa_nat_nat(suc,N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N2) ).

% Suc_eq_plus1_left
tff(fact_4112_mod__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat,P2: nat,M: nat] :
      ( aa_nat_o(Pa,N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P2)
         => ( ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),P2)
               => ( aa_nat_o(Pa,N5)
                 => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N5)),P2)) ) )
           => aa_nat_o(Pa,M) ) ) ) ) ).

% mod_induct
tff(fact_4113_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N2))),N2) ).

% mod_Suc_le_divisor
tff(fact_4114_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N2)) ).

% Suc_div_le_mono
tff(fact_4115_power__le__imp__le__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N2))),aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N2)))
     => ( 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_4116_power__le__imp__le__base,axiom,
    ! [A: rat,N2: 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,N2))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N2)))
     => ( 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_4117_power__le__imp__le__base,axiom,
    ! [A: nat,N2: 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,N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N2)))
     => ( 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_4118_power__le__imp__le__base,axiom,
    ! [A: int,N2: 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,N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,N2)))
     => ( 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_4119_power__inject__base,axiom,
    ! [A: code_integer,N2: nat,B: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(suc,N2)) = aa_nat_Code_integer(power_8256067586552552935nteger(B),aa_nat_nat(suc,N2)) )
     => ( 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_4120_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N2)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4121_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4122_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,N2)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,N2)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4123_power__gt1,axiom,
    ! [A: code_integer,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4124_power__gt1,axiom,
    ! [A: rat,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4125_power__gt1,axiom,
    ! [A: nat,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4126_power__gt1,axiom,
    ! [A: int,N2: 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,N2))) ) ).

% power_gt1
tff(fact_4127_ex__least__nat__less,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,N2)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),N2)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K)
               => ~ aa_nat_o(Pa,I4) )
            & aa_nat_o(Pa,aa_nat_nat(suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_4128_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M)) ) ) ).

% n_less_n_mult_m
tff(fact_4129_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ) ) ).

% n_less_m_mult_n
tff(fact_4130_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => ( 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),N2)) ) ) ).

% one_less_mult
tff(fact_4131_nat__induct__non__zero,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
             => ( aa_nat_o(Pa,N5)
               => aa_nat_o(Pa,aa_nat_nat(suc,N5)) ) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_induct_non_zero
tff(fact_4132_power__gt__expt,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),K2)) ) ).

% power_gt_expt
tff(fact_4133_nat__one__le__power,axiom,
    ! [I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N2)) ) ).

% nat_one_le_power
tff(fact_4134_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N5: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N5)))),E) ) ).

% nat_approx_posE
tff(fact_4135_power__Suc__le__self,axiom,
    ! [A: code_integer,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4136_power__Suc__le__self,axiom,
    ! [A: rat,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4137_power__Suc__le__self,axiom,
    ! [A: nat,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4138_power__Suc__le__self,axiom,
    ! [A: int,N2: 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,N2))),A) ) ) ).

% power_Suc_le_self
tff(fact_4139_power__Suc__less__one,axiom,
    ! [A: code_integer,N2: 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,N2))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_4140_power__Suc__less__one,axiom,
    ! [A: rat,N2: 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,N2))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_4141_power__Suc__less__one,axiom,
    ! [A: nat,N2: 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,N2))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_4142_power__Suc__less__one,axiom,
    ! [A: int,N2: 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,N2))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_4143_div__nat__eqI,axiom,
    ! [N2: nat,Q4: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),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,N2),aa_nat_nat(suc,Q4)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_4144_split__div_H,axiom,
    ! [Pa: fun_nat_o,M: nat,N2: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))
    <=> ( ( ( N2 = zero_zero_nat )
          & aa_nat_o(Pa,zero_zero_nat) )
        | ? [Q6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q6)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q6)))
            & aa_nat_o(Pa,Q6) ) ) ) ).

% split_div'
tff(fact_4145_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr5472464481163311559at_nat,X2: produc7388388658123137530it_nat] : size_o5858930521590659991it_nat(X,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro1613605141062480648at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4146_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr4403862266400233414at_nat,X2: produc3260487557148687353it_nat] : size_o1731029420616209814it_nat(X,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro545002926299402503at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4147_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr367382672041210069at_nat,X2: produc8664842809031399944it_nat] : size_o29782932136985253it_nat(X,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro5228051289423594006at_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4148_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X2: num] : aa_option_num_nat(size_option_num(X),aa_num_option_num(some_num,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_4149_Heap_Osize__gen,axiom,
    ! [Xa: fun_Product_unit_nat,X: fun_he5234484606919320760it_nat] : heap_T4142866422068808505t_unit(Xa,heap_T6183433275982383450t_unit(X)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size_gen
tff(fact_4150_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : aa_option_num_nat(size_option_num(X),none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_4151_Heap_Osize_I2_J,axiom,
    ! [X: fun_he5234484606919320760it_nat] : size_s1564113455978345259t_unit(heap_T6183433275982383450t_unit(X)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_4152_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q4: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),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) ),
              Q4 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4153_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,X) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_4154_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_4155_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( size_size_list_o(X) != size_size_list_o(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_4156_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_4157_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))
     => ( ! [I3: int,J3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J3))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J3)
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J3) )
             => aa_int_o(aa_int_fun_int_o(Pa,I3),J3) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4158_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_4159_option_Osize_I4_J,axiom,
    ! [X2: produc7388388658123137530it_nat] : size_s2866289034186623454it_nat(aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4160_option_Osize_I4_J,axiom,
    ! [X2: produc3260487557148687353it_nat] : size_s2363601347547812957it_nat(aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_4161_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_4162_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_4163_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_4164_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4165_int__power__div__base,axiom,
    ! [M: nat,K2: 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),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K2),M)),K2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K2),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_4166_one__less__nat__eq,axiom,
    ! [Z3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z3) ) ).

% one_less_nat_eq
tff(fact_4167_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_4168_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4169_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_4170_compl__le__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X) ) ).

% compl_le_compl_iff
tff(fact_4171_compl__le__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y))
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),X) ) ).

% compl_le_compl_iff
tff(fact_4172_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(X)),uminus1532241313380277803et_int(Y))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X) ) ).

% compl_le_compl_iff
tff(fact_4173_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_4174_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_4175_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_4176_add__diff__cancel__right_H,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_4177_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_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_4178_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_4179_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_4180_add__diff__cancel__right,axiom,
    ! [A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),C2),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),C2)) = minus_4286766774447292334at_nat(A,B) ).

% add_diff_cancel_right
tff(fact_4181_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_right
tff(fact_4182_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_4183_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_4184_add__diff__cancel__left_H,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_4185_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_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_4186_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_4187_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_4188_add__diff__cancel__left,axiom,
    ! [C2: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(C2),A),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(C2),B)) = minus_4286766774447292334at_nat(A,B) ).

% add_diff_cancel_left
tff(fact_4189_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_left
tff(fact_4190_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_4191_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_4192_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ).

% diff_add_cancel
tff(fact_4193_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_4194_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_4195_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_4196_compl__less__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),X) ) ).

% compl_less_compl_iff
tff(fact_4197_compl__less__compl__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(ord_le361264281704409273t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y))
    <=> aa_Product_unit_o(ord_le361264281704409273t_unit(Y),X) ) ).

% compl_less_compl_iff
tff(fact_4198_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_4199_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_4200_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_4201_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_4202_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_4203_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_4204_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_4205_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_4206_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_4207_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_4208_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_4209_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_4210_diff__0__eq__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N2) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_4211_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_4212_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% diff_Suc_Suc
tff(fact_4213_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),K2) ).

% Suc_diff_diff
tff(fact_4214_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_4215_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) ).

% diff_diff_left
tff(fact_4216_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_4217_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_4218_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_4219_zero__comp__diff__simps_I1_J,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) ) ).

% zero_comp_diff_simps(1)
tff(fact_4220_zero__comp__diff__simps_I1_J,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(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_4221_zero__comp__diff__simps_I1_J,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) ) ).

% zero_comp_diff_simps(1)
tff(fact_4222_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_4223_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_4224_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_4225_zero__comp__diff__simps_I2_J,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) ) ).

% zero_comp_diff_simps(2)
tff(fact_4226_zero__comp__diff__simps_I2_J,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(minus_minus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_4227_zero__comp__diff__simps_I2_J,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) ) ).

% zero_comp_diff_simps(2)
tff(fact_4228_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_4229_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_4230_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_4231_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_4232_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_4233_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_4234_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_4235_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_4236_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_4237_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_4238_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_4239_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_4240_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),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_4241_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_4242_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_4243_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,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_4244_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_4245_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_4246_diff__add__zero,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A),B)) = zero_z1048942125864253310at_nat ).

% diff_add_zero
tff(fact_4247_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_4248_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_4249_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_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_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_4260_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_4261_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_4262_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_4263_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_4264_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_4265_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_4266_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_4267_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_4268_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4269_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_4270_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_4271_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_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_4272_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_4273_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_4274_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) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% uminus_add_conv_diff
tff(fact_4275_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(X)),uminus6524753893492686040at_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4276_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : uminus103585356917985579_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,uminus103585356917985579_int_o(X)),uminus103585356917985579_int_o(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4277_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,uminus2330091110623919550at_nat(X)),uminus2330091110623919550at_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4278_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(X)),uminus935396558254630718at_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4279_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_nat,Y: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4280_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4281_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(X),Y)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_4282_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(X)),uminus6524753893492686040at_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4283_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : uminus103585356917985579_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,uminus103585356917985579_int_o(X)),uminus103585356917985579_int_o(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4284_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(X)),uminus2330091110623919550at_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4285_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(X)),uminus935396558254630718at_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4286_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_nat,Y: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4287_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4288_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(X),Y)) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_4289_zero__less__diff,axiom,
    ! [N2: 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,N2),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% zero_less_diff
tff(fact_4290_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4291_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% diff_is_0_eq
tff(fact_4292_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_4293_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4294_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_4295_diff__Suc__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N2)),one_one_nat) = N2 ).

% diff_Suc_1
tff(fact_4296_negative__zle,axiom,
    ! [N2: 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,N2))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_4297_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_4298_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_4299_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_4300_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_4301_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_4302_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_4303_Suc__pred,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))) = N2 ) ) ).

% Suc_pred
tff(fact_4304_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(suc,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4305_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( 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,J2),K2))),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4306_Suc__diff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_4307_nat__le__0,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),zero_zero_int)
     => ( aa_int_nat(nat2,Z3) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_4308_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa_int_nat(nat2,I2) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int) ) ).

% nat_0_iff
tff(fact_4309_zless__nat__conj,axiom,
    ! [W2: int,Z3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z3))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ) ).

% zless_nat_conj
tff(fact_4310_negative__zless,axiom,
    ! [N2: 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,N2)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_4311_int__nat__eq,axiom,
    ! [Z3: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z3)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3),Z3,zero_zero_int) ).

% int_nat_eq
tff(fact_4312_zle__diff1__eq,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ).

% zle_diff1_eq
tff(fact_4313_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) = N2 ) ) ).

% Suc_diff_1
tff(fact_4314_zero__less__nat__eq,axiom,
    ! [Z3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3) ) ).

% zero_less_nat_eq
tff(fact_4315_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_4316_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_4317_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_4318_diff__commute,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),K2)),J2) ).

% diff_commute
tff(fact_4319_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4320_nat__diff__distrib,axiom,
    ! [Z6: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z6),Z3)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z3)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_4321_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4322_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B4 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K2),B) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4323_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_4324_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_4325_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_4326_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_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_4327_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_4328_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_4329_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_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_4330_int__minus,axiom,
    ! [N2: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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,N2)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_4331_nat__minus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_minus_as_int
tff(fact_4332_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ) ) ).

% of_nat_diff
tff(fact_4333_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ).

% of_nat_diff
tff(fact_4334_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ) ) ).

% of_nat_diff
tff(fact_4335_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ) ) ).

% of_nat_diff
tff(fact_4336_diff__nat__eq__if,axiom,
    ! [Z3: int,Z6: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z3)),aa_int_nat(nat2,Z6)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z6),zero_zero_int),
        aa_int_nat(nat2,Z3),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),Z6),
          $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_4337_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),D2) )
     => ( 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,C2),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4338_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( 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,C2),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4339_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) ) ).

% diff_right_mono
tff(fact_4340_diff__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_right_mono
tff(fact_4341_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B)) ) ).

% diff_left_mono
tff(fact_4342_diff__left__mono,axiom,
    ! [B: int,A: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_left_mono
tff(fact_4343_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C2: 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,D2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_mono
tff(fact_4344_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C2: 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,D2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_mono
tff(fact_4345_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: 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(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) ) ).

% diff_strict_right_mono
tff(fact_4346_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: 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),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_strict_right_mono
tff(fact_4347_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B)) ) ).

% diff_strict_left_mono
tff(fact_4348_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: 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,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_strict_left_mono
tff(fact_4349_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4350_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4351_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C2: 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,D2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4352_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C2: 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,D2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4353_diff__diff__eq,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(minus_4286766774447292334at_nat(A,B),C2) = minus_4286766774447292334at_nat(A,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B),C2)) ).

% diff_diff_eq
tff(fact_4354_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ).

% diff_diff_eq
tff(fact_4355_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ).

% diff_diff_eq
tff(fact_4356_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ).

% diff_diff_eq
tff(fact_4357_add__implies__diff,axiom,
    ! [C2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(C2),B) = A )
     => ( C2 = minus_4286766774447292334at_nat(A,B) ) ) ).

% add_implies_diff
tff(fact_4358_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B) = A )
     => ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_4359_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4360_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4361_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4362_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4363_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B) ).

% diff_add_eq
tff(fact_4364_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_add_eq
tff(fact_4365_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C2)),B) ).

% diff_diff_eq2
tff(fact_4366_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ).

% diff_diff_eq2
tff(fact_4367_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C2) ).

% add_diff_eq
tff(fact_4368_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ).

% add_diff_eq
tff(fact_4369_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4370_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_4371_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C2 )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B) ) ) ).

% diff_eq_eq
tff(fact_4372_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) ) ) ).

% diff_eq_eq
tff(fact_4373_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4374_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4375_inf__period_I2_J,axiom,
    ! [Pa: fun_rat_o,D4: rat,Qa: fun_rat_o] :
      ( ! [X3: rat,K: rat] :
          ( aa_rat_o(Pa,X3)
        <=> aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D4))) )
     => ( ! [X3: rat,K: rat] :
            ( aa_rat_o(Qa,X3)
          <=> aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D4))) )
       => ! [X5: rat,K4: rat] :
            ( ( aa_rat_o(Pa,X5)
              | aa_rat_o(Qa,X5) )
          <=> ( aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4)))
              | aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_4376_inf__period_I2_J,axiom,
    ! [Pa: fun_int_o,D4: int,Qa: fun_int_o] :
      ( ! [X3: int,K: int] :
          ( aa_int_o(Pa,X3)
        <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
     => ( ! [X3: int,K: int] :
            ( aa_int_o(Qa,X3)
          <=> aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
       => ! [X5: int,K4: int] :
            ( ( aa_int_o(Pa,X5)
              | aa_int_o(Qa,X5) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))
              | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_4377_inf__period_I1_J,axiom,
    ! [Pa: fun_rat_o,D4: rat,Qa: fun_rat_o] :
      ( ! [X3: rat,K: rat] :
          ( aa_rat_o(Pa,X3)
        <=> aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D4))) )
     => ( ! [X3: rat,K: rat] :
            ( aa_rat_o(Qa,X3)
          <=> aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D4))) )
       => ! [X5: rat,K4: rat] :
            ( ( aa_rat_o(Pa,X5)
              & aa_rat_o(Qa,X5) )
          <=> ( aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4)))
              & aa_rat_o(Qa,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_4378_inf__period_I1_J,axiom,
    ! [Pa: fun_int_o,D4: int,Qa: fun_int_o] :
      ( ! [X3: int,K: int] :
          ( aa_int_o(Pa,X3)
        <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
     => ( ! [X3: int,K: int] :
            ( aa_int_o(Qa,X3)
          <=> aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
       => ! [X5: int,K4: int] :
            ( ( aa_int_o(Pa,X5)
              & aa_int_o(Qa,X5) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))
              & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_4379_compl__le__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,Y)),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).

% compl_le_swap2
tff(fact_4380_compl__le__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),X)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y) ) ).

% compl_le_swap2
tff(fact_4381_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(Y)),X)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(X)),Y) ) ).

% compl_le_swap2
tff(fact_4382_compl__le__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(uminus_uminus_assn,X))
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% compl_le_swap1
tff(fact_4383_compl__le__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X))
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% compl_le_swap1
tff(fact_4384_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),uminus1532241313380277803et_int(X))
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),uminus1532241313380277803et_int(Y)) ) ).

% compl_le_swap1
tff(fact_4385_compl__mono,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(uminus_uminus_assn,Y)),aa_assn_assn(uminus_uminus_assn,X)) ) ).

% compl_mono
tff(fact_4386_compl__mono,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(X),Y)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)) ) ).

% compl_mono
tff(fact_4387_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(X)) ) ).

% compl_mono
tff(fact_4388_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_4389_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_4390_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_4391_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_4392_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_4393_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_4394_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_4395_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_4396_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_4397_compl__less__swap2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,Y)),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(uminus_uminus_assn,X)),Y) ) ).

% compl_less_swap2
tff(fact_4398_compl__less__swap2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le361264281704409273t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),X)
     => aa_Product_unit_o(ord_le361264281704409273t_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y) ) ).

% compl_less_swap2
tff(fact_4399_compl__less__swap1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Y),aa_assn_assn(uminus_uminus_assn,X))
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% compl_less_swap1
tff(fact_4400_compl__less__swap1,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( aa_Product_unit_o(ord_le361264281704409273t_unit(Y),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X))
     => aa_Product_unit_o(ord_le361264281704409273t_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% compl_less_swap1
tff(fact_4401_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_4402_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_4403_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_4404_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_4405_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_4406_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_4407_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_4408_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_4409_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_4410_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_4411_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_4412_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_4413_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_4414_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_4415_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_4416_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_4417_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K2),A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_4418_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_4419_length__induct,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys)),aa_list_nat_nat(size_size_list_nat,Xs2))
             => aa_list_nat_o(Pa,Ys) )
         => aa_list_nat_o(Pa,Xs2) )
     => aa_list_nat_o(Pa,Xs) ) ).

% length_induct
tff(fact_4420_length__induct,axiom,
    ! [Pa: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys)),size_size_list_o(Xs2))
             => aa_list_o_o(Pa,Ys) )
         => aa_list_o_o(Pa,Xs2) )
     => aa_list_o_o(Pa,Xs) ) ).

% length_induct
tff(fact_4421_length__induct,axiom,
    ! [Pa: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys)),size_size_list_int(Xs2))
             => aa_list_int_o(Pa,Ys) )
         => aa_list_int_o(Pa,Xs2) )
     => aa_list_int_o(Pa,Xs) ) ).

% length_induct
tff(fact_4422_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_4423_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M) = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
tff(fact_4424_zero__induct__lemma,axiom,
    ! [Pa: fun_nat_o,K2: nat,I2: nat] :
      ( aa_nat_o(Pa,K2)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,aa_nat_nat(suc,N5))
           => aa_nat_o(Pa,N5) )
       => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),I2)) ) ) ).

% zero_induct_lemma
tff(fact_4425_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),N2)),K2) ) ).

% less_imp_diff_less
tff(fact_4426_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( 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),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_4427_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_4428_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_4429_diff__le__self,axiom,
    ! [M: nat,N2: 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),N2)),M) ).

% diff_le_self
tff(fact_4430_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => 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,N2),L)) ) ).

% diff_le_mono
tff(fact_4431_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4432_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ) ).

% le_diff_iff
tff(fact_4433_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2) )
        <=> ( M = N2 ) ) ) ) ).

% eq_diff_iff
tff(fact_4434_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% Nat.diff_cancel
tff(fact_4435_diff__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% diff_cancel2
tff(fact_4436_diff__add__inverse,axiom,
    ! [N2: 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,N2),M)),N2) = M ).

% diff_add_inverse
tff(fact_4437_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),N2) = M ).

% diff_add_inverse2
tff(fact_4438_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2)) ).

% diff_mult_distrib
tff(fact_4439_diff__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2)) ).

% diff_mult_distrib2
tff(fact_4440_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_4441_nat__mult__distrib__neg,axiom,
    ! [Z3: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z3),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z3))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_4442_zdiff__int__split,axiom,
    ! [Pa: fun_int_o,X: nat,Y: nat] :
      ( aa_int_o(Pa,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
         => aa_int_o(Pa,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_4443_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_4444_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_4445_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_4446_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_4447_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_4448_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_4449_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_4450_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)) ) ).

% diff_le_eq
tff(fact_4451_diff__le__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_le_eq
tff(fact_4452_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),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)),C2) ) ).

% le_diff_eq
tff(fact_4453_le__diff__eq,axiom,
    ! [A: int,C2: 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,C2),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)),C2) ) ).

% le_diff_eq
tff(fact_4454_ordered__cancel__comm__monoid__diff__class_Odiff__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 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_4455_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: 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,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A)) ) ).

% le_add_diff
tff(fact_4456_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: 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,C2),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,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4457_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: 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,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4458_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: 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,C2),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4459_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: 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)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4460_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: 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),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4461_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: 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,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4462_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_4463_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: 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) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4464_add__le__imp__le__diff,axiom,
    ! [I2: rat,K2: rat,N2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N2),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4465_add__le__imp__le__diff,axiom,
    ! [I2: nat,K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4466_add__le__imp__le__diff,axiom,
    ! [I2: int,K2: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,N2),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4467_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K2: rat,N2: rat,J2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),K2))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K2)),N2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J2),K2))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,N2),K2)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4468_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K2: nat,N2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),N2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4469_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K2: int,N2: int,J2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),K2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),N2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),K2))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N2),K2)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4470_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( minus_minus_set_o(X,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(X),Y) ) ).

% diff_shunt_var
tff(fact_4471_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% diff_shunt_var
tff(fact_4472_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y) ) ).

% diff_shunt_var
tff(fact_4473_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y) ) ).

% diff_shunt_var
tff(fact_4474_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,X),Y) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% diff_shunt_var
tff(fact_4475_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C2),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)),C2) ) ).

% less_diff_eq
tff(fact_4476_less__diff__eq,axiom,
    ! [A: int,C2: 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,C2),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)),C2) ) ).

% less_diff_eq
tff(fact_4477_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C2),B)) ) ).

% diff_less_eq
tff(fact_4478_diff__less__eq,axiom,
    ! [A: int,B: int,C2: 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)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_less_eq
tff(fact_4479_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),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4480_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_4481_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_4482_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2) )
    <=> ( 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(minus_minus_rat,A),B)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_4483_eq__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2) )
    <=> ( 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)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_4484_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: 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)),C2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2) )
    <=> ( C2 = 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(minus_minus_rat,B),A)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_4485_eq__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: 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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2) )
    <=> ( C2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_4486_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)) ).

% square_diff_square_factored
tff(fact_4487_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_4488_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_4489_ex__nat,axiom,
    ! [Pa: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(Pa,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_4490_all__nat,axiom,
    ! [Pa: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(Pa,X_12)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(Pa,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_4491_eq__nat__nat__iff,axiom,
    ! [Z3: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( ( aa_int_nat(nat2,Z3) = aa_int_nat(nat2,Z6) )
        <=> ( Z3 = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_4492_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_4493_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_4494_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_4495_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_4496_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_4497_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_4498_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_4499_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_4500_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_4501_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_4502_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_4503_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_4504_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_4505_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_4506_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_4507_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_4508_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_4509_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_4510_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_4511_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_4512_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_4513_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_4514_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_4515_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_4516_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_4517_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_4518_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_4519_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_4520_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_4521_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_4522_Suc__to__right,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_nat(suc,N2) = M )
     => ( N2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_4523_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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),N2)),M) ) ) ).

% diff_less
tff(fact_4524_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ) ) ).

% Suc_diff_Suc
tff(fact_4525_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_4526_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% Suc_diff_le
tff(fact_4527_diff__add__0,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_4528_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ) ) ).

% less_diff_iff
tff(fact_4529_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2)) ) ) ).

% diff_less_mono
tff(fact_4530_less__diff__conv,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2) ) ).

% less_diff_conv
tff(fact_4531_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_4532_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I2) = K2 )
      <=> ( J2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4533_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4534_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4535_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2) ) ) ).

% Nat.le_diff_conv2
tff(fact_4536_le__diff__conv,axiom,
    ! [J2: nat,K2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)) ) ).

% le_diff_conv
tff(fact_4537_int__le__induct,axiom,
    ! [I2: int,K2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2)
     => ( aa_int_o(Pa,K2)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K2)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_le_induct
tff(fact_4538_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2)) = 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)),N2) ).

% diff_Suc_eq_diff_pred
tff(fact_4539_int__less__induct,axiom,
    ! [I2: int,K2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K2)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_less_induct
tff(fact_4540_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ).

% mod_geq
tff(fact_4541_mod__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ).

% mod_if
tff(fact_4542_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ).

% le_mod_geq
tff(fact_4543_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_4544_le__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: 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)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
    <=> 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,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C2)),D2) ) ).

% le_add_iff1
tff(fact_4545_le__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: 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)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
    <=> 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)),C2)),D2) ) ).

% le_add_iff1
tff(fact_4546_le__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: 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)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),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(minus_minus_rat,B),A)),E)),D2)) ) ).

% le_add_iff2
tff(fact_4547_le__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: 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)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D2)) ) ).

% le_add_iff2
tff(fact_4548_less__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: 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)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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(minus_minus_rat,B),A)),E)),D2)) ) ).

% less_add_iff2
tff(fact_4549_less__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: 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)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D2)) ) ).

% less_add_iff2
tff(fact_4550_less__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: 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)),C2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D2))
    <=> 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,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E)),C2)),D2) ) ).

% less_add_iff1
tff(fact_4551_less__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: 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)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D2))
    <=> 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)),C2)),D2) ) ).

% less_add_iff1
tff(fact_4552_nat__mono__iff,axiom,
    ! [Z3: int,W2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ) ).

% nat_mono_iff
tff(fact_4553_square__diff__one__factored,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_4554_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),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,X),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4555_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_4556_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_4557_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_4558_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_4559_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_4560_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_4561_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_4562_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z3) ) ).

% zless_nat_eq_int_zless
tff(fact_4563_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_4564_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_4565_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_4566_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_4567_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_4568_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_4569_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),N2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% nat_le_iff
tff(fact_4570_int__eq__iff,axiom,
    ! [M: nat,Z3: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z3 )
    <=> ( ( M = aa_int_nat(nat2,Z3) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3) ) ) ).

% int_eq_iff
tff(fact_4571_nat__0__le,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z3)) = Z3 ) ) ).

% nat_0_le
tff(fact_4572_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_4573_inf__shunt,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = bot_bo3516217578726220278_int_o )
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),uminus103585356917985579_int_o(Y)) ) ).

% inf_shunt
tff(fact_4574_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),uminus6524753893492686040at_nat(Y)) ) ).

% inf_shunt
tff(fact_4575_inf__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o(X,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(X),uminus_uminus_set_o(Y)) ) ).

% inf_shunt
tff(fact_4576_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),uminus5710092332889474511et_nat(Y)) ) ).

% inf_shunt
tff(fact_4577_inf__shunt,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% inf_shunt
tff(fact_4578_inf__shunt,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(X),Y) = bot_bot_Product_unit )
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% inf_shunt
tff(fact_4579_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),uminus1532241313380277803et_int(Y)) ) ).

% inf_shunt
tff(fact_4580_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z3)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(Y)),Z3)) ) ).

% shunt1
tff(fact_4581_shunt1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z3)
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,uminus103585356917985579_int_o(Y)),Z3)) ) ).

% shunt1
tff(fact_4582_shunt1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),Y),Z3)
    <=> ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(Y)),Z3)) ) ).

% shunt1
tff(fact_4583_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),Z3)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(Y)),Z3)) ) ).

% shunt1
tff(fact_4584_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z3)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(Y)),Z3)) ) ).

% shunt1
tff(fact_4585_shunt1,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z3)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,Y)),Z3)) ) ).

% shunt1
tff(fact_4586_shunt1,axiom,
    ! [X: product_unit,Y: product_unit,Z3: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(X),Y)),Z3)
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)),Z3)) ) ).

% shunt1
tff(fact_4587_shunt1,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)),Z3)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(Y)),Z3)) ) ).

% shunt1
tff(fact_4588_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),uminus6524753893492686040at_nat(Y))),Z3)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z3)) ) ).

% shunt2
tff(fact_4589_shunt2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z3: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),uminus103585356917985579_int_o(Y))),Z3)
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z3)) ) ).

% shunt2
tff(fact_4590_shunt2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,X),uminus2330091110623919550at_nat(Y)),Z3)
    <=> ord_le8081472938463900775at_nat(X,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Y),Z3)) ) ).

% shunt2
tff(fact_4591_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),uminus935396558254630718at_nat(Y))),Z3)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z3)) ) ).

% shunt2
tff(fact_4592_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),uminus5710092332889474511et_nat(Y))),Z3)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z3)) ) ).

% shunt2
tff(fact_4593_shunt2,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y))),Z3)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z3)) ) ).

% shunt2
tff(fact_4594_shunt2,axiom,
    ! [X: product_unit,Y: product_unit,Z3: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y))),Z3)
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(X),aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(Y),Z3)) ) ).

% shunt2
tff(fact_4595_shunt2,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),uminus1532241313380277803et_int(Y))),Z3)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Y),Z3)) ) ).

% shunt2
tff(fact_4596_sup__neg__inf,axiom,
    ! [P2: set_Pr1261947904930325089at_nat,Q4: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,P2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q4),R))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P2),uminus6524753893492686040at_nat(Q4))),R) ) ).

% sup_neg_inf
tff(fact_4597_sup__neg__inf,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q4: fun_Pr3238374608701176162_int_o,R: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,P2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Q4),R))
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,P2),uminus103585356917985579_int_o(Q4))),R) ) ).

% sup_neg_inf
tff(fact_4598_sup__neg__inf,axiom,
    ! [P2: set_Pr8551490117392284871at_nat,Q4: set_Pr8551490117392284871at_nat,R: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(P2,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Q4),R))
    <=> ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,P2),uminus2330091110623919550at_nat(Q4)),R) ) ).

% sup_neg_inf
tff(fact_4599_sup__neg__inf,axiom,
    ! [P2: set_Pr4329608150637261639at_nat,Q4: set_Pr4329608150637261639at_nat,R: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(P2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Q4),R))
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,P2),uminus935396558254630718at_nat(Q4))),R) ) ).

% sup_neg_inf
tff(fact_4600_sup__neg__inf,axiom,
    ! [P2: set_nat,Q4: set_nat,R: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,P2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q4),R))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P2),uminus5710092332889474511et_nat(Q4))),R) ) ).

% sup_neg_inf
tff(fact_4601_sup__neg__inf,axiom,
    ! [P2: assn,Q4: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,P2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Q4),R))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P2),aa_assn_assn(uminus_uminus_assn,Q4))),R) ) ).

% sup_neg_inf
tff(fact_4602_sup__neg__inf,axiom,
    ! [P2: product_unit,Q4: product_unit,R: product_unit] :
      ( aa_Product_unit_o(ord_le3221252021190050221t_unit(P2),aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(Q4),R))
    <=> aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(P2),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Q4))),R) ) ).

% sup_neg_inf
tff(fact_4603_sup__neg__inf,axiom,
    ! [P2: set_int,Q4: set_int,R: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,P2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,Q4),R))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,P2),uminus1532241313380277803et_int(Q4))),R) ) ).

% sup_neg_inf
tff(fact_4604_nat__power__eq,axiom,
    ! [Z3: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z3),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z3)),N2) ) ) ).

% nat_power_eq
tff(fact_4605_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,I2))),N2) ) ).

% diff_Suc_less
tff(fact_4606_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) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ aa_nat_o(Pa,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_4607_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) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => aa_nat_o(Pa,D5) ) ) ) ).

% nat_diff_split
tff(fact_4608_less__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K2)),I2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)) ) ) ).

% less_diff_conv2
tff(fact_4609_minusinfinity,axiom,
    ! [D2: int,P1: fun_int_o,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2))) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
             => ( aa_int_o(Pa,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(Pa,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_4610_plusinfinity,axiom,
    ! [D2: int,P3: fun_int_o,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K: int] :
            ( aa_int_o(P3,X3)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2))) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
             => ( aa_int_o(Pa,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P3,X_13)
           => ? [X_1: int] : aa_int_o(Pa,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_4611_int__cases4,axiom,
    ! [M: int] :
      ( ! [N5: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N5)
     => ~ ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) ) ) ) ).

% int_cases4
tff(fact_4612_nat__eq__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),U)),M) = N2 ) ) ) ).

% nat_eq_add_iff1
tff(fact_4613_nat__eq__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2) )
      <=> ( 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,J2),I2)),U)),N2) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4614_nat__le__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),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(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),U)),M)),N2) ) ) ).

% nat_le_add_iff1
tff(fact_4615_nat__le__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2))
      <=> 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,J2),I2)),U)),N2)) ) ) ).

% nat_le_add_iff2
tff(fact_4616_nat__diff__add__eq1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),U)),M)),N2) ) ) ).

% nat_diff_add_eq1
tff(fact_4617_nat__diff__add__eq2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2)) = 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,J2),I2)),U)),N2)) ) ) ).

% nat_diff_add_eq2
tff(fact_4618_int__induct,axiom,
    ! [Pa: fun_int_o,K2: int,I2: int] :
      ( aa_int_o(Pa,K2)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I3)
           => ( aa_int_o(Pa,I3)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K2)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_induct
tff(fact_4619_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_4620_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ~ ! [N5: nat] : K2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) ) ).

% nonpos_int_cases
tff(fact_4621_negative__zle__0,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),zero_zero_int) ).

% negative_zle_0
tff(fact_4622_nat__plus__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_plus_as_int
tff(fact_4623_nat__times__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_times_as_int
tff(fact_4624_nat__div__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_div_as_int
tff(fact_4625_nat__mod__as__int,axiom,
    ! [X5: nat,Xa2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X5),Xa2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X5)),aa_nat_int(semiri1314217659103216013at_int,Xa2))) ).

% nat_mod_as_int
tff(fact_4626_frac__le__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z3))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4627_frac__less__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W2),Z3))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W2),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4628_nat__less__eq__zless,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ) ).

% nat_less_eq_zless
tff(fact_4629_nat__le__eq__zle,axiom,
    ! [W2: int,Z3: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z3))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z3) ) ) ).

% nat_le_eq_zle
tff(fact_4630_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_4631_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_4632_split__nat,axiom,
    ! [Pa: fun_nat_o,I2: int] :
      ( aa_nat_o(Pa,aa_int_nat(nat2,I2))
    <=> ( ! [N: nat] :
            ( ( I2 = aa_nat_int(semiri1314217659103216013at_int,N) )
           => aa_nat_o(Pa,N) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),zero_zero_int)
         => aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_4633_le__nat__iff,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N2)),K2) ) ) ).

% le_nat_iff
tff(fact_4634_nat__add__distrib,axiom,
    ! [Z3: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z3)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_4635_nat__mult__distrib,axiom,
    ! [Z3: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z3),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z3)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_4636_Suc__as__int,axiom,
    ! [X5: nat] : aa_nat_nat(suc,X5) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X5)),one_one_int)) ).

% Suc_as_int
tff(fact_4637_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),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),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4638_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4639_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4640_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> 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),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4641_minus__divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),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_4642_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_4643_minus__divide__add__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z3))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z3))),Z3) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4644_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z3: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z3))),B) = $ite(Z3 = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(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),Z3))),Z3)) ).

% add_divide_eq_if_simps(3)
tff(fact_4645_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_4646_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_4647_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_4648_power__diff,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ) ) ).

% power_diff
tff(fact_4649_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)) = 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),N2)) ) ) ) ).

% power_diff
tff(fact_4650_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)) = 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),N2)) ) ) ) ).

% power_diff
tff(fact_4651_power__diff,axiom,
    ! [A: code_natural,N2: nat,M: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N2)) ) ) ) ).

% power_diff
tff(fact_4652_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_diff
tff(fact_4653_nz__le__conv__less,axiom,
    ! [K2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_4654_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4655_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4656_add__eq__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = $ite(M = zero_zero_nat,N2,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)),N2))) ).

% add_eq_if
tff(fact_4657_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_4658_nat__less__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J2)),U)),M)),N2) ) ) ).

% nat_less_add_iff1
tff(fact_4659_nat__less__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),N2))
      <=> 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,J2),I2)),U)),N2)) ) ) ).

% nat_less_add_iff2
tff(fact_4660_decr__mult__lemma,axiom,
    ! [D2: int,Pa: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X5: int] :
              ( aa_int_o(Pa,X5)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4661_div__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | ( N2 = 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),N2)),N2)) ) ).

% div_if
tff(fact_4662_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = 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),N2)),N2)) ) ) ) ).

% div_geq
tff(fact_4663_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N5: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) )
       => ~ ! [N5: nat] :
              ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) ) ) ) ).

% int_cases3
tff(fact_4664_mult__eq__if,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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)),N2))) ).

% mult_eq_if
tff(fact_4665_not__zle__0__negative,axiom,
    ! [N2: 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,N2)))) ).

% not_zle_0_negative
tff(fact_4666_negD,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
     => ? [N5: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N5))) ) ).

% negD
tff(fact_4667_negative__zless__0,axiom,
    ! [N2: 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,N2)))),zero_zero_int) ).

% negative_zless_0
tff(fact_4668_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4669_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N2))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),N2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),N2)) ) ) ).

% verit_less_mono_div_int2
tff(fact_4670_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_4671_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_4672_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z3)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_4673_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_4674_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),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),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4675_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4676_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),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),C2)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4677_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
      <=> 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),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4678_minus__divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),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_4679_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_4680_power__diff__power__eq,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),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,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4681_power__diff__power__eq,axiom,
    ! [A: nat,M: nat,N2: 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),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)),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,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4682_power__diff__power__eq,axiom,
    ! [A: int,M: nat,N2: 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),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),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),N2)),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,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4683_power__diff__power__eq,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_nat_Code_natural(power_7079662738309270450atural(A),M),aa_nat_Code_natural(power_7079662738309270450atural(A),N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),divide5121882707175180666atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% power_diff_power_eq
tff(fact_4684_power__minus__mult,axiom,
    ! [N2: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N2) ) ) ).

% power_minus_mult
tff(fact_4685_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N2) ) ) ).

% power_minus_mult
tff(fact_4686_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) ) ) ).

% power_minus_mult
tff(fact_4687_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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,N2),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2) ) ) ).

% power_minus_mult
tff(fact_4688_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = 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),N2)),N2)) ) ) ) ).

% le_div_geq
tff(fact_4689_neg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ~ ! [N5: nat] :
            ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5) ) ) ).

% neg_int_cases
tff(fact_4690_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( 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,N2),K2)) = 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,N2),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4691_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4692_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( 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,N2),K2)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4693_pochhammer__code,axiom,
    ! [A: rat,N2: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = $ite(N2 = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ha(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_4694_pochhammer__code,axiom,
    ! [A: int,N2: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2) = $ite(N2 = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_hb(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_4695_pochhammer__code,axiom,
    ! [A: code_integer,N2: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2) = $ite(N2 = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_hc(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_4696_pochhammer__code,axiom,
    ! [A: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = $ite(N2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_hd(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_4697_size__union,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] : size_s8510653225128441779at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s8510653225128441779at_nat(M6)),size_s8510653225128441779at_nat(N7)) ).

% size_union
tff(fact_4698_normalize__negative,axiom,
    ! [Q4: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q4),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_4699_length__rule,axiom,
    ! [A: array_nat,Xs: list_nat] : hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),array_len_nat(A),aa_lis6041139530851742867t_assn(aTP_Lamm_he(A),Xs)) ).

% length_rule
tff(fact_4700_length__rule,axiom,
    ! [A: array_o,Xs: list_o] : hoare_3067605981109127869le_nat(snga_assn_o(A,Xs),array_len_o(A),aa_lis7996121028773434071t_assn(aTP_Lamm_hf(A),Xs)) ).

% length_rule
tff(fact_4701_length__rule,axiom,
    ! [A: array_int,Xs: list_int] : hoare_3067605981109127869le_nat(snga_assn_int(A,Xs),array_len_int(A),aa_lis2816288184639765743t_assn(aTP_Lamm_hg(A),Xs)) ).

% length_rule
tff(fact_4702_Diff__cancel,axiom,
    ! [A4: set_o] : minus_minus_set_o(A4,A4) = bot_bot_set_o ).

% Diff_cancel
tff(fact_4703_Diff__cancel,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),A4) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_4704_Diff__cancel,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),A4) = bot_bot_set_int ).

% Diff_cancel
tff(fact_4705_Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),A4) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_4706_empty__Diff,axiom,
    ! [A4: set_o] : minus_minus_set_o(bot_bot_set_o,A4) = bot_bot_set_o ).

% empty_Diff
tff(fact_4707_empty__Diff,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A4) = bot_bot_set_nat ).

% empty_Diff
tff(fact_4708_empty__Diff,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,bot_bot_set_int),A4) = bot_bot_set_int ).

% empty_Diff
tff(fact_4709_empty__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bot_bo2099793752762293965at_nat),A4) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_4710_Diff__empty,axiom,
    ! [A4: set_o] : minus_minus_set_o(A4,bot_bot_set_o) = A4 ).

% Diff_empty
tff(fact_4711_Diff__empty,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),bot_bot_set_nat) = A4 ).

% Diff_empty
tff(fact_4712_Diff__empty,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),bot_bot_set_int) = A4 ).

% Diff_empty
tff(fact_4713_Diff__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),bot_bo2099793752762293965at_nat) = A4 ).

% Diff_empty
tff(fact_4714_Compl__anti__mono,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(B4)),uminus1532241313380277803et_int(A4)) ) ).

% Compl_anti_mono
tff(fact_4715_Compl__subset__Compl__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,uminus1532241313380277803et_int(A4)),uminus1532241313380277803et_int(B4))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4) ) ).

% Compl_subset_Compl_iff
tff(fact_4716_Diff__Compl,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),uminus5710092332889474511et_nat(B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ).

% Diff_Compl
tff(fact_4717_Diff__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),uminus6524753893492686040at_nat(B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ).

% Diff_Compl
tff(fact_4718_inter__compl__diff__conv,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) ).

% inter_compl_diff_conv
tff(fact_4719_inter__compl__diff__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),uminus6524753893492686040at_nat(B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) ).

% inter_compl_diff_conv
tff(fact_4720_Compl__Diff__eq,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(A4)),B4) ).

% Compl_Diff_eq
tff(fact_4721_Compl__Diff__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A4)),B4) ).

% Compl_Diff_eq
tff(fact_4722_Compl__Diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A4)),B4) ).

% Compl_Diff_eq
tff(fact_4723_Compl__Diff__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(A4)),B4) ).

% Compl_Diff_eq
tff(fact_4724_Un__Diff__cancel,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B4),A4)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) ).

% Un_Diff_cancel
tff(fact_4725_Un__Diff__cancel,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),A4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) ).

% Un_Diff_cancel
tff(fact_4726_Un__Diff__cancel,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) ).

% Un_Diff_cancel
tff(fact_4727_Un__Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) ).

% Un_Diff_cancel
tff(fact_4728_Un__Diff__cancel2,axiom,
    ! [B4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B4),A4)),A4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),A4) ).

% Un_Diff_cancel2
tff(fact_4729_Un__Diff__cancel2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),A4)),A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),A4) ).

% Un_Diff_cancel2
tff(fact_4730_Un__Diff__cancel2,axiom,
    ! [B4: set_nat,A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)),A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A4) ).

% Un_Diff_cancel2
tff(fact_4731_Un__Diff__cancel2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)),A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),A4) ).

% Un_Diff_cancel2
tff(fact_4732_diff__diff__add__mset,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,Pa: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(minus_4286766774447292334at_nat(M6,N7),Pa) = minus_4286766774447292334at_nat(M6,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),Pa)) ).

% diff_diff_add_mset
tff(fact_4733_union__eq__empty,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7) = zero_z1048942125864253310at_nat )
    <=> ( ( M6 = zero_z1048942125864253310at_nat )
        & ( N7 = zero_z1048942125864253310at_nat ) ) ) ).

% union_eq_empty
tff(fact_4734_empty__eq__union,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7) )
    <=> ( ( M6 = zero_z1048942125864253310at_nat )
        & ( N7 = zero_z1048942125864253310at_nat ) ) ) ).

% empty_eq_union
tff(fact_4735_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(X),Y) )
    <=> ( ( X = zero_z1048942125864253310at_nat )
        & ( Y = zero_z1048942125864253310at_nat ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_4736_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(X),Y) = zero_z1048942125864253310at_nat )
    <=> ( ( X = zero_z1048942125864253310at_nat )
        & ( Y = zero_z1048942125864253310at_nat ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_4737_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A4,B4) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4738_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4739_Diff__eq__empty__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4740_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_4741_Diff__disjoint,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(A4,minus_minus_set_o(B4,A4)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_4742_Diff__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_4743_Diff__disjoint,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_4744_Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_4745_Compl__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),uminus6524753893492686040at_nat(A4)) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint
tff(fact_4746_Compl__disjoint,axiom,
    ! [A4: set_o] : inf_inf_set_o(A4,uminus_uminus_set_o(A4)) = bot_bot_set_o ).

% Compl_disjoint
tff(fact_4747_Compl__disjoint,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(A4)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_4748_Compl__disjoint,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(A4)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_4749_Compl__disjoint2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A4)),A4) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint2
tff(fact_4750_Compl__disjoint2,axiom,
    ! [A4: set_o] : inf_inf_set_o(uminus_uminus_set_o(A4),A4) = bot_bot_set_o ).

% Compl_disjoint2
tff(fact_4751_Compl__disjoint2,axiom,
    ! [A4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A4)),A4) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_4752_Compl__disjoint2,axiom,
    ! [A4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,uminus1532241313380277803et_int(A4)),A4) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_4753_mod__not__dist,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(uminus_uminus_assn,Pa)),Ha)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Ha)
        & ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha) ) ) ).

% mod_not_dist
tff(fact_4754_minus__assn__def,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(uminus_uminus_assn,B)) ).

% minus_assn_def
tff(fact_4755_Multiset_Odiff__add,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,Qa: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(M6,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),Qa)) = minus_4286766774447292334at_nat(minus_4286766774447292334at_nat(M6,N7),Qa) ).

% Multiset.diff_add
tff(fact_4756_diff__union__cancelL,axiom,
    ! [N7: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),M6),N7) = M6 ).

% diff_union_cancelL
tff(fact_4757_diff__union__cancelR,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7),N7) = M6 ).

% diff_union_cancelR
tff(fact_4758_union__diff__assoc,axiom,
    ! [C3: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat,A4: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat(C3,B4) = zero_z1048942125864253310at_nat )
     => ( minus_4286766774447292334at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A4),B4),C3) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A4),minus_4286766774447292334at_nat(B4,C3)) ) ) ).

% union_diff_assoc
tff(fact_4759_empty__neutral_I1_J,axiom,
    ! [X: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(zero_z1048942125864253310at_nat),X) = X ).

% empty_neutral(1)
tff(fact_4760_empty__neutral_I2_J,axiom,
    ! [X: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(X),zero_z1048942125864253310at_nat) = X ).

% empty_neutral(2)
tff(fact_4761_mset__distrib,axiom,
    ! [A4: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A4),B4) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7) )
     => ~ ! [Am: multis2468970476368604999at_nat,An: multis2468970476368604999at_nat] :
            ( ( A4 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Am),An) )
           => ! [Bm: multis2468970476368604999at_nat,Bn: multis2468970476368604999at_nat] :
                ( ( B4 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Bm),Bn) )
               => ( ( M6 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Am),Bm) )
                 => ( N7 != aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_4762_multi__union__self__other__eq,axiom,
    ! [A4: multis2468970476368604999at_nat,X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A4),X6) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A4),Y6) )
     => ( X6 = Y6 ) ) ).

% multi_union_self_other_eq
tff(fact_4763_union__right__cancel,axiom,
    ! [M6: multis2468970476368604999at_nat,K5: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),K5) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),K5) )
    <=> ( M6 = N7 ) ) ).

% union_right_cancel
tff(fact_4764_union__left__cancel,axiom,
    ! [K5: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(K5),M6) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(K5),N7) )
    <=> ( M6 = N7 ) ) ).

% union_left_cancel
tff(fact_4765_union__commute,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),M6) ).

% union_commute
tff(fact_4766_union__lcomm,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,K5: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),K5)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),K5)) ).

% union_lcomm
tff(fact_4767_union__assoc,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,K5: multis2468970476368604999at_nat] : aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),N7)),K5) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(M6),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(N7),K5)) ).

% union_assoc
tff(fact_4768_Diff__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),C3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,D4),B4)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,C3),D4)) ) ) ).

% Diff_mono
tff(fact_4769_Diff__mono,axiom,
    ! [A4: set_int,C3: set_int,D4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),C3)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,D4),B4)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C3),D4)) ) ) ).

% Diff_mono
tff(fact_4770_Diff__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),A4) ).

% Diff_subset
tff(fact_4771_Diff__subset,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)),A4) ).

% Diff_subset
tff(fact_4772_double__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),C3)
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,C3),A4)) = A4 ) ) ) ).

% double_diff
tff(fact_4773_double__diff,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C3)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,C3),A4)) = A4 ) ) ) ).

% double_diff
tff(fact_4774_Diff__Int__distrib2,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) ).

% Diff_Int_distrib2
tff(fact_4775_Diff__Int__distrib2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) ).

% Diff_Int_distrib2
tff(fact_4776_Diff__Int__distrib,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),A4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C3),B4)) ).

% Diff_Int_distrib
tff(fact_4777_Diff__Int__distrib,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),A4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C3),B4)) ).

% Diff_Int_distrib
tff(fact_4778_Diff__Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ).

% Diff_Diff_Int
tff(fact_4779_Diff__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ).

% Diff_Diff_Int
tff(fact_4780_Diff__Int2,axiom,
    ! [A4: set_nat,C3: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),C3)),B4) ).

% Diff_Int2
tff(fact_4781_Diff__Int2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),C3)),B4) ).

% Diff_Int2
tff(fact_4782_Int__Diff,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),C3)) ).

% Int_Diff
tff(fact_4783_Int__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),C3)) ).

% Int_Diff
tff(fact_4784_Un__Diff,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)),C3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),C3)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B4),C3)) ).

% Un_Diff
tff(fact_4785_Un__Diff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)),C3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),C3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),C3)) ).

% Un_Diff
tff(fact_4786_Un__Diff,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),C3)) ).

% Un_Diff
tff(fact_4787_Un__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),C3)) ).

% Un_Diff
tff(fact_4788_set__diff__diff__left,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4)),C3) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) ).

% set_diff_diff_left
tff(fact_4789_set__diff__diff__left,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),C3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) ).

% set_diff_diff_left
tff(fact_4790_set__diff__diff__left,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) ).

% set_diff_diff_left
tff(fact_4791_set__diff__diff__left,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)) ).

% set_diff_diff_left
tff(fact_4792_psubset__imp__ex__mem,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),B4)
     => ? [B3: $o] : aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_4793_psubset__imp__ex__mem,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A4),B4)
     => ? [B3: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B3),minus_2612819937483484256nt_int(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_4794_psubset__imp__ex__mem,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
     => ? [B3: set_nat] : aa_set_set_nat_o(member_set_nat(B3),minus_2163939370556025621et_nat(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_4795_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
     => ? [B3: nat] : aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_4796_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
     => ? [B3: int] : aa_set_int_o(member_int(B3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_4797_psubset__imp__ex__mem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),B4)
     => ? [B3: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_4798_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Uu) = product_Unity ).

% uminus_unit_def
tff(fact_4799_Collect__imp__eq,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] : collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_hh(Pa),Qa)) = sup_sup_set_list_nat(uminus3195874150345416415st_nat(collect_list_nat(Pa)),collect_list_nat(Qa)) ).

% Collect_imp_eq
tff(fact_4800_Collect__imp__eq,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] : collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_hi(Pa),Qa)) = sup_su2047564715030645325nt_int(uminus6423885277529793776nt_int(collec5210948495886036740nt_int(Pa)),collec5210948495886036740nt_int(Qa)) ).

% Collect_imp_eq
tff(fact_4801_Collect__imp__eq,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] : collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_hj(Pa),Qa)) = sup_sup_set_set_nat(uminus613421341184616069et_nat(collect_set_nat(Pa)),collect_set_nat(Qa)) ).

% Collect_imp_eq
tff(fact_4802_Collect__imp__eq,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] : collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_hk(Pa),Qa)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(collect_int(Pa))),collect_int(Qa)) ).

% Collect_imp_eq
tff(fact_4803_Collect__imp__eq,axiom,
    ! [Pa: fun_Pr1099658227000185396_nat_o,Qa: fun_Pr1099658227000185396_nat_o] : collec5204685387357076818at_nat(aa_fun9125434449681309094_nat_o(aTP_Lamm_hl(Pa),Qa)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(collec5204685387357076818at_nat(Pa))),collec5204685387357076818at_nat(Qa)) ).

% Collect_imp_eq
tff(fact_4804_Collect__imp__eq,axiom,
    ! [Pa: fun_Pr3059359313727237556_nat_o,Qa: fun_Pr3059359313727237556_nat_o] : collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_hm(Pa),Qa)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(collec6321179662152712658at_nat(Pa))),collec6321179662152712658at_nat(Qa)) ).

% Collect_imp_eq
tff(fact_4805_Collect__imp__eq,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] : collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_hn(Pa),Qa)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(collect_nat(Pa))),collect_nat(Qa)) ).

% Collect_imp_eq
tff(fact_4806_subset__minus__empty,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( minus_minus_set_o(A4,B4) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_4807_subset__minus__empty,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_4808_subset__minus__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
tff(fact_4809_subset__minus__empty,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_4810_subset__Compl__self__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(A4))
    <=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
tff(fact_4811_subset__Compl__self__eq,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(A4))
    <=> ( A4 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_4812_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(A4))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_4813_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),uminus1532241313380277803et_int(A4))
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_4814_disjoint__alt__simp2,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A4,B4) != A4 )
    <=> ( inf_inf_set_o(A4,B4) != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
tff(fact_4815_disjoint__alt__simp2,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) != A4 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_4816_disjoint__alt__simp2,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) != A4 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_4817_disjoint__alt__simp2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) != A4 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_4818_disjoint__alt__simp1,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A4,B4) = A4 )
    <=> ( inf_inf_set_o(A4,B4) = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
tff(fact_4819_disjoint__alt__simp1,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = A4 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_4820_disjoint__alt__simp1,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) = A4 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_4821_disjoint__alt__simp1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = A4 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_4822_Int__Diff__disjoint,axiom,
    ! [A4: set_o,B4: set_o] : inf_inf_set_o(inf_inf_set_o(A4,B4),minus_minus_set_o(A4,B4)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_4823_Int__Diff__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_4824_Int__Diff__disjoint,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_4825_Int__Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_4826_Diff__triv,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
     => ( minus_minus_set_o(A4,B4) = A4 ) ) ).

% Diff_triv
tff(fact_4827_Diff__triv,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = A4 ) ) ).

% Diff_triv
tff(fact_4828_Diff__triv,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) = A4 ) ) ).

% Diff_triv
tff(fact_4829_Diff__triv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = A4 ) ) ).

% Diff_triv
tff(fact_4830_Diff__subset__conv,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4),C3)
    <=> ord_le8081472938463900775at_nat(A4,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) ) ).

% Diff_subset_conv
tff(fact_4831_Diff__subset__conv,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),C3)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) ) ).

% Diff_subset_conv
tff(fact_4832_Diff__subset__conv,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),C3)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) ) ).

% Diff_subset_conv
tff(fact_4833_Diff__subset__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),C3)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)) ) ).

% Diff_subset_conv
tff(fact_4834_Diff__subset__conv,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)),C3)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B4),C3)) ) ).

% Diff_subset_conv
tff(fact_4835_Diff__partition,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ord_le8081472938463900775at_nat(A4,B4)
     => ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_4836_Diff__partition,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_4837_Diff__partition,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_4838_Diff__partition,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_4839_Diff__partition,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)) = B4 ) ) ).

% Diff_partition
tff(fact_4840_pochhammer__pos,axiom,
    ! [X: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(X),N2)) ) ).

% pochhammer_pos
tff(fact_4841_pochhammer__pos,axiom,
    ! [X: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N2)) ) ).

% pochhammer_pos
tff(fact_4842_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => 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,X),N2)) ) ).

% pochhammer_pos
tff(fact_4843_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => 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,X),N2)) ) ).

% pochhammer_pos
tff(fact_4844_Un__Diff__Int,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4845_Un__Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4846_Un__Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4847_Un__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) = A4 ).

% Un_Diff_Int
tff(fact_4848_Int__Diff__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4849_Int__Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4850_Int__Diff__Un,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4851_Int__Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = A4 ).

% Int_Diff_Un
tff(fact_4852_Diff__Int,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,B4),C3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),C3)) ).

% Diff_Int
tff(fact_4853_Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),C3)) ).

% Diff_Int
tff(fact_4854_Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),C3)) ).

% Diff_Int
tff(fact_4855_Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),C3)) ).

% Diff_Int
tff(fact_4856_Diff__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),B4)),aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(minus_5060654252129873198at_nat,A4),C3)) ).

% Diff_Un
tff(fact_4857_Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),C3)) ).

% Diff_Un
tff(fact_4858_Diff__Un,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),C3)) ).

% Diff_Un
tff(fact_4859_Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),C3)) ).

% Diff_Un
tff(fact_4860_Compl__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat(A4)),uminus6524753893492686040at_nat(B4)) ).

% Compl_Int
tff(fact_4861_Compl__Int,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,uminus2330091110623919550at_nat(A4)),uminus2330091110623919550at_nat(B4)) ).

% Compl_Int
tff(fact_4862_Compl__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,uminus935396558254630718at_nat(A4)),uminus935396558254630718at_nat(B4)) ).

% Compl_Int
tff(fact_4863_Compl__Int,axiom,
    ! [A4: set_nat,B4: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,uminus5710092332889474511et_nat(A4)),uminus5710092332889474511et_nat(B4)) ).

% Compl_Int
tff(fact_4864_Compl__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,uminus6524753893492686040at_nat(A4)),uminus6524753893492686040at_nat(B4)) ).

% Compl_Un
tff(fact_4865_Compl__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : uminus2330091110623919550at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,uminus2330091110623919550at_nat(A4)),uminus2330091110623919550at_nat(B4)) ).

% Compl_Un
tff(fact_4866_Compl__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,uminus935396558254630718at_nat(A4)),uminus935396558254630718at_nat(B4)) ).

% Compl_Un
tff(fact_4867_Compl__Un,axiom,
    ! [A4: set_nat,B4: set_nat] : uminus5710092332889474511et_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,uminus5710092332889474511et_nat(A4)),uminus5710092332889474511et_nat(B4)) ).

% Compl_Un
tff(fact_4868_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_4869_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_4870_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4871_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4872_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4873_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),uminus1532241313380277803et_int(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4874_pochhammer__nonneg,axiom,
    ! [X: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(comm_s8582702949713902594nteger(X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4875_pochhammer__nonneg,axiom,
    ! [X: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4876_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => 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,X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4877_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => 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,X),N2)) ) ).

% pochhammer_nonneg
tff(fact_4878_disjoint__alt__simp3,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(minus_minus_set_o(A4,B4)),A4)
    <=> ( inf_inf_set_o(A4,B4) != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
tff(fact_4879_disjoint__alt__simp3,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),A4)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_4880_disjoint__alt__simp3,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)),A4)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_4881_disjoint__alt__simp3,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),A4)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_4882_uminus__assn__def,axiom,
    ! [Pa: assn] : aa_assn_assn(uminus_uminus_assn,Pa) = abs_assn(aTP_Lamm_ho(Pa)) ).

% uminus_assn_def
tff(fact_4883_pochhammer__rec,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),N2)) ).

% pochhammer_rec
tff(fact_4884_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_4885_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_4886_pochhammer__rec,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_4887_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),N2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% pochhammer_Suc
tff(fact_4888_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N2))) ).

% pochhammer_Suc
tff(fact_4889_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N2))) ).

% pochhammer_Suc
tff(fact_4890_pochhammer__Suc,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))) ).

% pochhammer_Suc
tff(fact_4891_pochhammer__rec_H,axiom,
    ! [Z3: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z3),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),aa_nat_rat(semiri681578069525770553at_rat,N2))),aa_nat_rat(comm_s4028243227959126397er_rat(Z3),N2)) ).

% pochhammer_rec'
tff(fact_4892_pochhammer__rec_H,axiom,
    ! [Z3: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z3),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),aa_nat_int(semiri1314217659103216013at_int,N2))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z3),N2)) ).

% pochhammer_rec'
tff(fact_4893_pochhammer__rec_H,axiom,
    ! [Z3: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z3),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z3),aa_nat_nat(semiri1316708129612266289at_nat,N2))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z3),N2)) ).

% pochhammer_rec'
tff(fact_4894_pochhammer__rec_H,axiom,
    ! [Z3: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z3),aa_nat_nat(suc,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z3),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z3),N2)) ).

% pochhammer_rec'
tff(fact_4895_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),N2)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_4896_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K2) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4897_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K2) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4898_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K2) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4899_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K2) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4900_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K2) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4901_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K2) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4902_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4903_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4904_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K2) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4905_pochhammer__product_H,axiom,
    ! [Z3: rat,N2: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z3),N2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),aa_nat_rat(semiri681578069525770553at_rat,N2))),M)) ).

% pochhammer_product'
tff(fact_4906_pochhammer__product_H,axiom,
    ! [Z3: int,N2: nat,M: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z3),N2)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),aa_nat_int(semiri1314217659103216013at_int,N2))),M)) ).

% pochhammer_product'
tff(fact_4907_pochhammer__product_H,axiom,
    ! [Z3: nat,N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z3),N2)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z3),aa_nat_nat(semiri1316708129612266289at_nat,N2))),M)) ).

% pochhammer_product'
tff(fact_4908_pochhammer__product_H,axiom,
    ! [Z3: code_integer,N2: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z3),N2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z3),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),M)) ).

% pochhammer_product'
tff(fact_4909_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P2: int,Q4: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% normalize_denom_pos
tff(fact_4910_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z3),N2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z3),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4911_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z3),N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z3),M)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4912_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z3),N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z3),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z3),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4913_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z3),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z3),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z3),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_4914_pochhammer__absorb__comp,axiom,
    ! [R: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,R),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_4915_pochhammer__absorb__comp,axiom,
    ! [R: int,K2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_nat_int(semiri1314217659103216013at_int,K2))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,R)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_4916_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R),aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_4917_nonempty__has__size,axiom,
    ! [S: multis2468970476368604999at_nat] :
      ( ( S != zero_z1048942125864253310at_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s8510653225128441779at_nat(S)) ) ).

% nonempty_has_size
tff(fact_4918_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2) = 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)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_4919_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K2)),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,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_4920_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),one_one_Code_integer)),K2)) ).

% pochhammer_minus
tff(fact_4921_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = 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)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2)) ).

% pochhammer_minus'
tff(fact_4922_pochhammer__minus_H,axiom,
    ! [B: int,K2: 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,K2))),one_one_int)),K2) = 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)),K2)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_4923_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: 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,K2))),one_one_Code_integer)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2)) ).

% pochhammer_minus'
tff(fact_4924_nat0__intermed__int__val,axiom,
    ! [N2: nat,Fa: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => 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(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(Fa,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(Fa,N2))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N2)
              & ( aa_nat_int(Fa,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_4925_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_list_nat_nat(size_size_list_nat,slice_nat(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4926_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( size_size_list_o(slice_o(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4927_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( size_size_list_int(slice_int(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4928_nat__ivt__aux,axiom,
    ! [N2: nat,Fa: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => 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(Fa,aa_nat_nat(suc,I3))),aa_nat_int(Fa,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(Fa,N2))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N2)
              & ( aa_nat_int(Fa,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_4929_triangle__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N2)),aa_nat_nat(suc,N2)) ).

% triangle_Suc
tff(fact_4930_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_4931_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4932_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4933_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4934_fact__reduce,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_4935_DiffI,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),A4)
     => ( ~ aa_set_o_o(member_o((C2)),B4)
       => aa_set_o_o(member_o((C2)),minus_minus_set_o(A4,B4)) ) ) ).

% DiffI
tff(fact_4936_DiffI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4)
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),minus_2612819937483484256nt_int(A4,B4)) ) ) ).

% DiffI
tff(fact_4937_DiffI,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(C2),B4)
       => aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4)) ) ) ).

% DiffI
tff(fact_4938_DiffI,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A4)
     => ( ~ aa_set_nat_o(member_nat(C2),B4)
       => aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) ) ) ).

% DiffI
tff(fact_4939_DiffI,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),A4)
     => ( ~ aa_set_int_o(member_int(C2),B4)
       => aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)) ) ) ).

% DiffI
tff(fact_4940_DiffI,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) ) ) ).

% DiffI
tff(fact_4941_Diff__iff,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),minus_minus_set_o(A4,B4))
    <=> ( aa_set_o_o(member_o((C2)),A4)
        & ~ aa_set_o_o(member_o((C2)),B4) ) ) ).

% Diff_iff
tff(fact_4942_Diff__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),minus_2612819937483484256nt_int(A4,B4))
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
        & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% Diff_iff
tff(fact_4943_Diff__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A4)
        & ~ aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% Diff_iff
tff(fact_4944_Diff__iff,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
    <=> ( aa_set_nat_o(member_nat(C2),A4)
        & ~ aa_set_nat_o(member_nat(C2),B4) ) ) ).

% Diff_iff
tff(fact_4945_Diff__iff,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))
    <=> ( aa_set_int_o(member_int(C2),A4)
        & ~ aa_set_int_o(member_int(C2),B4) ) ) ).

% Diff_iff
tff(fact_4946_Diff__iff,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4) ) ) ).

% Diff_iff
tff(fact_4947_Diff__idemp,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) ).

% Diff_idemp
tff(fact_4948_ComplI,axiom,
    ! [C2: $o,A4: set_o] :
      ( ~ aa_set_o_o(member_o((C2)),A4)
     => aa_set_o_o(member_o((C2)),uminus_uminus_set_o(A4)) ) ).

% ComplI
tff(fact_4949_ComplI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),uminus6423885277529793776nt_int(A4)) ) ).

% ComplI
tff(fact_4950_ComplI,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(C2),A4)
     => aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A4)) ) ).

% ComplI
tff(fact_4951_ComplI,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C2),A4)
     => aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A4)) ) ).

% ComplI
tff(fact_4952_ComplI,axiom,
    ! [C2: int,A4: set_int] :
      ( ~ aa_set_int_o(member_int(C2),A4)
     => aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A4)) ) ).

% ComplI
tff(fact_4953_Compl__iff,axiom,
    ! [C2: $o,A4: set_o] :
      ( aa_set_o_o(member_o((C2)),uminus_uminus_set_o(A4))
    <=> ~ aa_set_o_o(member_o((C2)),A4) ) ).

% Compl_iff
tff(fact_4954_Compl__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),uminus6423885277529793776nt_int(A4))
    <=> ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4) ) ).

% Compl_iff
tff(fact_4955_Compl__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A4))
    <=> ~ aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% Compl_iff
tff(fact_4956_Compl__iff,axiom,
    ! [C2: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A4))
    <=> ~ aa_set_nat_o(member_nat(C2),A4) ) ).

% Compl_iff
tff(fact_4957_Compl__iff,axiom,
    ! [C2: int,A4: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A4))
    <=> ~ aa_set_int_o(member_int(C2),A4) ) ).

% Compl_iff
tff(fact_4958_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_4959_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_4960_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_4961_abs__of__nat,axiom,
    ! [N2: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)) = aa_nat_rat(semiri681578069525770553at_rat,N2) ).

% abs_of_nat
tff(fact_4962_abs__of__nat,axiom,
    ! [N2: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,N2) ).

% abs_of_nat
tff(fact_4963_abs__of__nat,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) ).

% abs_of_nat
tff(fact_4964_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_4965_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_4966_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_4967_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_4968_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_4969_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_4970_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_4971_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4972_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_4973_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_4974_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_4975_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_4976_slice__complete,axiom,
    ! [Xs: list_nat] : slice_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs),Xs) = Xs ).

% slice_complete
tff(fact_4977_slice__complete,axiom,
    ! [Xs: list_o] : slice_o(zero_zero_nat,size_size_list_o(Xs),Xs) = Xs ).

% slice_complete
tff(fact_4978_slice__complete,axiom,
    ! [Xs: list_int] : slice_int(zero_zero_nat,size_size_list_int(Xs),Xs) = Xs ).

% slice_complete
tff(fact_4979_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_4980_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_4981_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_4982_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4983_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4984_zabs__less__one__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z3)),one_one_int)
    <=> ( Z3 = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_4985_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: 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)),N2))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4986_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N2))
    <=> ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4987_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: 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)),N2))
    <=> ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4988_DiffE,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),minus_minus_set_o(A4,B4))
     => ~ ( aa_set_o_o(member_o((C2)),A4)
         => aa_set_o_o(member_o((C2)),B4) ) ) ).

% DiffE
tff(fact_4989_DiffE,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),minus_2612819937483484256nt_int(A4,B4))
     => ~ ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4)
         => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ) ).

% DiffE
tff(fact_4990_DiffE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
     => ~ ( aa_set_set_nat_o(member_set_nat(C2),A4)
         => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% DiffE
tff(fact_4991_DiffE,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
     => ~ ( aa_set_nat_o(member_nat(C2),A4)
         => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% DiffE
tff(fact_4992_DiffE,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))
     => ~ ( aa_set_int_o(member_int(C2),A4)
         => aa_set_int_o(member_int(C2),B4) ) ) ).

% DiffE
tff(fact_4993_DiffE,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))
     => ~ ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4) ) ) ).

% DiffE
tff(fact_4994_ComplD,axiom,
    ! [C2: $o,A4: set_o] :
      ( aa_set_o_o(member_o((C2)),uminus_uminus_set_o(A4))
     => ~ aa_set_o_o(member_o((C2)),A4) ) ).

% ComplD
tff(fact_4995_ComplD,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),uminus6423885277529793776nt_int(A4))
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4) ) ).

% ComplD
tff(fact_4996_ComplD,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A4))
     => ~ aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% ComplD
tff(fact_4997_ComplD,axiom,
    ! [C2: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A4))
     => ~ aa_set_nat_o(member_nat(C2),A4) ) ).

% ComplD
tff(fact_4998_ComplD,axiom,
    ! [C2: int,A4: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A4))
     => ~ aa_set_int_o(member_int(C2),A4) ) ).

% ComplD
tff(fact_4999_DiffD1,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),minus_minus_set_o(A4,B4))
     => aa_set_o_o(member_o((C2)),A4) ) ).

% DiffD1
tff(fact_5000_DiffD1,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),minus_2612819937483484256nt_int(A4,B4))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),A4) ) ).

% DiffD1
tff(fact_5001_DiffD1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
     => aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% DiffD1
tff(fact_5002_DiffD1,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
     => aa_set_nat_o(member_nat(C2),A4) ) ).

% DiffD1
tff(fact_5003_DiffD1,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))
     => aa_set_int_o(member_int(C2),A4) ) ).

% DiffD1
tff(fact_5004_DiffD1,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),A4) ) ).

% DiffD1
tff(fact_5005_DiffD2,axiom,
    ! [C2: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C2)),minus_minus_set_o(A4,B4))
     => ~ aa_set_o_o(member_o((C2)),B4) ) ).

% DiffD2
tff(fact_5006_DiffD2,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),minus_2612819937483484256nt_int(A4,B4))
     => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(C2),B4) ) ).

% DiffD2
tff(fact_5007_DiffD2,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
     => ~ aa_set_set_nat_o(member_set_nat(C2),B4) ) ).

% DiffD2
tff(fact_5008_DiffD2,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))
     => ~ aa_set_nat_o(member_nat(C2),B4) ) ).

% DiffD2
tff(fact_5009_DiffD2,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))
     => ~ aa_set_int_o(member_int(C2),B4) ) ).

% DiffD2
tff(fact_5010_DiffD2,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))
     => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C2),B4) ) ).

% DiffD2
tff(fact_5011_uminus__set__def,axiom,
    ! [A4: set_o] : uminus_uminus_set_o(A4) = collect_o(uminus6910356288409981324un_o_o(aTP_Lamm_a(A4))) ).

% uminus_set_def
tff(fact_5012_uminus__set__def,axiom,
    ! [A4: set_list_nat] : uminus3195874150345416415st_nat(A4) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_aa(A4))) ).

% uminus_set_def
tff(fact_5013_uminus__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int] : uminus6423885277529793776nt_int(A4) = collec5210948495886036740nt_int(uminus3838094161332292555_int_o(aTP_Lamm_ab(A4))) ).

% uminus_set_def
tff(fact_5014_uminus__set__def,axiom,
    ! [A4: set_set_nat] : uminus613421341184616069et_nat(A4) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ac(A4))) ).

% uminus_set_def
tff(fact_5015_uminus__set__def,axiom,
    ! [A4: set_nat] : uminus5710092332889474511et_nat(A4) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ad(A4))) ).

% uminus_set_def
tff(fact_5016_uminus__set__def,axiom,
    ! [A4: set_int] : uminus1532241313380277803et_int(A4) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ae(A4))) ).

% uminus_set_def
tff(fact_5017_minus__set__def,axiom,
    ! [A4: set_o,B4: set_o] : minus_minus_set_o(A4,B4) = collect_o(minus_minus_fun_o_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% minus_set_def
tff(fact_5018_minus__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ).

% minus_set_def
tff(fact_5019_minus__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] : minus_2612819937483484256nt_int(A4,B4) = collec5210948495886036740nt_int(minus_3219807633925007451_int_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% minus_set_def
tff(fact_5020_minus__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% minus_set_def
tff(fact_5021_minus__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% minus_set_def
tff(fact_5022_minus__set__def,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% minus_set_def
tff(fact_5023_minus__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(minus_2386458686304230003_nat_o(aTP_Lamm_gq(A4)),aTP_Lamm_gq(B4))) ).

% minus_set_def
tff(fact_5024_Collect__neg__eq,axiom,
    ! [Pa: fun_list_nat_o] : collect_list_nat(aTP_Lamm_hp(Pa)) = uminus3195874150345416415st_nat(collect_list_nat(Pa)) ).

% Collect_neg_eq
tff(fact_5025_Collect__neg__eq,axiom,
    ! [Pa: fun_se7249947736802449410_int_o] : collec5210948495886036740nt_int(aTP_Lamm_hq(Pa)) = uminus6423885277529793776nt_int(collec5210948495886036740nt_int(Pa)) ).

% Collect_neg_eq
tff(fact_5026_Collect__neg__eq,axiom,
    ! [Pa: fun_set_nat_o] : collect_set_nat(aTP_Lamm_hr(Pa)) = uminus613421341184616069et_nat(collect_set_nat(Pa)) ).

% Collect_neg_eq
tff(fact_5027_Collect__neg__eq,axiom,
    ! [Pa: fun_nat_o] : collect_nat(aTP_Lamm_hs(Pa)) = uminus5710092332889474511et_nat(collect_nat(Pa)) ).

% Collect_neg_eq
tff(fact_5028_Collect__neg__eq,axiom,
    ! [Pa: fun_int_o] : collect_int(aTP_Lamm_ht(Pa)) = uminus1532241313380277803et_int(collect_int(Pa)) ).

% Collect_neg_eq
tff(fact_5029_Compl__eq,axiom,
    ! [A4: set_o] : uminus_uminus_set_o(A4) = collect_o(aTP_Lamm_hu(A4)) ).

% Compl_eq
tff(fact_5030_Compl__eq,axiom,
    ! [A4: set_list_nat] : uminus3195874150345416415st_nat(A4) = collect_list_nat(aTP_Lamm_hv(A4)) ).

% Compl_eq
tff(fact_5031_Compl__eq,axiom,
    ! [A4: set_se6260736226359567993nt_int] : uminus6423885277529793776nt_int(A4) = collec5210948495886036740nt_int(aTP_Lamm_hw(A4)) ).

% Compl_eq
tff(fact_5032_Compl__eq,axiom,
    ! [A4: set_set_nat] : uminus613421341184616069et_nat(A4) = collect_set_nat(aTP_Lamm_hx(A4)) ).

% Compl_eq
tff(fact_5033_Compl__eq,axiom,
    ! [A4: set_nat] : uminus5710092332889474511et_nat(A4) = collect_nat(aTP_Lamm_hy(A4)) ).

% Compl_eq
tff(fact_5034_Compl__eq,axiom,
    ! [A4: set_int] : uminus1532241313380277803et_int(A4) = collect_int(aTP_Lamm_hz(A4)) ).

% Compl_eq
tff(fact_5035_set__diff__eq,axiom,
    ! [A4: set_o,B4: set_o] : minus_minus_set_o(A4,B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_ia(A4),B4)) ).

% set_diff_eq
tff(fact_5036_set__diff__eq,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ib(A4),B4)) ).

% set_diff_eq
tff(fact_5037_set__diff__eq,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] : minus_2612819937483484256nt_int(A4,B4) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_ic(A4),B4)) ).

% set_diff_eq
tff(fact_5038_set__diff__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_id(A4),B4)) ).

% set_diff_eq
tff(fact_5039_set__diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ie(A4),B4)) ).

% set_diff_eq
tff(fact_5040_set__diff__eq,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_if(A4),B4)) ).

% set_diff_eq
tff(fact_5041_set__diff__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_ig(A4),B4)) ).

% set_diff_eq
tff(fact_5042_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_5043_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_5044_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_5045_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_5046_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_5047_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_5048_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ).

% fact_mono_nat
tff(fact_5049_fact__ge__self,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_self
tff(fact_5050_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_5051_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_5052_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_5053_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_5054_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_5055_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_5056_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_5057_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_5058_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_5059_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_5060_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_5061_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_5062_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_5063_abs__triangle__ineq2__sym,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(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5064_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_5065_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_5066_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq3
tff(fact_5067_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_5068_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_5069_abs__triangle__ineq2,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(minus_minus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq2
tff(fact_5070_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_5071_abs__mult__less,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),C2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B)),D2)
       => 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,C2),D2)) ) ) ).

% abs_mult_less
tff(fact_5072_abs__mult__less,axiom,
    ! [A: rat,C2: rat,B: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D2)
       => 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(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),D2)) ) ) ).

% abs_mult_less
tff(fact_5073_abs__mult__less,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D2)
       => 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,C2),D2)) ) ) ).

% abs_mult_less
tff(fact_5074_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_5075_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_5076_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_5077_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_5078_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_5079_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_5080_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_5081_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_5082_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_5083_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_5084_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_5085_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_5086_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_5087_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_5088_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_5089_fact__less__mono__nat,axiom,
    ! [M: nat,N2: 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),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ) ).

% fact_less_mono_nat
tff(fact_5090_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: 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,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_5091_fact__ge__zero,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_ge_zero
tff(fact_5092_fact__ge__zero,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_ge_zero
tff(fact_5093_fact__ge__zero,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_ge_zero
tff(fact_5094_fact__ge__zero,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_zero
tff(fact_5095_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),zero_z3403309356797280102nteger) ).

% fact_not_neg
tff(fact_5096_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),zero_zero_rat) ).

% fact_not_neg
tff(fact_5097_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),zero_zero_int) ).

% fact_not_neg
tff(fact_5098_fact__not__neg,axiom,
    ! [N2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),zero_zero_nat) ).

% fact_not_neg
tff(fact_5099_fact__gt__zero,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_gt_zero
tff(fact_5100_fact__gt__zero,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_gt_zero
tff(fact_5101_fact__gt__zero,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_gt_zero
tff(fact_5102_fact__gt__zero,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_gt_zero
tff(fact_5103_fact__ge__1,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% fact_ge_1
tff(fact_5104_fact__ge__1,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_ge_1
tff(fact_5105_fact__ge__1,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_ge_1
tff(fact_5106_fact__ge__1,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_ge_1
tff(fact_5107_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ) ).

% fact_mono
tff(fact_5108_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N2)) ) ).

% fact_mono
tff(fact_5109_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ).

% fact_mono
tff(fact_5110_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer'
tff(fact_5111_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ).

% gbinomial_addition_formula
tff(fact_5112_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K2)) = 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,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_5113_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),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,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_5114_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E2) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5115_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),X) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5116_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5117_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5118_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_5119_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_5120_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_5121_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(gbinomial_rat(A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_5122_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_5123_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5124_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_5125_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_5126_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5127_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_5128_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_5129_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_5130_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_5131_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_5132_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_5133_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_5134_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: 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,C2),D2)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),C2))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_5135_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_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,C2),D2)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_5136_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)))),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),C2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_5137_abs__diff__le__iff,axiom,
    ! [X: 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,X),A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_5138_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_5139_abs__diff__le__iff,axiom,
    ! [X: 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,X),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)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_5140_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_5141_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_5142_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_5143_abs__if__raw,axiom,
    ! [X5: int] :
      aa_int_int(abs_abs_int,X5) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X5),zero_zero_int),aa_int_int(uminus_uminus_int,X5),X5) ).

% abs_if_raw
tff(fact_5144_abs__if__raw,axiom,
    ! [X5: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X5) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X5),X5) ).

% abs_if_raw
tff(fact_5145_abs__if__raw,axiom,
    ! [X5: rat] :
      aa_rat_rat(abs_abs_rat,X5) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X5),X5) ).

% abs_if_raw
tff(fact_5146_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_5147_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_5148_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_5149_abs__diff__less__iff,axiom,
    ! [X: 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,X),A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),R)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_5150_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),R)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_5151_abs__diff__less__iff,axiom,
    ! [X: 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,X),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)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_5152_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% abs_div_pos
tff(fact_5153_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: 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)),N2)) ).

% zero_le_power_abs
tff(fact_5154_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N2)) ).

% zero_le_power_abs
tff(fact_5155_zero__le__power__abs,axiom,
    ! [A: int,N2: 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)),N2)) ).

% zero_le_power_abs
tff(fact_5156_fact__ge__Suc__0__nat,axiom,
    ! [N2: 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,N2)) ).

% fact_ge_Suc_0_nat
tff(fact_5157_zabs__def,axiom,
    ! [I2: int] :
      aa_int_int(abs_abs_int,I2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I2),zero_zero_int),aa_int_int(uminus_uminus_int,I2),I2) ).

% zabs_def
tff(fact_5158_fact__less__mono,axiom,
    ! [M: nat,N2: 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),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ) ) ).

% fact_less_mono
tff(fact_5159_fact__less__mono,axiom,
    ! [M: nat,N2: 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),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N2)) ) ) ).

% fact_less_mono
tff(fact_5160_fact__less__mono,axiom,
    ! [M: nat,N2: 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),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ) ).

% fact_less_mono
tff(fact_5161_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_5162_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_5163_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_5164_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( modulo8411746178871703098atural(aa_nat_Code_natural(semiri2447717529341329178atural,N2),aa_nat_Code_natural(semiri2447717529341329178atural,M)) = zero_z2226904508553997617atural ) ) ).

% fact_mod
tff(fact_5165_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_5166_fact__le__power,axiom,
    ! [N2: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,N2)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5167_fact__le__power,axiom,
    ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5168_fact__le__power,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5169_fact__le__power,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2))) ).

% fact_le_power
tff(fact_5170_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2))) = 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),K2)) ).

% Suc_times_gbinomial
tff(fact_5171_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_5172_abs__add__one__gt__zero,axiom,
    ! [X: 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,X))) ).

% abs_add_one_gt_zero
tff(fact_5173_abs__add__one__gt__zero,axiom,
    ! [X: 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),aa_rat_rat(abs_abs_rat,X))) ).

% abs_add_one_gt_zero
tff(fact_5174_abs__add__one__gt__zero,axiom,
    ! [X: 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,X))) ).

% abs_add_one_gt_zero
tff(fact_5175_gbinomial__code,axiom,
    ! [A: rat,K2: nat] :
      aa_nat_rat(gbinomial_rat(A),K2) = $ite(K2 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ih(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K2))) ).

% gbinomial_code
tff(fact_5176_nat__abs__triangle__ineq,axiom,
    ! [K2: 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,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_5177_fact__div__fact__le__pow,axiom,
    ! [R: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),R)) ) ).

% fact_div_fact_le_pow
tff(fact_5178_gbinomial__factors,axiom,
    ! [A: rat,K2: 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,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_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,K2)))),aa_nat_rat(gbinomial_rat(A),K2)) ).

% gbinomial_factors
tff(fact_5179_gbinomial__rec,axiom,
    ! [A: rat,K2: 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,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_rat_rat(aa_rat_fun_rat_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,K2)))) ).

% gbinomial_rec
tff(fact_5180_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K2) = 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)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% gbinomial_minus
tff(fact_5181_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_5182_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_5183_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2) )
         => 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(Fa,aa_nat_nat(suc,I3))),aa_nat_int(Fa,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,M)),K2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(Fa,N2))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N2)
                & ( aa_nat_int(Fa,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5184_decr__lemma,axiom,
    ! [D2: int,X: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z3))),one_one_int)),D2))),Z3) ) ).

% decr_lemma
tff(fact_5185_incr__lemma,axiom,
    ! [D2: int,Z3: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z3))),one_one_int)),D2))) ) ).

% incr_lemma
tff(fact_5186_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) ) )
        | ? [L2: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),zero_zero_int) )
            & ( L2 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L2) ) )
        | ? [R3: int,L2: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q6),R3) )
            & ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,L2) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,L2))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L2)),R3) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5187_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_5188_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5189_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => ( rotate1_o(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5190_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5191_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_5192_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_o,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_o(nth_o(slice_o(From,To,Xs)),I2)
          <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_5193_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I2) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_5194_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K2: int,Q4: 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))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L)),R) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5195_dbl__dec__def,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).

% dbl_dec_def
tff(fact_5196_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_5197_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_5198_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_5199_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_5200_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_5201_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_5202_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_5203_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_5204_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_5205_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_5206_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_5207_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_5208_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_5209_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_5210_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_5211_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(sgn_sgn_rat,A) ) ) ).

% same_sgn_sgn_add
tff(fact_5212_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_5213_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I) = aa_nat_nat(nth_nat(Ys2),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5214_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I)
            <=> aa_nat_o(nth_o(Ys2),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5215_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I) = aa_nat_int(nth_int(Ys2),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5216_Skolem__list__nth,axiom,
    ! [K2: nat,Pa: fun_nat_fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,I),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_nat_o(aa_nat_fun_nat_o(Pa,I),aa_nat_nat(nth_nat(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_5217_Skolem__list__nth,axiom,
    ! [K2: nat,Pa: fun_nat_fun_o_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(Pa,I),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_o_o(aa_nat_fun_o_o(Pa,I),aa_nat_o(nth_o(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_5218_Skolem__list__nth,axiom,
    ! [K2: nat,Pa: fun_nat_fun_int_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(Pa,I),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_int_o(aa_nat_fun_int_o(Pa,I),aa_nat_int(nth_int(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_5219_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_5220_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_5221_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_5222_obtain__list__from__elements,axiom,
    ! [N2: nat,Pa: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(Pa,Li),I3) )
     => ~ ! [L3: list_nat] :
            ( ( aa_list_nat_nat(size_size_list_nat,L3) = N2 )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
                 => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_5223_obtain__list__from__elements,axiom,
    ! [N2: nat,Pa: fun_o_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(Pa,(Li)),I3) )
     => ~ ! [L3: list_o] :
            ( ( size_size_list_o(L3) = N2 )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
                 => aa_nat_o(aa_o_fun_nat_o(Pa,aa_nat_o(nth_o(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_5224_obtain__list__from__elements,axiom,
    ! [N2: nat,Pa: fun_int_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N2)
         => ? [Li: int] : aa_nat_o(aa_int_fun_nat_o(Pa,Li),I3) )
     => ~ ! [L3: list_int] :
            ( ( size_size_list_int(L3) = N2 )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N2)
                 => aa_nat_o(aa_int_fun_nat_o(Pa,aa_nat_int(nth_int(L3),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_5225_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_5226_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_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,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% same_sgn_abs_add
tff(fact_5227_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_5228_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),N2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).

% nth_rotate1
tff(fact_5229_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(rotate1_o(Xs)),N2)
      <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),size_size_list_o(Xs))) ) ) ).

% nth_rotate1
tff(fact_5230_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),N2) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_5231_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_5232_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_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_5233_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_5234_zsgn__def,axiom,
    ! [I2: int] :
      aa_int_int(sgn_sgn_int,I2) = $ite(
        I2 = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I2),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_5235_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_5236_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_5237_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_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_5238_sgn__if,axiom,
    ! [X: int] :
      aa_int_int(sgn_sgn_int,X) = $ite(
        X = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_5239_sgn__if,axiom,
    ! [X: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = $ite(
        X = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_5240_sgn__if,axiom,
    ! [X: rat] :
      aa_rat_rat(sgn_sgn_rat,X) = $ite(
        X = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_if
tff(fact_5241_nth__rule,axiom,
    ! [I2: nat,Xs: list_nat,A: array_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
     => hoare_3067605981109127869le_nat(snga_assn_nat(A,Xs),array_nth_nat(A,I2),aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ii(I2),Xs),A)) ) ).

% nth_rule
tff(fact_5242_nth__rule,axiom,
    ! [I2: nat,Xs: list_o,A: array_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => hoare_hoare_triple_o(snga_assn_o(A,Xs),array_nth_o(A,I2),aa_arr2544151649598861476o_assn(aa_lis966790859614740569o_assn(aTP_Lamm_ij(I2),Xs),A)) ) ).

% nth_rule
tff(fact_5243_nth__rule,axiom,
    ! [I2: nat,Xs: list_int,A: array_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => hoare_3065115510600077593le_int(snga_assn_int(A,Xs),array_nth_int(A,I2),aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_ik(I2),Xs),A)) ) ).

% nth_rule
tff(fact_5244_nth__rule,axiom,
    ! [I2: nat,Xs: list_Product_unit,A: array_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s245203480648594047t_unit(Xs))
     => hoare_8945653483474564448t_unit(snga_a4522542871529764173t_unit(A,Xs),array_7872002506669749220t_unit(A,I2),aa_arr8761957006411741156t_assn(aa_lis5628126305658872932t_assn(aTP_Lamm_il(I2),Xs),A)) ) ).

% nth_rule
tff(fact_5245_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys2),N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_5246_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr112076138515278198_nat_o(product_nat_o(Xs,Ys2),N2) = product_Pair_nat_o(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_5247_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys2),N2) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_5248_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys2),N2) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_5249_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys2)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys2),N2) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_5250_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys2),N2) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_5251_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),aa_list_nat_nat(size_size_list_nat,Ys2)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys2),N2) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys2)))) ) ) ).

% product_nth
tff(fact_5252_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr7514405829937366042_int_o(product_int_o(Xs,Ys2),N2) = product_Pair_int_o(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_5253_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),N2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2)))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_5254_product__nth,axiom,
    ! [N2: nat,Xs: list_b,Ys2: list_P131111800688179804it_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_b(Xs)),size_s341701280123345136it_nat(Ys2)))
     => ( nth_Pr3591576263217728691it_nat(produc1870533552098665926it_nat(Xs,Ys2),N2) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,nth_b(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_s341701280123345136it_nat(Ys2)))),nth_Pr4561283279332054789it_nat(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_s341701280123345136it_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_5255_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( 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),size_size_list_o(L))
         => ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),L)
           => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),I2)),(X))
             => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_nat_o(nth_o(L),J2))
               => ~ ! [K: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),J2)
                       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),K)),(X))
                         => ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_nat_o(nth_o(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5256_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_rat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( 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),size_size_list_rat(L))
         => ( sorted_wrt_rat(ord_less_eq_rat,L)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I2)),X)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(nth_rat(L),J2))
               => ~ ! [K: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),J2)
                       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),K)),X)
                         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(nth_rat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5257_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_num,X: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( 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),size_size_list_num(L))
         => ( sorted_wrt_num(ord_less_eq_num,L)
           => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),I2)),X)
             => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),aa_nat_num(nth_num(L),J2))
               => ~ ! [K: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),J2)
                       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),K)),X)
                         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),aa_nat_num(nth_num(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5258_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( 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),aa_list_nat_nat(size_size_list_nat,L))
         => ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I2)),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(nth_nat(L),J2))
               => ~ ! [K: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),J2)
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),K)),X)
                         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5259_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I2)
     => ( 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),size_size_list_int(L))
         => ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),L)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),I2)),X)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(nth_int(L),J2))
               => ~ ! [K: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),J2)
                       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),K)),X)
                         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5260_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N2,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5261_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_o(Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(N2,Xs),M) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M),aa_nat_o(nth_o(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5262_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N2,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5263_array__of__list__make,axiom,
    ! [Xs: list_nat] : array_of_list_nat(Xs) = array_make_nat(aa_list_nat_nat(size_size_list_nat,Xs),nth_nat(Xs)) ).

% array_of_list_make
tff(fact_5264_array__of__list__make,axiom,
    ! [Xs: list_o] : array_of_list_o(Xs) = array_make_o(size_size_list_o(Xs),nth_o(Xs)) ).

% array_of_list_make
tff(fact_5265_array__of__list__make,axiom,
    ! [Xs: list_int] : array_of_list_int(Xs) = array_make_int(size_size_list_int(Xs),nth_int(Xs)) ).

% array_of_list_make
tff(fact_5266_nth__step__trancl,axiom,
    ! [Xs: list_o,Ra: set_Product_prod_o_o,N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_o(Xs)),one_one_nat))
         => member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(suc,N5)),aa_nat_o(nth_o(Xs),N5)),Ra) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
         => member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),N2),aa_nat_o(nth_o(Xs),M)),transitive_trancl_o(Ra)) ) ) ) ).

% nth_step_trancl
tff(fact_5267_nth__step__trancl,axiom,
    ! [Xs: list_int,Ra: set_Pr958786334691620121nt_int,N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat))
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),aa_nat_nat(suc,N5))),aa_nat_int(nth_int(Xs),N5))),Ra) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),N2)),aa_nat_int(nth_int(Xs),M))),transi6261509568448316235cl_int(Ra)) ) ) ) ).

% nth_step_trancl
tff(fact_5268_nth__step__trancl,axiom,
    ! [Xs: list_nat,Ra: set_Pr1261947904930325089at_nat,N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,N5))),aa_nat_nat(nth_nat(Xs),N5))),Ra) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N2)),aa_nat_nat(nth_nat(Xs),M))),transi6264000038957366511cl_nat(Ra)) ) ) ) ).

% nth_step_trancl
tff(fact_5269_nth__rotate,axiom,
    ! [N2: nat,Xs: list_nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(rotate_nat(M,Xs)),N2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_list_nat_nat(size_size_list_nat,Xs))) ) ) ).

% nth_rotate
tff(fact_5270_nth__rotate,axiom,
    ! [N2: nat,Xs: list_o,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(rotate_o(M,Xs)),N2)
      <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),size_size_list_o(Xs))) ) ) ).

% nth_rotate
tff(fact_5271_nth__rotate,axiom,
    ! [N2: nat,Xs: list_int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rotate_int(M,Xs)),N2) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),size_size_list_int(Xs))) ) ) ).

% nth_rotate
tff(fact_5272_rotate__length01,axiom,
    ! [Xs: list_nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => ( rotate_nat(N2,Xs) = Xs ) ) ).

% rotate_length01
tff(fact_5273_rotate__length01,axiom,
    ! [Xs: list_o,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => ( rotate_o(N2,Xs) = Xs ) ) ).

% rotate_length01
tff(fact_5274_rotate__length01,axiom,
    ! [Xs: list_int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => ( rotate_int(N2,Xs) = Xs ) ) ).

% rotate_length01
tff(fact_5275_sorted__wrt__true,axiom,
    ! [Xs: list_nat] : aa_list_nat_o(sorted_wrt_nat(aTP_Lamm_im),Xs) ).

% sorted_wrt_true
tff(fact_5276_sorted__wrt__true,axiom,
    ! [Xs: list_int] : aa_list_int_o(sorted_wrt_int(aTP_Lamm_in),Xs) ).

% sorted_wrt_true
tff(fact_5277_trancl_Ocases,axiom,
    ! [A1: int,A22: int,R: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A1),A22)),transi6261509568448316235cl_int(R))
     => ( ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A1),A22)),R)
       => ~ ! [B3: int] :
              ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A1),B3)),transi6261509568448316235cl_int(R))
             => ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B3),A22)),R) ) ) ) ).

% trancl.cases
tff(fact_5278_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22)),transi6264000038957366511cl_nat(R))
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22)),R)
       => ~ ! [B3: nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),B3)),transi6264000038957366511cl_nat(R))
             => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),A22)),R) ) ) ) ).

% trancl.cases
tff(fact_5279_trancl_Osimps,axiom,
    ! [A1: int,A22: int,R: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A1),A22)),transi6261509568448316235cl_int(R))
    <=> ( ? [A6: int,B6: int] :
            ( ( A1 = A6 )
            & ( A22 = B6 )
            & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A6),B6)),R) )
        | ? [A6: int,B6: int,C4: int] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A6),B6)),transi6261509568448316235cl_int(R))
            & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B6),C4)),R) ) ) ) ).

% trancl.simps
tff(fact_5280_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22)),transi6264000038957366511cl_nat(R))
    <=> ( ? [A6: nat,B6: nat] :
            ( ( A1 = A6 )
            & ( A22 = B6 )
            & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A6),B6)),R) )
        | ? [A6: nat,B6: nat,C4: nat] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A6),B6)),transi6264000038957366511cl_nat(R))
            & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B6),C4)),R) ) ) ) ).

% trancl.simps
tff(fact_5281_trancl_Or__into__trancl,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),R)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),transi6261509568448316235cl_int(R)) ) ).

% trancl.r_into_trancl
tff(fact_5282_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),R)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),transi6264000038957366511cl_nat(R)) ) ).

% trancl.r_into_trancl
tff(fact_5283_tranclE,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),transi6261509568448316235cl_int(R))
     => ( ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),R)
       => ~ ! [C: int] :
              ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),C)),transi6261509568448316235cl_int(R))
             => ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,C),B)),R) ) ) ) ).

% tranclE
tff(fact_5284_tranclE,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),transi6264000038957366511cl_nat(R))
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),R)
       => ~ ! [C: nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C)),transi6264000038957366511cl_nat(R))
             => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(C),B)),R) ) ) ) ).

% tranclE
tff(fact_5285_trancl__trans,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Z3: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),transi6261509568448316235cl_int(R))
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y),Z3)),transi6261509568448316235cl_int(R))
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Z3)),transi6261509568448316235cl_int(R)) ) ) ).

% trancl_trans
tff(fact_5286_trancl__trans,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z3: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),transi6264000038957366511cl_nat(R))
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z3)),transi6264000038957366511cl_nat(R))
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z3)),transi6264000038957366511cl_nat(R)) ) ) ).

% trancl_trans
tff(fact_5287_trancl__induct,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,Pa: fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),transi6261509568448316235cl_int(R))
     => ( ! [Y3: int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),Y3)),R)
           => aa_int_o(Pa,Y3) )
       => ( ! [Y3: int,Z2: int] :
              ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),Y3)),transi6261509568448316235cl_int(R))
             => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),Z2)),R)
               => ( aa_int_o(Pa,Y3)
                 => aa_int_o(Pa,Z2) ) ) )
         => aa_int_o(Pa,B) ) ) ) ).

% trancl_induct
tff(fact_5288_trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,Pa: fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),transi6264000038957366511cl_nat(R))
     => ( ! [Y3: nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Y3)),R)
           => aa_nat_o(Pa,Y3) )
       => ( ! [Y3: nat,Z2: nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Y3)),transi6264000038957366511cl_nat(R))
             => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z2)),R)
               => ( aa_nat_o(Pa,Y3)
                 => aa_nat_o(Pa,Z2) ) ) )
         => aa_nat_o(Pa,B) ) ) ) ).

% trancl_induct
tff(fact_5289_r__r__into__trancl,axiom,
    ! [A: int,B: int,Ra: set_Pr958786334691620121nt_int,C2: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),Ra)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),C2)),Ra)
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),C2)),transi6261509568448316235cl_int(Ra)) ) ) ).

% r_r_into_trancl
tff(fact_5290_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,Ra: set_Pr1261947904930325089at_nat,C2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C2)),Ra)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C2)),transi6264000038957366511cl_nat(Ra)) ) ) ).

% r_r_into_trancl
tff(fact_5291_converse__tranclE,axiom,
    ! [X: int,Z3: int,R: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Z3)),transi6261509568448316235cl_int(R))
     => ( ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Z3)),R)
       => ~ ! [Y3: int] :
              ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y3)),R)
             => ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),Z3)),transi6261509568448316235cl_int(R)) ) ) ) ).

% converse_tranclE
tff(fact_5292_converse__tranclE,axiom,
    ! [X: nat,Z3: nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z3)),transi6264000038957366511cl_nat(R))
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z3)),R)
       => ~ ! [Y3: nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3)),R)
             => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z3)),transi6264000038957366511cl_nat(R)) ) ) ) ).

% converse_tranclE
tff(fact_5293_irrefl__trancl__rD,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ! [X3: int] : ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),X3)),transi6261509568448316235cl_int(R))
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),R)
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5294_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3)),transi6264000038957366511cl_nat(R))
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),R)
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5295_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,C2: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),transi6261509568448316235cl_int(R))
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),C2)),R)
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),C2)),transi6261509568448316235cl_int(R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5296_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),transi6264000038957366511cl_nat(R))
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C2)),R)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C2)),transi6264000038957366511cl_nat(R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5297_trancl__into__trancl2,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,C2: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),R)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),C2)),transi6261509568448316235cl_int(R))
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),C2)),transi6261509568448316235cl_int(R)) ) ) ).

% trancl_into_trancl2
tff(fact_5298_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),R)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C2)),transi6264000038957366511cl_nat(R))
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C2)),transi6264000038957366511cl_nat(R)) ) ) ).

% trancl_into_trancl2
tff(fact_5299_trancl__trans__induct,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Pa: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),transi6261509568448316235cl_int(R))
     => ( ! [X3: int,Y3: int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)),R)
           => aa_int_o(aa_int_fun_int_o(Pa,X3),Y3) )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)),transi6261509568448316235cl_int(R))
             => ( aa_int_o(aa_int_fun_int_o(Pa,X3),Y3)
               => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),Z2)),transi6261509568448316235cl_int(R))
                 => ( aa_int_o(aa_int_fun_int_o(Pa,Y3),Z2)
                   => aa_int_o(aa_int_fun_int_o(Pa,X3),Z2) ) ) ) )
         => aa_int_o(aa_int_fun_int_o(Pa,X),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_5300_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Pa: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),transi6264000038957366511cl_nat(R))
     => ( ! [X3: nat,Y3: nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)),R)
           => aa_nat_o(aa_nat_fun_nat_o(Pa,X3),Y3) )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3)),transi6264000038957366511cl_nat(R))
             => ( aa_nat_o(aa_nat_fun_nat_o(Pa,X3),Y3)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z2)),transi6264000038957366511cl_nat(R))
                 => ( aa_nat_o(aa_nat_fun_nat_o(Pa,Y3),Z2)
                   => aa_nat_o(aa_nat_fun_nat_o(Pa,X3),Z2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(Pa,X),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_5301_converse__trancl__induct,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,Pa: fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),transi6261509568448316235cl_int(R))
     => ( ! [Y3: int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),B)),R)
           => aa_int_o(Pa,Y3) )
       => ( ! [Y3: int,Z2: int] :
              ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y3),Z2)),R)
             => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Z2),B)),transi6261509568448316235cl_int(R))
               => ( aa_int_o(Pa,Z2)
                 => aa_int_o(Pa,Y3) ) ) )
         => aa_int_o(Pa,A) ) ) ) ).

% converse_trancl_induct
tff(fact_5302_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,Pa: fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),transi6264000038957366511cl_nat(R))
     => ( ! [Y3: nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),B)),R)
           => aa_nat_o(Pa,Y3) )
       => ( ! [Y3: nat,Z2: nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z2)),R)
             => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z2),B)),transi6264000038957366511cl_nat(R))
               => ( aa_nat_o(Pa,Z2)
                 => aa_nat_o(Pa,Y3) ) ) )
         => aa_nat_o(Pa,A) ) ) ) ).

% converse_trancl_induct
tff(fact_5303_trancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R: set_Pr2560585780119916871nt_int,Pa: fun_int_fun_int_o] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ax),Ay),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Bx),By)),transi6288783178788033498nt_int(R))
     => ( ! [A3: int,B3: int] :
            ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ax),Ay),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3)),R)
           => aa_int_o(aa_int_fun_int_o(Pa,A3),B3) )
       => ( ! [A3: int,B3: int,Aa: int,Ba: int] :
              ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ax),Ay),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3)),transi6288783178788033498nt_int(R))
             => ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Aa),Ba)),R)
               => ( aa_int_o(aa_int_fun_int_o(Pa,A3),B3)
                 => aa_int_o(aa_int_fun_int_o(Pa,Aa),Ba) ) ) )
         => aa_int_o(aa_int_fun_int_o(Pa,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5304_trancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc3658429121746597890et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc3658429121746597890et_nat,R: set_Pr9066344408807579265et_nat,Pa: fun_fu8419763957888848259_nat_o] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Ax),Ay),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Bx),By)),transi7952567189410240703et_nat(R))
     => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
            ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Ax),Ay),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3)),R)
           => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Pa,A3),B3) )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc3658429121746597890et_nat] :
              ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Ax),Ay),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3)),transi7952567189410240703et_nat(R))
             => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Aa),Ba)),R)
               => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Pa,A3),B3)
                 => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Pa,Aa),Ba) ) ) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Pa,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5305_trancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc8799533983703829792et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc8799533983703829792et_nat,R: set_Pr7057080801110549565et_nat,Pa: fun_fu2611741329380808165_nat_o] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Ax),Ay),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Bx),By)),transi2167408805420631517et_nat(R))
     => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
            ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Ax),Ay),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3)),R)
           => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Pa,A3),B3) )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc8799533983703829792et_nat] :
              ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Ax),Ay),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3)),transi2167408805420631517et_nat(R))
             => ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Aa),Ba)),R)
               => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Pa,A3),B3)
                 => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Pa,Aa),Ba) ) ) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Pa,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5306_trancl__induct2,axiom,
    ! [Ax: b,Ay: produc6653097349344004940it_nat,Bx: b,By: produc6653097349344004940it_nat,R: set_Pr5508209795250834101it_nat,Pa: fun_b_540864168722188329_nat_o] :
      ( member6820296301096096254it_nat(produc6853161671299316109it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Ax),Ay),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Bx),By)),transi2518514244971309593it_nat(R))
     => ( ! [A3: b,B3: produc6653097349344004940it_nat] :
            ( member6820296301096096254it_nat(produc6853161671299316109it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Ax),Ay),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3)),R)
           => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Pa,A3),B3) )
       => ( ! [A3: b,B3: produc6653097349344004940it_nat,Aa: b,Ba: produc6653097349344004940it_nat] :
              ( member6820296301096096254it_nat(produc6853161671299316109it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Ax),Ay),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3)),transi2518514244971309593it_nat(R))
             => ( member6820296301096096254it_nat(produc6853161671299316109it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Aa),Ba)),R)
               => ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Pa,A3),B3)
                 => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Pa,Aa),Ba) ) ) )
         => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Pa,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5307_trancl__induct2,axiom,
    ! [Ax: a,Ay: produc6653097349344004940it_nat,Bx: a,By: produc6653097349344004940it_nat,R: set_Pr2819221443900773171it_nat,Pa: fun_a_2467521563176467050_nat_o] :
      ( member5335690527091456380it_nat(produc1743342482959036555it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ax),Ay),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Bx),By)),transi7613985180851635224it_nat(R))
     => ( ! [A3: a,B3: produc6653097349344004940it_nat] :
            ( member5335690527091456380it_nat(produc1743342482959036555it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ax),Ay),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3)),R)
           => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,A3),B3) )
       => ( ! [A3: a,B3: produc6653097349344004940it_nat,Aa: a,Ba: produc6653097349344004940it_nat] :
              ( member5335690527091456380it_nat(produc1743342482959036555it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Ax),Ay),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3)),transi7613985180851635224it_nat(R))
             => ( member5335690527091456380it_nat(produc1743342482959036555it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Aa),Ba)),R)
               => ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,A3),B3)
                 => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,Aa),Ba) ) ) )
         => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Pa,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5308_trancl__mono,axiom,
    ! [P2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(P2),transi6264000038957366511cl_nat(R))
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,R),S2)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(P2),transi6264000038957366511cl_nat(S2)) ) ) ).

% trancl_mono
tff(fact_5309_trancl__mono__mp,axiom,
    ! [U2: set_Pr1261947904930325089at_nat,V2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,U2),V2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),transi6264000038957366511cl_nat(U2))
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),transi6264000038957366511cl_nat(V2)) ) ) ).

% trancl_mono_mp
tff(fact_5310_trancl__sub,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),transi6264000038957366511cl_nat(Ra)) ).

% trancl_sub
tff(fact_5311_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5312_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5313_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5314_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_int),Xs)
     => aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5315_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2)) ) ) ).

% sorted_wrt_less_idx
tff(fact_5316_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(Pa),Xs)
    <=> ! [I: nat,J: 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),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5317_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_o_fun_o_o,Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(Pa),Xs)
    <=> ! [I: nat,J: 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),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(Pa,aa_nat_o(nth_o(Xs),I)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5318_sorted__wrt__iff__nth__less,axiom,
    ! [Pa: fun_int_fun_int_o,Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(Pa),Xs)
    <=> ! [I: nat,J: 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),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(Pa,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5319_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_nat_fun_nat_o,Xs: list_nat,I2: nat,J2: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(Pa),Xs)
     => ( 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),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J2)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5320_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_o_fun_o_o,Xs: list_o,I2: nat,J2: nat] :
      ( aa_list_o_o(sorted_wrt_o(Pa),Xs)
     => ( 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),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(Pa,aa_nat_o(nth_o(Xs),I2)),aa_nat_o(nth_o(Xs),J2)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5321_sorted__wrt__nth__less,axiom,
    ! [Pa: fun_int_fun_int_o,Xs: list_int,I2: nat,J2: nat] :
      ( aa_list_int_o(sorted_wrt_int(Pa),Xs)
     => ( 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),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(Pa,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J2)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5322_sorted__wrt01,axiom,
    ! [Xs: list_nat,Pa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_5323_sorted__wrt01,axiom,
    ! [Xs: list_o,Pa: fun_o_fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => aa_list_o_o(sorted_wrt_o(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_5324_sorted__wrt01,axiom,
    ! [Xs: list_int,Pa: fun_int_fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => aa_list_int_o(sorted_wrt_int(Pa),Xs) ) ).

% sorted_wrt01
tff(fact_5325_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I: nat,J: 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),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5326_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I: nat,J: 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),size_size_list_rat(Xs))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I)),aa_nat_rat(nth_rat(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5327_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I: nat,J: 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),size_size_list_num(Xs))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I)),aa_nat_num(nth_num(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5328_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I: nat,J: 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),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5329_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
    <=> ! [I: nat,J: 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),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5330_sorted01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs) ) ).

% sorted01
tff(fact_5331_sorted01,axiom,
    ! [Xs: list_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_rat(Xs)),one_one_nat)
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% sorted01
tff(fact_5332_sorted01,axiom,
    ! [Xs: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_num(Xs)),one_one_nat)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% sorted01
tff(fact_5333_sorted01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% sorted01
tff(fact_5334_sorted01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs) ) ).

% sorted01
tff(fact_5335_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I)),aa_nat_o(nth_o(Xs),aa_nat_nat(suc,I))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5336_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),size_size_list_rat(Xs))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I)),aa_nat_rat(nth_rat(Xs),aa_nat_nat(suc,I))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5337_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),size_size_list_num(Xs))
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I)),aa_nat_num(nth_num(Xs),aa_nat_nat(suc,I))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5338_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5339_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5340_sorted__iff__nth__mono,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [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_nat,J),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5341_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [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_nat,J),size_size_list_rat(Xs))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I)),aa_nat_rat(nth_rat(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5342_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [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_nat,J),size_size_list_num(Xs))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I)),aa_nat_num(nth_num(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5343_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [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_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5344_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
    <=> ! [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_nat,J),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5345_sorted__nth__mono,axiom,
    ! [Xs: list_o,I2: nat,J2: nat] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I2)),aa_nat_o(nth_o(Xs),J2)) ) ) ) ).

% sorted_nth_mono
tff(fact_5346_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I2: nat,J2: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_rat(Xs))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I2)),aa_nat_rat(nth_rat(Xs),J2)) ) ) ) ).

% sorted_nth_mono
tff(fact_5347_sorted__nth__mono,axiom,
    ! [Xs: list_num,I2: nat,J2: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_num(Xs))
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I2)),aa_nat_num(nth_num(Xs),J2)) ) ) ) ).

% sorted_nth_mono
tff(fact_5348_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I2: nat,J2: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Xs),J2)) ) ) ) ).

% sorted_nth_mono
tff(fact_5349_sorted__nth__mono,axiom,
    ! [Xs: list_int,I2: nat,J2: nat] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xs),J2)) ) ) ) ).

% sorted_nth_mono
tff(fact_5350_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ys2))
       => ( nth_Pr7617993195940197384at_nat(zip_nat_nat(Xs,Ys2),I2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5351_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Ys2))
       => ( nth_Pr112076138515278198_nat_o(zip_nat_o(Xs,Ys2),I2) = product_Pair_nat_o(aa_nat_nat(nth_nat(Xs),I2),aa_nat_o(nth_o(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5352_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Ys2))
       => ( nth_Pr3440142176431000676at_int(zip_nat_int(Xs,Ys2),I2) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),I2),aa_nat_int(nth_int(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5353_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ys2))
       => ( nth_Pr5826913651314560976_o_nat(zip_o_nat(Xs,Ys2),I2) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),I2),aa_nat_nat(nth_nat(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5354_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Ys2))
       => ( nth_Product_prod_o_o(zip_o_o(Xs,Ys2),I2) = product_Pair_o_o(aa_nat_o(nth_o(Xs),I2),aa_nat_o(nth_o(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5355_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Ys2))
       => ( nth_Pr1649062631805364268_o_int(zip_o_int(Xs,Ys2),I2) = product_Pair_o_int(aa_nat_o(nth_o(Xs),I2),aa_nat_int(nth_int(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5356_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ys2))
       => ( nth_Pr8617346907841251940nt_nat(zip_int_nat(Xs,Ys2),I2) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),I2),aa_nat_nat(nth_nat(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5357_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Ys2))
       => ( nth_Pr7514405829937366042_int_o(zip_int_o(Xs,Ys2),I2) = product_Pair_int_o(aa_nat_int(nth_int(Xs),I2),aa_nat_o(nth_o(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5358_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Ys2))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys2),I2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_5359_nth__zip,axiom,
    ! [I2: nat,Xs: list_b,Ys2: list_P131111800688179804it_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_b(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s341701280123345136it_nat(Ys2))
       => ( nth_Pr3591576263217728691it_nat(zip_b_5987488365222434772it_nat(Xs,Ys2),I2) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,nth_b(Xs,I2)),nth_Pr4561283279332054789it_nat(Ys2,I2)) ) ) ) ).

% nth_zip
tff(fact_5360_nth__Cons__pos,axiom,
    ! [N2: nat,X: int,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_int(nth_int(cons_int(X,Xs)),N2) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5361_nth__Cons__pos,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N2) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5362_find__Some__iff,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,Xs: list_P7438302566501821706it_nat,X: produc7388388658123137530it_nat] :
      ( ( find_P8943372834735446238it_nat(Pa,Xs) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4572066133376685982it_nat(Xs))
          & aa_Pro1537530206668979808_nat_o(Pa,nth_Pr3591576263217728691it_nat(Xs,I))
          & ( X = nth_Pr3591576263217728691it_nat(Xs,I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_Pro1537530206668979808_nat_o(Pa,nth_Pr3591576263217728691it_nat(Xs,J)) ) ) ) ).

% find_Some_iff
tff(fact_5363_find__Some__iff,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,Xs: list_P6935614879863011209it_nat,X: produc3260487557148687353it_nat] :
      ( ( find_P4815471733760996061it_nat(Pa,Xs) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4069378446737875485it_nat(Xs))
          & aa_Pro6426884464393220769_nat_o(Pa,nth_Pr8687047199098054322it_nat(Xs,I))
          & ( X = nth_Pr8687047199098054322it_nat(Xs,I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_Pro6426884464393220769_nat_o(Pa,nth_Pr8687047199098054322it_nat(Xs,J)) ) ) ) ).

% find_Some_iff
tff(fact_5364_find__Some__iff,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,Xs: list_P626663023886443800it_nat,X: produc8664842809031399944it_nat] :
      ( ( find_P5317947125398478060it_nat(Pa,Xs) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4881146867534228396it_nat(Xs))
          & aa_Pro4744234731172327378_nat_o(Pa,nth_Pr2956344619228612545it_nat(Xs,I))
          & ( X = nth_Pr2956344619228612545it_nat(Xs,I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_Pro4744234731172327378_nat_o(Pa,nth_Pr2956344619228612545it_nat(Xs,J)) ) ) ) ).

% find_Some_iff
tff(fact_5365_find__Some__iff,axiom,
    ! [Pa: fun_num_o,Xs: list_num,X: num] :
      ( ( find_num(Pa,Xs) = aa_num_option_num(some_num,X) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_num(Xs))
          & aa_num_o(Pa,aa_nat_num(nth_num(Xs),I))
          & ( X = aa_nat_num(nth_num(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_num_o(Pa,aa_nat_num(nth_num(Xs),J)) ) ) ) ).

% find_Some_iff
tff(fact_5366_find__Some__iff,axiom,
    ! [Pa: fun_nat_o,Xs: list_nat,X: nat] :
      ( ( find_nat(Pa,Xs) = aa_nat_option_nat(some_nat,X) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
          & aa_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I))
          & ( X = aa_nat_nat(nth_nat(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_nat_o(Pa,aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% find_Some_iff
tff(fact_5367_find__Some__iff,axiom,
    ! [Pa: fun_o_o,Xs: list_o,X: $o] :
      ( ( find_o(Pa,Xs) = some_o((X)) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
          & aa_o_o(Pa,aa_nat_o(nth_o(Xs),I))
          & ( (X)
          <=> aa_nat_o(nth_o(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_o_o(Pa,aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% find_Some_iff
tff(fact_5368_find__Some__iff,axiom,
    ! [Pa: fun_int_o,Xs: list_int,X: int] :
      ( ( find_int(Pa,Xs) = aa_int_option_int(some_int,X) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
          & aa_int_o(Pa,aa_nat_int(nth_int(Xs),I))
          & ( X = aa_nat_int(nth_int(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_int_o(Pa,aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% find_Some_iff
tff(fact_5369_find__Some__iff2,axiom,
    ! [X: produc7388388658123137530it_nat,Pa: fun_Pr856651894144386551_nat_o,Xs: list_P7438302566501821706it_nat] :
      ( ( aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) = find_P8943372834735446238it_nat(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4572066133376685982it_nat(Xs))
          & aa_Pro1537530206668979808_nat_o(Pa,nth_Pr3591576263217728691it_nat(Xs,I))
          & ( X = nth_Pr3591576263217728691it_nat(Xs,I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_Pro1537530206668979808_nat_o(Pa,nth_Pr3591576263217728691it_nat(Xs,J)) ) ) ) ).

% find_Some_iff2
tff(fact_5370_find__Some__iff2,axiom,
    ! [X: produc3260487557148687353it_nat,Pa: fun_Pr5746006151868627512_nat_o,Xs: list_P6935614879863011209it_nat] :
      ( ( aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) = find_P4815471733760996061it_nat(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4069378446737875485it_nat(Xs))
          & aa_Pro6426884464393220769_nat_o(Pa,nth_Pr8687047199098054322it_nat(Xs,I))
          & ( X = nth_Pr8687047199098054322it_nat(Xs,I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_Pro6426884464393220769_nat_o(Pa,nth_Pr8687047199098054322it_nat(Xs,J)) ) ) ) ).

% find_Some_iff2
tff(fact_5371_find__Some__iff2,axiom,
    ! [X: produc8664842809031399944it_nat,Pa: fun_Pr8032424337255632745_nat_o,Xs: list_P626663023886443800it_nat] :
      ( ( aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) = find_P5317947125398478060it_nat(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4881146867534228396it_nat(Xs))
          & aa_Pro4744234731172327378_nat_o(Pa,nth_Pr2956344619228612545it_nat(Xs,I))
          & ( X = nth_Pr2956344619228612545it_nat(Xs,I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_Pro4744234731172327378_nat_o(Pa,nth_Pr2956344619228612545it_nat(Xs,J)) ) ) ) ).

% find_Some_iff2
tff(fact_5372_find__Some__iff2,axiom,
    ! [X: num,Pa: fun_num_o,Xs: list_num] :
      ( ( aa_num_option_num(some_num,X) = find_num(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_num(Xs))
          & aa_num_o(Pa,aa_nat_num(nth_num(Xs),I))
          & ( X = aa_nat_num(nth_num(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_num_o(Pa,aa_nat_num(nth_num(Xs),J)) ) ) ) ).

% find_Some_iff2
tff(fact_5373_find__Some__iff2,axiom,
    ! [X: nat,Pa: fun_nat_o,Xs: list_nat] :
      ( ( aa_nat_option_nat(some_nat,X) = find_nat(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
          & aa_nat_o(Pa,aa_nat_nat(nth_nat(Xs),I))
          & ( X = aa_nat_nat(nth_nat(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_nat_o(Pa,aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% find_Some_iff2
tff(fact_5374_find__Some__iff2,axiom,
    ! [X: $o,Pa: fun_o_o,Xs: list_o] :
      ( ( some_o((X)) = find_o(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
          & aa_o_o(Pa,aa_nat_o(nth_o(Xs),I))
          & ( (X)
          <=> aa_nat_o(nth_o(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_o_o(Pa,aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% find_Some_iff2
tff(fact_5375_find__Some__iff2,axiom,
    ! [X: int,Pa: fun_int_o,Xs: list_int] :
      ( ( aa_int_option_int(some_int,X) = find_int(Pa,Xs) )
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
          & aa_int_o(Pa,aa_nat_int(nth_int(Xs),I))
          & ( X = aa_nat_int(nth_int(Xs),I) )
          & ! [J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),I)
             => ~ aa_int_o(Pa,aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% find_Some_iff2
tff(fact_5376_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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,aa_num_rat(numeral_numeral_rat,W2))),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5377_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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,aa_num_rat(numeral_numeral_rat,W2))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5378_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,M)),aa_num_Code_natural(numera5444537566228673987atural,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_5379_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_5380_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_5381_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_5382_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% numeral_le_iff
tff(fact_5383_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_5384_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_5385_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_5386_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,M)),aa_num_Code_natural(numera5444537566228673987atural,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_5387_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% numeral_less_iff
tff(fact_5388_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% numeral_plus_numeral
tff(fact_5389_numeral__plus__numeral,axiom,
    ! [M: num,N2: 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,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% numeral_plus_numeral
tff(fact_5390_numeral__plus__numeral,axiom,
    ! [M: num,N2: 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,N2)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% numeral_plus_numeral
tff(fact_5391_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,M)),aa_num_Code_natural(numera5444537566228673987atural,N2)) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% numeral_plus_numeral
tff(fact_5392_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% numeral_plus_numeral
tff(fact_5393_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W2)),Z3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z3) ).

% add_numeral_left
tff(fact_5394_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W2)),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z3) ).

% add_numeral_left
tff(fact_5395_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W2)),Z3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z3) ).

% add_numeral_left
tff(fact_5396_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,V)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,W2)),Z3)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z3) ).

% add_numeral_left
tff(fact_5397_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: 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,W2)),Z3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z3) ).

% add_numeral_left
tff(fact_5398_power__add__numeral2,axiom,
    ! [A: assn,M: num,N2: num,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,M))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_5399_power__add__numeral2,axiom,
    ! [A: rat,M: num,N2: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_5400_power__add__numeral2,axiom,
    ! [A: nat,M: num,N2: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_5401_power__add__numeral2,axiom,
    ! [A: int,M: num,N2: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,N2))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)))),B) ).

% power_add_numeral2
tff(fact_5402_power__add__numeral,axiom,
    ! [A: assn,M: num,N2: num] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_assn(power_power_assn(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_5403_power__add__numeral,axiom,
    ! [A: rat,M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_5404_power__add__numeral,axiom,
    ! [A: nat,M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_5405_power__add__numeral,axiom,
    ! [A: int,M: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,N2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% power_add_numeral
tff(fact_5406_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_nat_nat(A,B) = zip_nat_nat(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5407_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_o,A2: list_nat,B2: list_o] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = size_size_list_o(B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A2) = size_size_list_o(B2) )
       => ( ( zip_nat_o(A,B) = zip_nat_o(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5408_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_int,A2: list_nat,B2: list_int] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = size_size_list_int(B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A2) = size_size_list_int(B2) )
       => ( ( zip_nat_int(A,B) = zip_nat_int(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5409_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_nat,A2: list_o,B2: list_nat] :
      ( ( size_size_list_o(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_size_list_o(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_o_nat(A,B) = zip_o_nat(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5410_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_o,A2: list_o,B2: list_o] :
      ( ( size_size_list_o(A) = size_size_list_o(B) )
     => ( ( size_size_list_o(A2) = size_size_list_o(B2) )
       => ( ( zip_o_o(A,B) = zip_o_o(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5411_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_int,A2: list_o,B2: list_int] :
      ( ( size_size_list_o(A) = size_size_list_int(B) )
     => ( ( size_size_list_o(A2) = size_size_list_int(B2) )
       => ( ( zip_o_int(A,B) = zip_o_int(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5412_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_nat,A2: list_int,B2: list_nat] :
      ( ( size_size_list_int(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_size_list_int(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_int_nat(A,B) = zip_int_nat(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5413_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_o,A2: list_int,B2: list_o] :
      ( ( size_size_list_int(A) = size_size_list_o(B) )
     => ( ( size_size_list_int(A2) = size_size_list_o(B2) )
       => ( ( zip_int_o(A,B) = zip_int_o(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5414_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_int,A2: list_int,B2: list_int] :
      ( ( size_size_list_int(A) = size_size_list_int(B) )
     => ( ( size_size_list_int(A2) = size_size_list_int(B2) )
       => ( ( zip_int_int(A,B) = zip_int_int(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5415_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),M) ) ).

% neg_numeral_le_iff
tff(fact_5416_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),M) ) ).

% neg_numeral_le_iff
tff(fact_5417_neg__numeral__le__iff,axiom,
    ! [M: num,N2: 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,aa_num_int(numeral_numeral_int,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),M) ) ).

% neg_numeral_le_iff
tff(fact_5418_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% distrib_left_numeral
tff(fact_5419_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C2)) ).

% distrib_left_numeral
tff(fact_5420_distrib__left__numeral,axiom,
    ! [V: num,B: int,C2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C2)) ).

% distrib_left_numeral
tff(fact_5421_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,C2: code_natural] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C2)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),B)),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,V),C2)) ).

% distrib_left_numeral
tff(fact_5422_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C2)) = 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)),C2)) ).

% distrib_left_numeral
tff(fact_5423_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)),aa_num_rat(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),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_5424_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_5425_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_5426_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V: num] : times_2397367101498566445atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),aa_num_Code_natural(numera5444537566228673987atural,V)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(A,aa_num_Code_natural(numera5444537566228673987atural,V))),times_2397367101498566445atural(B,aa_num_Code_natural(numera5444537566228673987atural,V))) ).

% distrib_right_numeral
tff(fact_5427_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_5428_neg__numeral__less__iff,axiom,
    ! [M: num,N2: 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,aa_num_int(numeral_numeral_int,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N2),M) ) ).

% neg_numeral_less_iff
tff(fact_5429_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N2),M) ) ).

% neg_numeral_less_iff
tff(fact_5430_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N2),M) ) ).

% neg_numeral_less_iff
tff(fact_5431_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: 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,N2))) = 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,N2))) ).

% add_neg_numeral_simps(3)
tff(fact_5432_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% add_neg_numeral_simps(3)
tff(fact_5433_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2))) ).

% add_neg_numeral_simps(3)
tff(fact_5434_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_5435_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: 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,W2))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_5436_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_5437_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% diff_numeral_simps(3)
tff(fact_5438_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% diff_numeral_simps(3)
tff(fact_5439_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2))) ).

% diff_numeral_simps(3)
tff(fact_5440_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% diff_numeral_simps(2)
tff(fact_5441_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% diff_numeral_simps(2)
tff(fact_5442_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% diff_numeral_simps(2)
tff(fact_5443_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys2: list_nat] : zip_int_nat(cons_int(X,Xs),cons_nat(Y,Ys2)) = cons_P7512249878480867347nt_nat(product_Pair_int_nat(X,Y),zip_int_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5444_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys2: list_int] : zip_nat_int(cons_nat(X,Xs),cons_int(Y,Ys2)) = cons_P2335045147070616083at_int(product_Pair_nat_int(X,Y),zip_nat_int(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5445_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] : zip_nat_nat(cons_nat(X,Xs),cons_nat(Y,Ys2)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),zip_nat_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5446_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys2: list_int] : zip_int_int(cons_int(X,Xs),cons_int(Y,Ys2)) = cons_P3334398858971670639nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y),zip_int_int(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5447_zip__Cons__Cons,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xs: list_f5222602229612168821_nat_o,Y: produc3658429121746597890et_nat,Ys2: list_P9062070895058802706et_nat] : zip_fu1498620252286093946et_nat(cons_f8040665141538986917_nat_o(X,Xs),cons_P2869003950965171916et_nat(Y,Ys2)) = cons_P8169920869477312234et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y),zip_fu1498620252286093946et_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5448_zip__Cons__Cons,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xs: list_f5222602229612168821_nat_o,Y: produc8799533983703829792et_nat,Ys2: list_P8836981409019564976et_nat] : zip_fu1756369540273021848et_nat(cons_f8040665141538986917_nat_o(X,Xs),cons_P8169920869477312234et_nat(Y,Ys2)) = cons_P5637129515692979848et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y),zip_fu1756369540273021848et_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5449_zip__Cons__Cons,axiom,
    ! [X: b,Xs: list_b,Y: produc6653097349344004940it_nat,Ys2: list_P131111800688179804it_nat] : zip_b_5987488365222434772it_nat(cons_b(X,Xs),cons_P7873165156130251286it_nat(Y,Ys2)) = cons_P6467324686191779524it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),Y),zip_b_5987488365222434772it_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5450_zip__Cons__Cons,axiom,
    ! [X: a,Xs: list_a,Y: produc6653097349344004940it_nat,Ys2: list_P131111800688179804it_nat] : zip_a_1859587264247984595it_nat(cons_a(X,Xs),cons_P7873165156130251286it_nat(Y,Ys2)) = cons_P2339423585217329347it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),Y),zip_a_1859587264247984595it_nat(Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_5451_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_5452_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2))),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),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_5453_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W2))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_5454_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_num_rat(numeral_numeral_rat,W2))),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),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_5455_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> 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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5456_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5457_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5458_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5459_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X)),N2))
    <=> 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,X)),N2)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_5460_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: 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,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_5461_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X)),N2))
    <=> 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,X)),N2)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_5462_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: 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,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_5463_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5464_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5465_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5466_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5467_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5468_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5469_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2)),aa_nat_Code_natural(semiri3763490453095760265atural,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5470_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5471_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5472_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5473_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(semiri3763490453095760265atural,X)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5474_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5475_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5476_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5477_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5478_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,I2)),N2)),aa_nat_Code_natural(semiri3763490453095760265atural,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5479_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),N2)),aa_nat_Code_integer(semiri4939895301339042750nteger,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5480_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),N2)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5481_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5482_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),N2)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),N2)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5483_nat__numeral__as__int,axiom,
    ! [X5: num] : aa_num_nat(numeral_numeral_nat,X5) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X5)) ).

% nat_numeral_as_int
tff(fact_5484_int__ops_I3_J,axiom,
    ! [N2: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_int(numeral_numeral_int,N2) ).

% int_ops(3)
tff(fact_5485_list__tail__coinc,axiom,
    ! [N1: int,R12: list_int,N22: int,R23: list_int] :
      ( ( cons_int(N1,R12) = cons_int(N22,R23) )
     => ( ( N1 = N22 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
tff(fact_5486_list__tail__coinc,axiom,
    ! [N1: nat,R12: list_nat,N22: nat,R23: list_nat] :
      ( ( cons_nat(N1,R12) = cons_nat(N22,R23) )
     => ( ( N1 = N22 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
tff(fact_5487_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys2: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( zip_int_nat(Xs,Ys2) = cons_P7512249878480867347nt_nat(Xy,Xys) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs = cons_int(X3,Xs4) )
           => ! [Y3: nat,Ys3: list_nat] :
                ( ( Ys2 = cons_nat(Y3,Ys3) )
               => ( ( Xy = product_Pair_int_nat(X3,Y3) )
                 => ( Xys != zip_int_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5488_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys2: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( zip_nat_int(Xs,Ys2) = cons_P2335045147070616083at_int(Xy,Xys) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs = cons_nat(X3,Xs4) )
           => ! [Y3: int,Ys3: list_int] :
                ( ( Ys2 = cons_int(Y3,Ys3) )
               => ( ( Xy = product_Pair_nat_int(X3,Y3) )
                 => ( Xys != zip_nat_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5489_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat(Xs,Ys2) = cons_P6512896166579812791at_nat(Xy,Xys) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs = cons_nat(X3,Xs4) )
           => ! [Y3: nat,Ys3: list_nat] :
                ( ( Ys2 = cons_nat(Y3,Ys3) )
               => ( ( Xy = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
                 => ( Xys != zip_nat_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5490_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys2: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( zip_int_int(Xs,Ys2) = cons_P3334398858971670639nt_int(Xy,Xys) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs = cons_int(X3,Xs4) )
           => ! [Y3: int,Ys3: list_int] :
                ( ( Ys2 = cons_int(Y3,Ys3) )
               => ( ( Xy = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
                 => ( Xys != zip_int_int(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5491_zip__eq__ConsE,axiom,
    ! [Xs: list_f5222602229612168821_nat_o,Ys2: list_P9062070895058802706et_nat,Xy: produc8799533983703829792et_nat,Xys: list_P8836981409019564976et_nat] :
      ( ( zip_fu1498620252286093946et_nat(Xs,Ys2) = cons_P8169920869477312234et_nat(Xy,Xys) )
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Xs4: list_f5222602229612168821_nat_o] :
            ( ( Xs = cons_f8040665141538986917_nat_o(X3,Xs4) )
           => ! [Y3: produc3658429121746597890et_nat,Ys3: list_P9062070895058802706et_nat] :
                ( ( Ys2 = cons_P2869003950965171916et_nat(Y3,Ys3) )
               => ( ( Xy = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3) )
                 => ( Xys != zip_fu1498620252286093946et_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5492_zip__eq__ConsE,axiom,
    ! [Xs: list_f5222602229612168821_nat_o,Ys2: list_P8836981409019564976et_nat,Xy: produc3112576180687729086et_nat,Xys: list_P4783563340545138382et_nat] :
      ( ( zip_fu1756369540273021848et_nat(Xs,Ys2) = cons_P5637129515692979848et_nat(Xy,Xys) )
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Xs4: list_f5222602229612168821_nat_o] :
            ( ( Xs = cons_f8040665141538986917_nat_o(X3,Xs4) )
           => ! [Y3: produc8799533983703829792et_nat,Ys3: list_P8836981409019564976et_nat] :
                ( ( Ys2 = cons_P8169920869477312234et_nat(Y3,Ys3) )
               => ( ( Xy = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3) )
                 => ( Xys != zip_fu1756369540273021848et_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5493_zip__eq__ConsE,axiom,
    ! [Xs: list_b,Ys2: list_P131111800688179804it_nat,Xy: produc7388388658123137530it_nat,Xys: list_P7438302566501821706it_nat] :
      ( ( zip_b_5987488365222434772it_nat(Xs,Ys2) = cons_P6467324686191779524it_nat(Xy,Xys) )
     => ~ ! [X3: b,Xs4: list_b] :
            ( ( Xs = cons_b(X3,Xs4) )
           => ! [Y3: produc6653097349344004940it_nat,Ys3: list_P131111800688179804it_nat] :
                ( ( Ys2 = cons_P7873165156130251286it_nat(Y3,Ys3) )
               => ( ( Xy = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3) )
                 => ( Xys != zip_b_5987488365222434772it_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5494_zip__eq__ConsE,axiom,
    ! [Xs: list_a,Ys2: list_P131111800688179804it_nat,Xy: produc3260487557148687353it_nat,Xys: list_P6935614879863011209it_nat] :
      ( ( zip_a_1859587264247984595it_nat(Xs,Ys2) = cons_P2339423585217329347it_nat(Xy,Xys) )
     => ~ ! [X3: a,Xs4: list_a] :
            ( ( Xs = cons_a(X3,Xs4) )
           => ! [Y3: produc6653097349344004940it_nat,Ys3: list_P131111800688179804it_nat] :
                ( ( Ys2 = cons_P7873165156130251286it_nat(Y3,Ys3) )
               => ( ( Xy = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
                 => ( Xys != zip_a_1859587264247984595it_nat(Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5495_find_Osimps_I2_J,axiom,
    ! [Pa: fun_int_o,X: int,Xs: list_int] :
      find_int(Pa,cons_int(X,Xs)) = $ite(aa_int_o(Pa,X),aa_int_option_int(some_int,X),find_int(Pa,Xs)) ).

% find.simps(2)
tff(fact_5496_find_Osimps_I2_J,axiom,
    ! [Pa: fun_nat_o,X: nat,Xs: list_nat] :
      find_nat(Pa,cons_nat(X,Xs)) = $ite(aa_nat_o(Pa,X),aa_nat_option_nat(some_nat,X),find_nat(Pa,Xs)) ).

% find.simps(2)
tff(fact_5497_find_Osimps_I2_J,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,X: produc7388388658123137530it_nat,Xs: list_P7438302566501821706it_nat] :
      find_P8943372834735446238it_nat(Pa,cons_P6467324686191779524it_nat(X,Xs)) = $ite(aa_Pro1537530206668979808_nat_o(Pa,X),aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X),find_P8943372834735446238it_nat(Pa,Xs)) ).

% find.simps(2)
tff(fact_5498_find_Osimps_I2_J,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,X: produc3260487557148687353it_nat,Xs: list_P6935614879863011209it_nat] :
      find_P4815471733760996061it_nat(Pa,cons_P2339423585217329347it_nat(X,Xs)) = $ite(aa_Pro6426884464393220769_nat_o(Pa,X),aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X),find_P4815471733760996061it_nat(Pa,Xs)) ).

% find.simps(2)
tff(fact_5499_find_Osimps_I2_J,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,X: produc8664842809031399944it_nat,Xs: list_P626663023886443800it_nat] :
      find_P5317947125398478060it_nat(Pa,cons_P4136552807766418642it_nat(X,Xs)) = $ite(aa_Pro4744234731172327378_nat_o(Pa,X),aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X),find_P5317947125398478060it_nat(Pa,Xs)) ).

% find.simps(2)
tff(fact_5500_find_Osimps_I2_J,axiom,
    ! [Pa: fun_num_o,X: num,Xs: list_num] :
      find_num(Pa,cons_num(X,Xs)) = $ite(aa_num_o(Pa,X),aa_num_option_num(some_num,X),find_num(Pa,Xs)) ).

% find.simps(2)
tff(fact_5501_pair__list__split,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ~ ! [L1: list_nat,L22: list_nat] :
          ( ( L = zip_nat_nat(L1,L22) )
         => ( ( aa_list_nat_nat(size_size_list_nat,L1) = aa_list_nat_nat(size_size_list_nat,L22) )
           => ( size_s5460976970255530739at_nat(L) != aa_list_nat_nat(size_size_list_nat,L22) ) ) ) ).

% pair_list_split
tff(fact_5502_pair__list__split,axiom,
    ! [L: list_P7333126701944960589_nat_o] :
      ~ ! [L1: list_nat,L22: list_o] :
          ( ( L = zip_nat_o(L1,L22) )
         => ( ( aa_list_nat_nat(size_size_list_nat,L1) = size_size_list_o(L22) )
           => ( size_s6491369823275344609_nat_o(L) != size_size_list_o(L22) ) ) ) ).

% pair_list_split
tff(fact_5503_pair__list__split,axiom,
    ! [L: list_P3521021558325789923at_int] :
      ~ ! [L1: list_nat,L22: list_int] :
          ( ( L = zip_nat_int(L1,L22) )
         => ( ( aa_list_nat_nat(size_size_list_nat,L1) = size_size_list_int(L22) )
           => ( size_s2970893825323803983at_int(L) != size_size_list_int(L22) ) ) ) ).

% pair_list_split
tff(fact_5504_pair__list__split,axiom,
    ! [L: list_P6285523579766656935_o_nat] :
      ~ ! [L1: list_o,L22: list_nat] :
          ( ( L = zip_o_nat(L1,L22) )
         => ( ( size_size_list_o(L1) = aa_list_nat_nat(size_size_list_nat,L22) )
           => ( size_s5443766701097040955_o_nat(L) != aa_list_nat_nat(size_size_list_nat,L22) ) ) ) ).

% pair_list_split
tff(fact_5505_pair__list__split,axiom,
    ! [L: list_P4002435161011370285od_o_o] :
      ~ ! [L1: list_o,L22: list_o] :
          ( ( L = zip_o_o(L1,L22) )
         => ( ( size_size_list_o(L1) = size_size_list_o(L22) )
           => ( size_s1515746228057227161od_o_o(L) != size_size_list_o(L22) ) ) ) ).

% pair_list_split
tff(fact_5506_pair__list__split,axiom,
    ! [L: list_P3795440434834930179_o_int] :
      ~ ! [L1: list_o,L22: list_int] :
          ( ( L = zip_o_int(L1,L22) )
         => ( ( size_size_list_o(L1) = size_size_list_int(L22) )
           => ( size_s2953683556165314199_o_int(L) != size_size_list_int(L22) ) ) ) ).

% pair_list_split
tff(fact_5507_pair__list__split,axiom,
    ! [L: list_P8198026277950538467nt_nat] :
      ~ ! [L1: list_int,L22: list_nat] :
          ( ( L = zip_int_nat(L1,L22) )
         => ( ( size_size_list_int(L1) = aa_list_nat_nat(size_size_list_nat,L22) )
           => ( size_s7647898544948552527nt_nat(L) != aa_list_nat_nat(size_size_list_nat,L22) ) ) ) ).

% pair_list_split
tff(fact_5508_pair__list__split,axiom,
    ! [L: list_P5087981734274514673_int_o] :
      ~ ! [L1: list_int,L22: list_o] :
          ( ( L = zip_int_o(L1,L22) )
         => ( ( size_size_list_int(L1) = size_size_list_o(L22) )
           => ( size_s4246224855604898693_int_o(L) != size_size_list_o(L22) ) ) ) ).

% pair_list_split
tff(fact_5509_pair__list__split,axiom,
    ! [L: list_P5707943133018811711nt_int] :
      ~ ! [L1: list_int,L22: list_int] :
          ( ( L = zip_int_int(L1,L22) )
         => ( ( size_size_list_int(L1) = size_size_list_int(L22) )
           => ( size_s5157815400016825771nt_int(L) != size_size_list_int(L22) ) ) ) ).

% pair_list_split
tff(fact_5510_zip__inj,axiom,
    ! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_nat_nat(A,B) = zip_nat_nat(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5511_zip__inj,axiom,
    ! [A: list_nat,B: list_o,A2: list_nat,B2: list_o] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = size_size_list_o(B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A2) = size_size_list_o(B2) )
       => ( ( zip_nat_o(A,B) = zip_nat_o(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5512_zip__inj,axiom,
    ! [A: list_nat,B: list_int,A2: list_nat,B2: list_int] :
      ( ( aa_list_nat_nat(size_size_list_nat,A) = size_size_list_int(B) )
     => ( ( aa_list_nat_nat(size_size_list_nat,A2) = size_size_list_int(B2) )
       => ( ( zip_nat_int(A,B) = zip_nat_int(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5513_zip__inj,axiom,
    ! [A: list_o,B: list_nat,A2: list_o,B2: list_nat] :
      ( ( size_size_list_o(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_size_list_o(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_o_nat(A,B) = zip_o_nat(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5514_zip__inj,axiom,
    ! [A: list_o,B: list_o,A2: list_o,B2: list_o] :
      ( ( size_size_list_o(A) = size_size_list_o(B) )
     => ( ( size_size_list_o(A2) = size_size_list_o(B2) )
       => ( ( zip_o_o(A,B) = zip_o_o(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5515_zip__inj,axiom,
    ! [A: list_o,B: list_int,A2: list_o,B2: list_int] :
      ( ( size_size_list_o(A) = size_size_list_int(B) )
     => ( ( size_size_list_o(A2) = size_size_list_int(B2) )
       => ( ( zip_o_int(A,B) = zip_o_int(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5516_zip__inj,axiom,
    ! [A: list_int,B: list_nat,A2: list_int,B2: list_nat] :
      ( ( size_size_list_int(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_size_list_int(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_int_nat(A,B) = zip_int_nat(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5517_zip__inj,axiom,
    ! [A: list_int,B: list_o,A2: list_int,B2: list_o] :
      ( ( size_size_list_int(A) = size_size_list_o(B) )
     => ( ( size_size_list_int(A2) = size_size_list_o(B2) )
       => ( ( zip_int_o(A,B) = zip_int_o(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5518_zip__inj,axiom,
    ! [A: list_int,B: list_int,A2: list_int,B2: list_int] :
      ( ( size_size_list_int(A) = size_size_list_int(B) )
     => ( ( size_size_list_int(A2) = size_size_list_int(B2) )
       => ( ( zip_int_int(A,B) = zip_int_int(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_5519_find__SomeD_I1_J,axiom,
    ! [Pa: fun_Pr856651894144386551_nat_o,Xs: list_P7438302566501821706it_nat,X: produc7388388658123137530it_nat] :
      ( ( find_P8943372834735446238it_nat(Pa,Xs) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X) )
     => aa_Pro1537530206668979808_nat_o(Pa,X) ) ).

% find_SomeD(1)
tff(fact_5520_find__SomeD_I1_J,axiom,
    ! [Pa: fun_Pr5746006151868627512_nat_o,Xs: list_P6935614879863011209it_nat,X: produc3260487557148687353it_nat] :
      ( ( find_P4815471733760996061it_nat(Pa,Xs) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X) )
     => aa_Pro6426884464393220769_nat_o(Pa,X) ) ).

% find_SomeD(1)
tff(fact_5521_find__SomeD_I1_J,axiom,
    ! [Pa: fun_Pr8032424337255632745_nat_o,Xs: list_P626663023886443800it_nat,X: produc8664842809031399944it_nat] :
      ( ( find_P5317947125398478060it_nat(Pa,Xs) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X) )
     => aa_Pro4744234731172327378_nat_o(Pa,X) ) ).

% find_SomeD(1)
tff(fact_5522_find__SomeD_I1_J,axiom,
    ! [Pa: fun_num_o,Xs: list_num,X: num] :
      ( ( find_num(Pa,Xs) = aa_num_option_num(some_num,X) )
     => aa_num_o(Pa,X) ) ).

% find_SomeD(1)
tff(fact_5523_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),zero_z2226904508553997617atural) ).

% not_numeral_le_zero
tff(fact_5524_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),zero_z3403309356797280102nteger) ).

% not_numeral_le_zero
tff(fact_5525_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_5526_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_5527_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_5528_zero__le__numeral,axiom,
    ! [N2: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% zero_le_numeral
tff(fact_5529_zero__le__numeral,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% zero_le_numeral
tff(fact_5530_zero__le__numeral,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N2)) ).

% zero_le_numeral
tff(fact_5531_zero__le__numeral,axiom,
    ! [N2: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N2)) ).

% zero_le_numeral
tff(fact_5532_zero__le__numeral,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N2)) ).

% zero_le_numeral
tff(fact_5533_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_5534_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N2)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_5535_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_5536_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),zero_z2226904508553997617atural) ).

% not_numeral_less_zero
tff(fact_5537_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),zero_z3403309356797280102nteger) ).

% not_numeral_less_zero
tff(fact_5538_zero__less__numeral,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N2)) ).

% zero_less_numeral
tff(fact_5539_zero__less__numeral,axiom,
    ! [N2: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N2)) ).

% zero_less_numeral
tff(fact_5540_zero__less__numeral,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N2)) ).

% zero_less_numeral
tff(fact_5541_zero__less__numeral,axiom,
    ! [N2: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% zero_less_numeral
tff(fact_5542_zero__less__numeral,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% zero_less_numeral
tff(fact_5543_one__le__numeral,axiom,
    ! [N2: num] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% one_le_numeral
tff(fact_5544_one__le__numeral,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% one_le_numeral
tff(fact_5545_one__le__numeral,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2)) ).

% one_le_numeral
tff(fact_5546_one__le__numeral,axiom,
    ! [N2: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2)) ).

% one_le_numeral
tff(fact_5547_one__le__numeral,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N2)) ).

% one_le_numeral
tff(fact_5548_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat) ).

% not_numeral_less_one
tff(fact_5549_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat) ).

% not_numeral_less_one
tff(fact_5550_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),one_one_int) ).

% not_numeral_less_one
tff(fact_5551_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),one_one_Code_natural) ).

% not_numeral_less_one
tff(fact_5552_not__numeral__less__one,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_5553_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_numeral_le_neg_numeral
tff(fact_5554_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_numeral_le_neg_numeral
tff(fact_5555_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: 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,aa_num_int(numeral_numeral_int,N2))) ).

% not_numeral_le_neg_numeral
tff(fact_5556_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% neg_numeral_le_numeral
tff(fact_5557_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) ).

% neg_numeral_le_numeral
tff(fact_5558_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: 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_num_int(numeral_numeral_int,N2)) ).

% neg_numeral_le_numeral
tff(fact_5559_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_5560_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_5561_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_5562_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,X)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,X)),one_one_Code_natural) ).

% one_plus_numeral_commute
tff(fact_5563_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_5564_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: 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,aa_num_int(numeral_numeral_int,N2))) ).

% not_numeral_less_neg_numeral
tff(fact_5565_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_numeral_less_neg_numeral
tff(fact_5566_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_numeral_less_neg_numeral
tff(fact_5567_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: 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_num_int(numeral_numeral_int,N2)) ).

% neg_numeral_less_numeral
tff(fact_5568_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% neg_numeral_less_numeral
tff(fact_5569_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) ).

% neg_numeral_less_numeral
tff(fact_5570_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] : aa_list_nat_nat(size_size_list_nat,cons_nat(X,Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_Cons
tff(fact_5571_length__Cons,axiom,
    ! [X: $o,Xs: list_o] : size_size_list_o(cons_o((X),Xs)) = aa_nat_nat(suc,size_size_list_o(Xs)) ).

% length_Cons
tff(fact_5572_length__Cons,axiom,
    ! [X: int,Xs: list_int] : size_size_list_int(cons_int(X,Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ).

% length_Cons
tff(fact_5573_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys2: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys2))
     => ( Xs != cons_nat(X,Ys2) ) ) ).

% impossible_Cons
tff(fact_5574_impossible__Cons,axiom,
    ! [Xs: list_o,Ys2: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),size_size_list_o(Ys2))
     => ( Xs != cons_o((X),Ys2) ) ) ).

% impossible_Cons
tff(fact_5575_impossible__Cons,axiom,
    ! [Xs: list_int,Ys2: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),size_size_list_int(Ys2))
     => ( Xs != cons_int(X,Ys2) ) ) ).

% impossible_Cons
tff(fact_5576_sorted2,axiom,
    ! [X: rat,Y: rat,Zs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,cons_rat(X,cons_rat(Y,Zs)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
        & sorted_wrt_rat(ord_less_eq_rat,cons_rat(Y,Zs)) ) ) ).

% sorted2
tff(fact_5577_sorted2,axiom,
    ! [X: num,Y: num,Zs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,cons_num(X,cons_num(Y,Zs)))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
        & sorted_wrt_num(ord_less_eq_num,cons_num(Y,Zs)) ) ) ).

% sorted2
tff(fact_5578_sorted2,axiom,
    ! [X: nat,Y: nat,Zs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),cons_nat(X,cons_nat(Y,Zs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),cons_nat(Y,Zs)) ) ) ).

% sorted2
tff(fact_5579_sorted2,axiom,
    ! [X: int,Y: int,Zs: list_int] :
      ( aa_list_int_o(sorted_wrt_int(ord_less_eq_int),cons_int(X,cons_int(Y,Zs)))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_list_int_o(sorted_wrt_int(ord_less_eq_int),cons_int(Y,Zs)) ) ) ).

% sorted2
tff(fact_5580_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_zero_le_neg_numeral
tff(fact_5581_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_zero_le_neg_numeral
tff(fact_5582_not__zero__le__neg__numeral,axiom,
    ! [N2: 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,N2))) ).

% not_zero_le_neg_numeral
tff(fact_5583_neg__numeral__le__zero,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5584_neg__numeral__le__zero,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5585_neg__numeral__le__zero,axiom,
    ! [N2: 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,N2))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5586_not__zero__less__neg__numeral,axiom,
    ! [N2: 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,N2))) ).

% not_zero_less_neg_numeral
tff(fact_5587_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% not_zero_less_neg_numeral
tff(fact_5588_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) ).

% not_zero_less_neg_numeral
tff(fact_5589_neg__numeral__less__zero,axiom,
    ! [N2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5590_neg__numeral__less__zero,axiom,
    ! [N2: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5591_neg__numeral__less__zero,axiom,
    ! [N2: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5592_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_5593_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,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_5594_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_5595_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_5596_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)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_le_numeral
tff(fact_5597_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_5598_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_5599_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,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_5600_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_5601_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_5602_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_5603_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_5604_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_5605_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,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_le_neg_numeral
tff(fact_5606_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_5607_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_5608_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_5609_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_5610_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_5611_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_5612_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)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_less_numeral
tff(fact_5613_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_5614_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_5615_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_5616_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_5617_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_5618_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,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_less_neg_numeral
tff(fact_5619_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_5620_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_5621_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,aa_num_rat(numeral_numeral_rat,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_5622_Suc__le__length__iff,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_list_nat_nat(size_size_list_nat,Xs))
    <=> ? [X4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(X4,Ys4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_list_nat_nat(size_size_list_nat,Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_5623_Suc__le__length__iff,axiom,
    ! [N2: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),size_size_list_o(Xs))
    <=> ? [X4: $o,Ys4: list_o] :
          ( ( Xs = cons_o((X4),Ys4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),size_size_list_o(Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_5624_Suc__le__length__iff,axiom,
    ! [N2: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),size_size_list_int(Xs))
    <=> ? [X4: int,Ys4: list_int] :
          ( ( Xs = cons_int(X4,Ys4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),size_size_list_int(Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_5625_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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_num_rat(numeral_numeral_rat,W2)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_5626_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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_num_rat(numeral_numeral_rat,W2)),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_5627_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] : aa_list_nat_nat(size_size_list_nat,cons_nat(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5628_list_Osize_I4_J,axiom,
    ! [X21: $o,X222: list_o] : size_size_list_o(cons_o((X21),X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5629_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] : size_size_list_int(cons_int(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_5630_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W2)),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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_num_rat(numeral_numeral_rat,W2)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_5631_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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_num_rat(numeral_numeral_rat,W2)),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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_num_rat(numeral_numeral_rat,W2)),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_5632_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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,aa_num_rat(numeral_numeral_rat,W2))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5633_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          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,aa_num_rat(numeral_numeral_rat,W2))),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),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,aa_num_rat(numeral_numeral_rat,W2))),C2)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5634_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N2: nat] :
      ( ( X != Y )
     => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N2) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5635_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N2: nat] :
      ( ( X != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N2) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5636_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      slice_int(Begin,End,cons_int(X,Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        cons_int(X,slice_int(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_5637_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      slice_nat(Begin,End,cons_nat(X,Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        cons_nat(X,slice_nat(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_5638_same__fstI,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o,X: fun_Pr6460944699146269935_nat_o,Y7: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat,Ra: fun_fu5004560472337415769et_nat] :
      ( aa_fun1675921005893230629at_o_o(Pa,X)
     => ( member6099555550032318734et_nat(produc8199053930788261021et_nat(Y7,Y),aa_fun8297425419345108306et_nat(Ra,X))
       => member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y7),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y)),same_f842378046369478405et_nat(Pa,Ra)) ) ) ).

% same_fstI
tff(fact_5639_same__fstI,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o,X: fun_Pr6460944699146269935_nat_o,Y7: produc8799533983703829792et_nat,Y: produc8799533983703829792et_nat,Ra: fun_fu951349246861503509et_nat] :
      ( aa_fun1675921005893230629at_o_o(Pa,X)
     => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(Y7,Y),aa_fun313470179692725518et_nat(Ra,X))
       => member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y7),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y)),same_f7796809703074202275et_nat(Pa,Ra)) ) ) ).

% same_fstI
tff(fact_5640_same__fstI,axiom,
    ! [Pa: fun_b_o,X: b,Y7: produc6653097349344004940it_nat,Y: produc6653097349344004940it_nat,Ra: fun_b_4666577566277263197it_nat] :
      ( aa_b_o(Pa,X)
     => ( member5617269971687963298it_nat(produc3130510018828335921it_nat(Y7,Y),aa_b_s8207871714854413398it_nat(Ra,X))
       => member6820296301096096254it_nat(produc6853161671299316109it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),Y7),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),Y)),same_f3722904161141537503it_nat(Pa,Ra)) ) ) ).

% same_fstI
tff(fact_5641_same__fstI,axiom,
    ! [Pa: fun_a_o,X: a,Y7: produc6653097349344004940it_nat,Y: produc6653097349344004940it_nat,Ra: fun_a_728177123528609950it_nat] :
      ( aa_a_o(Pa,X)
     => ( member5617269971687963298it_nat(produc3130510018828335921it_nat(Y7,Y),aa_a_s4269471272105760151it_nat(Ra,X))
       => member5335690527091456380it_nat(produc1743342482959036555it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),Y7),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),Y)),same_f8818375097021863134it_nat(Pa,Ra)) ) ) ).

% same_fstI
tff(fact_5642_same__fstI,axiom,
    ! [Pa: fun_int_o,X: int,Y7: int,Y: int,Ra: fun_in7209865251430457670nt_int] :
      ( aa_int_o(Pa,X)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y7),Y)),aa_int8519581412571446831nt_int(Ra,X))
       => member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y7),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),same_fst_int_int(Pa,Ra)) ) ) ).

% same_fstI
tff(fact_5643_upto__aux__rec,axiom,
    ! [I2: int,J2: int,Js: list_int] :
      upto_aux(I2,J2,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I2),Js,upto_aux(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J2),one_one_int),cons_int(J2,Js))) ).

% upto_aux_rec
tff(fact_5644_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: 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))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_5645_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),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,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_5646_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X))),N2))
    <=> 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,X))),N2)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5647_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X))),N2))
    <=> 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,X))),N2)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5648_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,aa_num_rat(numeral_numeral_rat,X))),N2))
    <=> 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,X))),N2)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5649_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,X))),N2)),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,X))),N2)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5650_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,X))),N2)),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,X))),N2)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5651_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,aa_num_rat(numeral_numeral_rat,X))),N2)),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,X))),N2)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5652_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X))),N2))
    <=> 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,X))),N2)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5653_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,aa_num_rat(numeral_numeral_rat,X))),N2))
    <=> 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,X))),N2)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5654_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X))),N2))
    <=> 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,X))),N2)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5655_of__int__le__iff,axiom,
    ! [W2: int,Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z3) ) ).

% of_int_le_iff
tff(fact_5656_of__int__le__iff,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z3) ) ).

% of_int_le_iff
tff(fact_5657_of__int__less__iff,axiom,
    ! [W2: int,Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ).

% of_int_less_iff
tff(fact_5658_of__int__less__iff,axiom,
    ! [W2: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z3) ) ).

% of_int_less_iff
tff(fact_5659_of__int__add,axiom,
    ! [W2: int,Z3: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z3)) ).

% of_int_add
tff(fact_5660_of__int__add,axiom,
    ! [W2: int,Z3: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% of_int_add
tff(fact_5661_ceiling__add__of__int,axiom,
    ! [X: rat,Z3: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z3) ).

% ceiling_add_of_int
tff(fact_5662_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z3)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5663_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5664_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z3)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5665_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3) ) ).

% of_int_0_le_iff
tff(fact_5666_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3) ) ).

% of_int_0_le_iff
tff(fact_5667_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3) ) ).

% of_int_0_le_iff
tff(fact_5668_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z3)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5669_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5670_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z3)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5671_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3) ) ).

% of_int_0_less_iff
tff(fact_5672_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3) ) ).

% of_int_0_less_iff
tff(fact_5673_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3) ) ).

% of_int_0_less_iff
tff(fact_5674_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z3) ) ).

% of_int_numeral_le_iff
tff(fact_5675_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z3) ) ).

% of_int_numeral_le_iff
tff(fact_5676_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z3) ) ).

% of_int_numeral_le_iff
tff(fact_5677_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z3)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_le_numeral_iff
tff(fact_5678_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_le_numeral_iff
tff(fact_5679_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z3)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_le_numeral_iff
tff(fact_5680_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z3) ) ).

% of_int_numeral_less_iff
tff(fact_5681_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z3) ) ).

% of_int_numeral_less_iff
tff(fact_5682_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z3) ) ).

% of_int_numeral_less_iff
tff(fact_5683_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_less_numeral_iff
tff(fact_5684_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z3)),aa_num_int(numeral_numeral_int,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_less_numeral_iff
tff(fact_5685_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z3)),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_num_int(numeral_numeral_int,N2)) ) ).

% of_int_less_numeral_iff
tff(fact_5686_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z3)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5687_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5688_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z3)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5689_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z3) ) ).

% of_int_1_le_iff
tff(fact_5690_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z3) ) ).

% of_int_1_le_iff
tff(fact_5691_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z3) ) ).

% of_int_1_le_iff
tff(fact_5692_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z3)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5693_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5694_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z3)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5695_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z3) ) ).

% of_int_1_less_iff
tff(fact_5696_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z3) ) ).

% of_int_1_less_iff
tff(fact_5697_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z3))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z3) ) ).

% of_int_1_less_iff
tff(fact_5698_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_5699_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% zero_less_ceiling
tff(fact_5700_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_5701_ceiling__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_5702_one__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% one_le_ceiling
tff(fact_5703_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_less_ceiling
tff(fact_5704_ceiling__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ).

% ceiling_le_one
tff(fact_5705_one__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ).

% one_less_ceiling
tff(fact_5706_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5707_ceiling__add__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_add_one
tff(fact_5708_of__nat__nat,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z3)) = aa_int_rat(ring_1_of_int_rat,Z3) ) ) ).

% of_nat_nat
tff(fact_5709_of__nat__nat,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z3)) = aa_int_int(ring_1_of_int_int,Z3) ) ) ).

% of_nat_nat
tff(fact_5710_of__nat__nat,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,Z3)) = aa_int_Code_integer(ring_18347121197199848620nteger,Z3) ) ) ).

% of_nat_nat
tff(fact_5711_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5712_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5713_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5714_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5715_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5716_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: 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)),W2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5717_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5718_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5719_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_5720_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X) ) ).

% zero_le_ceiling
tff(fact_5721_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_5722_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_le_ceiling
tff(fact_5723_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),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,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5724_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: 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))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_5725_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5726_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5727_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5728_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X)),N2))
    <=> 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,X)),N2)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5729_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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_num_rat(numeral_numeral_rat,X)),N2))
    <=> 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,X)),N2)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5730_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X)),N2))
    <=> 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,X)),N2)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5731_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5732_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5733_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,X)),N2)),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,X)),N2)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5734_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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_num_rat(numeral_numeral_rat,X)),N2))
    <=> 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,X)),N2)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5735_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X)),N2))
    <=> 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,X)),N2)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5736_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: 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,X)),N2))
    <=> 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,X)),N2)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5737_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,X))),N2)),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,X))),N2)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5738_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,aa_num_rat(numeral_numeral_rat,X))),N2)),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,X))),N2)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5739_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: 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,X))),N2)),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,X))),N2)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5740_le__of__int__ceiling,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ).

% le_of_int_ceiling
tff(fact_5741_ceiling__le__iff,axiom,
    ! [X: rat,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z3)) ) ).

% ceiling_le_iff
tff(fact_5742_less__ceiling__iff,axiom,
    ! [Z3: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X) ) ).

% less_ceiling_iff
tff(fact_5743_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_5744_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X) ).

% ex_of_int_less
tff(fact_5745_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_5746_ceiling__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_correct
tff(fact_5747_ceiling__unique,axiom,
    ! [Z3: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z3 ) ) ) ).

% ceiling_unique
tff(fact_5748_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_5749_ceiling__split,axiom,
    ! [Pa: fun_int_o,T6: rat] :
      ( aa_int_o(Pa,aa_rat_int(archim2889992004027027881ng_rat,T6))
    <=> ! [I: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,I)),one_one_rat)),T6)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T6),aa_int_rat(ring_1_of_int_rat,I)) )
         => aa_int_o(Pa,I) ) ) ).

% ceiling_split
tff(fact_5750_ceiling__less__iff,axiom,
    ! [X: rat,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_5751_le__ceiling__iff,axiom,
    ! [Z3: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat)),X) ) ).

% le_ceiling_iff
tff(fact_5752_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,X)) ) ).

% ceiling_mono
tff(fact_5753_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% ceiling_less_cancel
tff(fact_5754_ceiling__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q4)))),Q4)) ) ).

% ceiling_divide_upper
tff(fact_5755_ceiling__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => 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(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q4)))),one_one_rat)),Q4)),P2) ) ).

% ceiling_divide_lower
tff(fact_5756_of__nat__ceiling,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R)))) ).

% of_nat_ceiling
tff(fact_5757_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_5758_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z3)) ) ).

% of_int_nonneg
tff(fact_5759_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z3)) ) ).

% of_int_nonneg
tff(fact_5760_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z3)) ) ).

% of_int_nonneg
tff(fact_5761_of__int__pos,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z3)) ) ).

% of_int_pos
tff(fact_5762_of__int__pos,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z3)) ) ).

% of_int_pos
tff(fact_5763_of__int__pos,axiom,
    ! [Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z3)) ) ).

% of_int_pos
tff(fact_5764_of__int__leD,axiom,
    ! [N2: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).

% of_int_leD
tff(fact_5765_of__int__leD,axiom,
    ! [N2: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% of_int_leD
tff(fact_5766_of__int__leD,axiom,
    ! [N2: int,X: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).

% of_int_leD
tff(fact_5767_of__int__lessD,axiom,
    ! [N2: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).

% of_int_lessD
tff(fact_5768_of__int__lessD,axiom,
    ! [N2: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% of_int_lessD
tff(fact_5769_of__int__lessD,axiom,
    ! [N2: int,X: 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,N2))),X)
     => ( ( N2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).

% of_int_lessD
tff(fact_5770_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_5771_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y5)),X)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
tff(fact_5772_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_5773_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_5774_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_int_Code_integer(ring_18347121197199848620nteger,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_5775_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,aa_rat_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,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).

% mult_ceiling_le
tff(fact_5776_of__int__of__nat,axiom,
    ! [K2: int] :
      aa_int_rat(ring_1_of_int_rat,K2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K2)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K2))) ).

% of_int_of_nat
tff(fact_5777_of__int__of__nat,axiom,
    ! [K2: int] :
      aa_int_int(ring_1_of_int_int,K2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K2)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K2))) ).

% of_int_of_nat
tff(fact_5778_of__int__of__nat,axiom,
    ! [K2: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K2)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K2))) ).

% of_int_of_nat
tff(fact_5779_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_5780_neg__numeral__less__floor,axiom,
    ! [V: num,X: 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))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> 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,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_less_floor
tff(fact_5781_in__lex__prod,axiom,
    ! [A: int,B: int,A2: int,B2: int,R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2)),lex_prod_int_int(R,S2))
    <=> ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),A2)),R)
        | ( ( A = A2 )
          & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),B2)),S2) ) ) ) ).

% in_lex_prod
tff(fact_5782_in__lex__prod,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat,R: set_Pr2890355626087285447_nat_o,S2: set_Pr719794911490849221et_nat] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A2),B2)),lex_pr6479962325641483138et_nat(R,S2))
    <=> ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A,A2),R)
        | ( ( A = A2 )
          & member6099555550032318734et_nat(produc8199053930788261021et_nat(B,B2),S2) ) ) ) ).

% in_lex_prod
tff(fact_5783_in__lex__prod,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat,R: set_Pr2890355626087285447_nat_o,S2: set_Pr9066344408807579265et_nat] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A2),B2)),lex_pr1264321499788159136et_nat(R,S2))
    <=> ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A,A2),R)
        | ( ( A = A2 )
          & member2216856746449346506et_nat(produc8050396023069295193et_nat(B,B2),S2) ) ) ) ).

% in_lex_prod
tff(fact_5784_in__lex__prod,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,A2: b,B2: produc6653097349344004940it_nat,R: set_Product_prod_b_b,S2: set_Pr4389693562480114009it_nat] :
      ( member6820296301096096254it_nat(produc6853161671299316109it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A2),B2)),lex_pr2603589247317408988it_nat(R,S2))
    <=> ( member7862447936710763792od_b_b(product_Pair_b_b(A,A2),R)
        | ( ( A = A2 )
          & member5617269971687963298it_nat(produc3130510018828335921it_nat(B,B2),S2) ) ) ) ).

% in_lex_prod
tff(fact_5785_in__lex__prod,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat,R: set_Product_prod_a_a,S2: set_Pr4389693562480114009it_nat] :
      ( member5335690527091456380it_nat(produc1743342482959036555it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A2),B2)),lex_pr7699060183197734619it_nat(R,S2))
    <=> ( member1426531477525435216od_a_a(product_Pair_a_a(A,A2),R)
        | ( ( A = A2 )
          & member5617269971687963298it_nat(produc3130510018828335921it_nat(B,B2),S2) ) ) ) ).

% in_lex_prod
tff(fact_5786_in__lex__prod,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),lex_prod_nat_nat(R,S2))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),A2)),R)
        | ( ( A = A2 )
          & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),B2)),S2) ) ) ) ).

% in_lex_prod
tff(fact_5787_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)
     => ( aa_set_rat_o(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_rat_int(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,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5788_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)
     => ( aa_set_rat_o(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_rat_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,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5789_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_neg_numeral
tff(fact_5790_neg__numeral__le__floor,axiom,
    ! [V: num,X: 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))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_le_floor
tff(fact_5791_floor__add2,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_set_rat_o(member_rat(X),ring_1_Ints_rat)
        | aa_set_rat_o(member_rat(Y),ring_1_Ints_rat) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ) ).

% floor_add2
tff(fact_5792_zero__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X) ) ).

% zero_le_floor
tff(fact_5793_floor__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_5794_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_le_floor
tff(fact_5795_zero__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% zero_less_floor
tff(fact_5796_floor__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_le_zero
tff(fact_5797_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% floor_less_numeral
tff(fact_5798_one__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% one_le_floor
tff(fact_5799_floor__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_less_one
tff(fact_5800_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_less_floor
tff(fact_5801_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_5802_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( aa_set_rat_o(member_rat(B),ring_1_Ints_rat)
       => aa_set_rat_o(member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_5803_Ints__add,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_set_int_o(member_int(B),ring_1_Ints_int)
       => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),ring_1_Ints_int) ) ) ).

% Ints_add
tff(fact_5804_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ).

% floor_mono
tff(fact_5805_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ).

% floor_less_cancel
tff(fact_5806_of__int__floor__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X) ).

% of_int_floor_le
tff(fact_5807_floor__le__ceiling,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X)) ).

% floor_le_ceiling
tff(fact_5808_Ints__double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( aa_set5643387332196575111eger_o(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_5809_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(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_5810_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( aa_set_int_o(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_5811_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)
     => ( aa_set_rat_o(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,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_5812_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)
     => ( aa_set_rat_o(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,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_int(ring_1_of_int_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_5813_le__floor__iff,axiom,
    ! [Z3: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X) ) ).

% le_floor_iff
tff(fact_5814_le__floor__add,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y))) ).

% le_floor_add
tff(fact_5815_floor__less__iff,axiom,
    ! [X: rat,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z3)) ) ).

% floor_less_iff
tff(fact_5816_floor__add__int,axiom,
    ! [X: rat,Z3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z3) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).

% floor_add_int
tff(fact_5817_int__add__floor,axiom,
    ! [Z3: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X)) ).

% int_add_floor
tff(fact_5818_Ints__odd__nonzero,axiom,
    ! [A: code_integer] :
      ( aa_set5643387332196575111eger_o(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_5819_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(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_5820_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( aa_set_int_o(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_5821_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,aa_rat_int(archim3151403230148437115or_rat,R)))),R) ) ).

% of_nat_floor
tff(fact_5822_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) ).

% one_add_floor
tff(fact_5823_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_5824_ceiling__altdef,axiom,
    ! [X: rat] :
      aa_rat_int(archim2889992004027027881ng_rat,X) = $ite(X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)) ).

% ceiling_altdef
tff(fact_5825_Ints__odd__less__0,axiom,
    ! [A: code_integer] :
      ( aa_set5643387332196575111eger_o(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_5826_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(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_5827_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( aa_set_int_o(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_5828_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5829_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( aa_set_rat_o(member_rat(X),ring_1_Ints_rat)
     => ( ( X != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5830_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( aa_set_int_o(member_int(X),ring_1_Ints_int)
     => ( ( X != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5831_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),one_one_Code_integer)
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5832_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( aa_set_rat_o(member_rat(X),ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5833_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( aa_set_int_o(member_int(X),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5834_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),ring_11222124179247155820nteger)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),ring_11222124179247155820nteger)
       => ( ( X = Y )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Y))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5835_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( aa_set_rat_o(member_rat(X),ring_1_Ints_rat)
     => ( aa_set_rat_o(member_rat(Y),ring_1_Ints_rat)
       => ( ( X = Y )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5836_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(member_int(X),ring_1_Ints_int)
     => ( aa_set_int_o(member_int(Y),ring_1_Ints_int)
       => ( ( X = Y )
        <=> 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,X),Y))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5837_floor__unique,axiom,
    ! [Z3: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z3 ) ) ) ).

% floor_unique
tff(fact_5838_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),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_5839_floor__split,axiom,
    ! [Pa: fun_int_o,T6: rat] :
      ( aa_int_o(Pa,aa_rat_int(archim3151403230148437115or_rat,T6))
    <=> ! [I: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I)),T6)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T6),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I)),one_one_rat)) )
         => aa_int_o(Pa,I) ) ) ).

% floor_split
tff(fact_5840_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,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% le_mult_floor
tff(fact_5841_less__floor__iff,axiom,
    ! [Z3: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> 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,Z3)),one_one_rat)),X) ) ).

% less_floor_iff
tff(fact_5842_floor__le__iff,axiom,
    ! [X: rat,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z3)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z3)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_5843_floor__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))) ) ).

% floor_correct
tff(fact_5844_floor__divide__lower,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => 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,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q4)))),Q4)),P2) ) ).

% floor_divide_lower
tff(fact_5845_floor__divide__upper,axiom,
    ! [Q4: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),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,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q4)))),one_one_rat)),Q4)) ) ).

% floor_divide_upper
tff(fact_5846_floor__add,axiom,
    ! [X: rat,Y: rat] :
      aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(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(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int)) ).

% floor_add
tff(fact_5847_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = A )
    <=> ( aa_set_rat_o(member_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),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_5848_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(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_5849_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(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_5850_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_5851_diff__numeral__special_I3_J,axiom,
    ! [N2: 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,N2))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% diff_numeral_special(3)
tff(fact_5852_diff__numeral__special_I3_J,axiom,
    ! [N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% diff_numeral_special(3)
tff(fact_5853_diff__numeral__special_I3_J,axiom,
    ! [N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% diff_numeral_special(3)
tff(fact_5854_add__neg__numeral__special_I5_J,axiom,
    ! [N2: 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,N2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,N2))) ).

% add_neg_numeral_special(5)
tff(fact_5855_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,N2))) ).

% add_neg_numeral_special(5)
tff(fact_5856_add__neg__numeral__special_I5_J,axiom,
    ! [N2: 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,aa_num_rat(numeral_numeral_rat,N2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,N2))) ).

% add_neg_numeral_special(5)
tff(fact_5857_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_5858_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_5859_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,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_5860_semiring__norm_I68_J,axiom,
    ! [N2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),N2) ).

% semiring_norm(68)
tff(fact_5861_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_5862_Suc__numeral,axiom,
    ! [N2: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% Suc_numeral
tff(fact_5863_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))
    <=> ~ aa_set_rat_o(member_rat(X),ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_5864_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_5865_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,aa_num_rat(numeral_numeral_rat,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5866_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_5867_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_5868_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_5869_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,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5870_one__plus__numeral,axiom,
    ! [N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% one_plus_numeral
tff(fact_5871_one__plus__numeral,axiom,
    ! [N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% one_plus_numeral
tff(fact_5872_one__plus__numeral,axiom,
    ! [N2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N2)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% one_plus_numeral
tff(fact_5873_one__plus__numeral,axiom,
    ! [N2: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N2)) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% one_plus_numeral
tff(fact_5874_one__plus__numeral,axiom,
    ! [N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2)) ).

% one_plus_numeral
tff(fact_5875_numeral__plus__one,axiom,
    ! [N2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% numeral_plus_one
tff(fact_5876_numeral__plus__one,axiom,
    ! [N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% numeral_plus_one
tff(fact_5877_numeral__plus__one,axiom,
    ! [N2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N2)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% numeral_plus_one
tff(fact_5878_numeral__plus__one,axiom,
    ! [N2: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),one_one_Code_natural) = aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% numeral_plus_one
tff(fact_5879_numeral__plus__one,axiom,
    ! [N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% numeral_plus_one
tff(fact_5880_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),one_one_Code_natural)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_5881_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),one_one_Code_integer)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_5882_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_5883_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_5884_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N2),one) ) ).

% numeral_le_one_iff
tff(fact_5885_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_5886_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_5887_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_5888_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_num_Code_natural(numera5444537566228673987atural,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_5889_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N2) ) ).

% one_less_numeral_iff
tff(fact_5890_add__One,axiom,
    ! [X: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),one) = aa_num_num(inc,X) ).

% add_One
tff(fact_5891_add__One__commute,axiom,
    ! [N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N2) = aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one) ).

% add_One_commute
tff(fact_5892_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),one)
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_5893_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) ).

% add_inc
tff(fact_5894_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,X),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)),X) ).

% mult_inc
tff(fact_5895_frac__ge__0,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)) ).

% frac_ge_0
tff(fact_5896_frac__lt__1,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat) ).

% frac_lt_1
tff(fact_5897_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_5898_numeral__inc,axiom,
    ! [X: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% numeral_inc
tff(fact_5899_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% numeral_inc
tff(fact_5900_numeral__inc,axiom,
    ! [X: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% numeral_inc
tff(fact_5901_numeral__inc,axiom,
    ! [X: num] : aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(inc,X)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,X)),one_one_Code_natural) ).

% numeral_inc
tff(fact_5902_numeral__inc,axiom,
    ! [X: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,X)),one_one_Code_integer) ).

% numeral_inc
tff(fact_5903_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),N2) ).

% Suc_nat_number_of_add
tff(fact_5904_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% frac_eq
tff(fact_5905_frac__add,axiom,
    ! [X: rat,Y: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) = $ite(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(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat)) ).

% frac_add
tff(fact_5906_pochhammer__double,axiom,
    ! [Z3: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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))),N2)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z3),N2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N2)) ).

% pochhammer_double
tff(fact_5907_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: 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,Q4),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,Q4),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_5908_execute__change,axiom,
    ! [Fa: fun_Pr2191293291710260995t_unit,R: ref_Product_unit,Ha: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(ref_ch7259622376331601608t_unit(Fa,R)),Ha) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_Pro8294289767544744956t_unit(Fa,ref_get_Product_unit(Ha,R)),produc584006145561248582it_nat(ref_set_Product_unit(R,aa_Pro8294289767544744956t_unit(Fa,ref_get_Product_unit(Ha,R)),Ha),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))) ).

% execute_change
tff(fact_5909_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_5910_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_5911_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_5912_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: 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,Q4),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,Q4),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_5913_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5914_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_5915_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_5916_semiring__norm_I6_J,axiom,
    ! [M: num,N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,N2)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% semiring_norm(6)
tff(fact_5917_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(78)
tff(fact_5918_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(71)
tff(fact_5919_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(80)
tff(fact_5920_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(73)
tff(fact_5921_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_5922_semiring__norm_I9_J,axiom,
    ! [M: num,N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% semiring_norm(9)
tff(fact_5923_semiring__norm_I7_J,axiom,
    ! [M: num,N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit1,N2)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)) ).

% semiring_norm(7)
tff(fact_5924_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_5925_semiring__norm_I76_J,axiom,
    ! [N2: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,N2)) ).

% semiring_norm(76)
tff(fact_5926_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(72)
tff(fact_5927_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(81)
tff(fact_5928_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_5929_semiring__norm_I77_J,axiom,
    ! [N2: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,N2)) ).

% semiring_norm(77)
tff(fact_5930_semiring__norm_I10_J,axiom,
    ! [M: num,N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)),one)) ).

% semiring_norm(10)
tff(fact_5931_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_5932_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_5933_semiring__norm_I4_J,axiom,
    ! [N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N2)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,N2),one)) ).

% semiring_norm(4)
tff(fact_5934_semiring__norm_I3_J,axiom,
    ! [N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,N2)) = aa_num_num(bit1,N2) ).

% semiring_norm(3)
tff(fact_5935_semiring__norm_I16_J,axiom,
    ! [M: num,N2: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N2)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N2)))) ).

% semiring_norm(16)
tff(fact_5936_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2) ) ).

% semiring_norm(79)
tff(fact_5937_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% semiring_norm(74)
tff(fact_5938_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_5939_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_5940_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_5941_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_5942_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_5943_add__2__eq__Suc_H,axiom,
    ! [N2: nat] : 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_nat(suc,aa_nat_nat(suc,N2)) ).

% add_2_eq_Suc'
tff(fact_5944_add__2__eq__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = aa_nat_nat(suc,aa_nat_nat(suc,N2)) ).

% add_2_eq_Suc
tff(fact_5945_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_5946_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_5947_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_5948_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,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5949_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_5950_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_5951_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_5952_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5953_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5954_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5955_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_5956_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_5957_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_5958_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_5959_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_5960_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: 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,N2)))) = 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))),N2)) ).

% mod_Suc_eq_mod_add3
tff(fact_5961_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_5962_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: 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,N2)))) = 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))),N2)) ).

% div_Suc_eq_div_add3
tff(fact_5963_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_5964_one__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_5965_floor__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_5966_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_5967_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W2)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_5968_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_5969_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_5970_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_5971_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_num_rat(numeral_numeral_rat,N2)) ).

% numeral_Bit0
tff(fact_5972_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,N2)) ).

% numeral_Bit0
tff(fact_5973_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,N2)) ).

% numeral_Bit0
tff(fact_5974_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,N2)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),aa_num_Code_natural(numera5444537566228673987atural,N2)) ).

% numeral_Bit0
tff(fact_5975_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_num_Code_integer(numera6620942414471956472nteger,N2)) ).

% numeral_Bit0
tff(fact_5976_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N2)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,N2),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m),m) ) ).

% numeral_code(2)
tff(fact_5977_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N2),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_5978_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N2),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_5979_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,N2)) = $let(
        m: code_natural,
        m:= aa_num_Code_natural(numera5444537566228673987atural,N2),
        aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,m),m) ) ).

% numeral_code(2)
tff(fact_5980_numeral__code_I2_J,axiom,
    ! [N2: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,N2)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,N2),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m) ) ).

% numeral_code(2)
tff(fact_5981_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_num_rat(numeral_numeral_rat,N2))),one_one_rat) ).

% numeral_Bit1
tff(fact_5982_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2)) = 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,N2)),aa_num_nat(numeral_numeral_nat,N2))),one_one_nat) ).

% numeral_Bit1
tff(fact_5983_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)) = 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,N2)),aa_num_int(numeral_numeral_int,N2))),one_one_int) ).

% numeral_Bit1
tff(fact_5984_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,N2)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_num_Code_natural(numera5444537566228673987atural,N2)),aa_num_Code_natural(numera5444537566228673987atural,N2))),one_one_Code_natural) ).

% numeral_Bit1
tff(fact_5985_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_num_Code_integer(numera6620942414471956472nteger,N2))),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_5986_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_5987_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_5988_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N2)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,N2),
        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_5989_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N2),
        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_5990_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N2),
        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_5991_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,N2)) = $let(
        m: code_natural,
        m:= aa_num_Code_natural(numera5444537566228673987atural,N2),
        aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,m),m)),one_one_Code_natural) ) ).

% numeral_code(3)
tff(fact_5992_numeral__code_I3_J,axiom,
    ! [N2: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,N2)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,N2),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m)),one_one_Code_integer) ) ).

% numeral_code(3)
tff(fact_5993_power__numeral__odd,axiom,
    ! [Z3: assn,W2: num] :
      aa_nat_assn(power_power_assn(Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Z3),w)),w) ) ).

% power_numeral_odd
tff(fact_5994_power__numeral__odd,axiom,
    ! [Z3: rat,W2: num] :
      aa_nat_rat(power_power_rat(Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),w)),w) ) ).

% power_numeral_odd
tff(fact_5995_power__numeral__odd,axiom,
    ! [Z3: nat,W2: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z3),w)),w) ) ).

% power_numeral_odd
tff(fact_5996_power__numeral__odd,axiom,
    ! [Z3: int,W2: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z3),w)),w) ) ).

% power_numeral_odd
tff(fact_5997_power__numeral__even,axiom,
    ! [Z3: assn,W2: num] :
      aa_nat_assn(power_power_assn(Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,w),w) ) ).

% power_numeral_even
tff(fact_5998_power__numeral__even,axiom,
    ! [Z3: rat,W2: num] :
      aa_nat_rat(power_power_rat(Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_5999_power__numeral__even,axiom,
    ! [Z3: nat,W2: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_6000_power__numeral__even,axiom,
    ! [Z3: int,W2: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z3),aa_num_nat(numeral_numeral_nat,W2)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_6001_Suc3__eq__add__3,axiom,
    ! [N2: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N2) ).

% Suc3_eq_add_3
tff(fact_6002_mult__2,axiom,
    ! [Z3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),Z3) ).

% mult_2
tff(fact_6003_mult__2,axiom,
    ! [Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z3),Z3) ).

% mult_2
tff(fact_6004_mult__2,axiom,
    ! [Z3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),Z3) ).

% mult_2
tff(fact_6005_mult__2,axiom,
    ! [Z3: code_natural] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),Z3) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z3),Z3) ).

% mult_2
tff(fact_6006_mult__2,axiom,
    ! [Z3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Z3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z3),Z3) ).

% mult_2
tff(fact_6007_mult__2__right,axiom,
    ! [Z3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),Z3) ).

% mult_2_right
tff(fact_6008_mult__2__right,axiom,
    ! [Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z3),Z3) ).

% mult_2_right
tff(fact_6009_mult__2__right,axiom,
    ! [Z3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),Z3) ).

% mult_2_right
tff(fact_6010_mult__2__right,axiom,
    ! [Z3: code_natural] : times_2397367101498566445atural(Z3,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z3),Z3) ).

% mult_2_right
tff(fact_6011_mult__2__right,axiom,
    ! [Z3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z3),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z3),Z3) ).

% mult_2_right
tff(fact_6012_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,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_6013_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_6014_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_6015_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A)),B) ).

% left_add_twice
tff(fact_6016_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_6017_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_6018_less__exp,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% less_exp
tff(fact_6019_power2__nat__le__imp__le,axiom,
    ! [M: nat,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,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% power2_nat_le_imp_le
tff(fact_6020_power2__nat__le__eq__le,axiom,
    ! [M: nat,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,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% power2_nat_le_eq_le
tff(fact_6021_self__le__ge2__pow,axiom,
    ! [K2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M)) ) ).

% self_le_ge2_pow
tff(fact_6022_half__gt__zero__iff,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(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_6023_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_6024_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_6025_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_6026_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_6027_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6028_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6029_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6030_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_6031_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6032_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6033_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6034_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% power2_le_imp_le
tff(fact_6035_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_6036_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_6037_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_6038_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: 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)))),N2) = 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)),N2) ).

% Suc_mod_eq_add3_mod
tff(fact_6039_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: 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)))),N2) = 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)),N2) ).

% Suc_div_eq_add3_div
tff(fact_6040_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger )
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_6041_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_6042_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_6043_less__2__cases,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( N2 = zero_zero_nat )
        | ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_6044_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( N2 = zero_zero_nat )
        | ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_6045_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6046_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6047_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6048_nat__induct2,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N5: nat] :
              ( aa_nat_o(Pa,N5)
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_induct2
tff(fact_6049_diff__le__diff__pow,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),N2))) ) ).

% diff_le_diff_pow
tff(fact_6050_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_6051_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_6052_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_6053_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6054_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6055_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6056_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% power2_less_imp_less
tff(fact_6057_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6058_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6059_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_6060_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_6061_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_6062_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_6063_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6064_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6065_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_6066_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_6067_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: 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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_6068_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_6069_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6070_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6071_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6072_power2__sum,axiom,
    ! [X: code_natural,Y: code_natural] : aa_nat_Code_natural(power_7079662738309270450atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(power_7079662738309270450atural(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_natural(power_7079662738309270450atural(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),times_2397367101498566445atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),X),Y)) ).

% power2_sum
tff(fact_6073_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_6074_square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6075_square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6076_square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6077_zero__le__even__power_H,axiom,
    ! [A: code_integer,N2: 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))),N2))) ).

% zero_le_even_power'
tff(fact_6078_zero__le__even__power_H,axiom,
    ! [A: rat,N2: 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))),N2))) ).

% zero_le_even_power'
tff(fact_6079_zero__le__even__power_H,axiom,
    ! [A: int,N2: 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))),N2))) ).

% zero_le_even_power'
tff(fact_6080_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6081_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6082_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6083_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),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,X)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6084_abs__square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6085_abs__square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),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,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6086_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),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,X)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6087_abs__square__less__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6088_abs__square__less__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),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,X)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6089_Suc__n__div__2__gt__zero,axiom,
    ! [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(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_6090_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => 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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_6091_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_6092_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_6093_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_6094_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6095_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6096_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Y)),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(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_6097_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N2: 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))),N2))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_6098_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: 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))),N2))))
     => 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_6099_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: 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))),N2))))
     => 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_6100_odd__power__less__zero,axiom,
    ! [A: code_integer,N2: 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))),N2)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_6101_odd__power__less__zero,axiom,
    ! [A: rat,N2: 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))),N2)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_6102_odd__power__less__zero,axiom,
    ! [A: int,N2: 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))),N2)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_6103_ex__power__ivl1,axiom,
    ! [B: nat,K2: 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),K2)
       => ? [N5: 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),N5)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_6104_ex__power__ivl2,axiom,
    ! [B: nat,K2: 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))),K2)
       => ? [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N5)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_6105_mod__double__modulus,axiom,
    ! [M: code_integer,X: 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),X)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,X),M) )
          | ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,X),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,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6106_mod__double__modulus,axiom,
    ! [M: nat,X: 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),X)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,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,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6107_mod__double__modulus,axiom,
    ! [M: int,X: 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),X)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6108_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_6109_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_6110_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_6111_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_6112_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_6113_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_6114_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_6115_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% half_nonnegative_int_iff
tff(fact_6116_divmod__step__eq,axiom,
    ! [L: num,Q4: nat,R: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),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))),Q4)),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))),Q4)),R)) ).

% divmod_step_eq
tff(fact_6117_divmod__step__eq,axiom,
    ! [L: num,Q4: int,R: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),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))),Q4)),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))),Q4)),R)) ).

% divmod_step_eq
tff(fact_6118_divmod__step__eq,axiom,
    ! [L: num,Q4: code_integer,R: code_integer] :
      aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),produc1086072967326762835nteger(Q4,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))),Q4)),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))),Q4),R)) ).

% divmod_step_eq
tff(fact_6119_int__bit__induct,axiom,
    ! [Pa: fun_int_o,K2: int] :
      ( aa_int_o(Pa,zero_zero_int)
     => ( aa_int_o(Pa,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K: int] :
              ( aa_int_o(Pa,K)
             => ( ( K != zero_zero_int )
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K: int] :
                ( aa_int_o(Pa,K)
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(Pa,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,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(Pa,K2) ) ) ) ) ).

% int_bit_induct
tff(fact_6120_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,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))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,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(minus_minus_nat,N2),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6121_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6122_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6123_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( modulo8411746178871703098atural(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))),N2)) = times_2397367101498566445atural(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(minus_minus_nat,N2),M))),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6124_nat__bit__induct,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( ! [N5: nat] :
            ( aa_nat_o(Pa,N5)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
             => 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))),N5)) ) )
       => ( ! [N5: nat] :
              ( aa_nat_o(Pa,N5)
             => 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))),N5))) )
         => aa_nat_o(Pa,N2) ) ) ) ).

% nat_bit_induct
tff(fact_6125_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: 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),N2)) != 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))),N2) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_6126_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: 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),N2)) != 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))),N2) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_6127_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: 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),N2)) != zero_z2226904508553997617atural )
     => ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2) != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_right
tff(fact_6128_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: 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),N2)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_6129_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: 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),N2)) != 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_6130_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: 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),N2)) != 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_6131_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: 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),N2)) != 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_6132_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: 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),N2)) != 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_6133_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N2: 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))),N2)) = 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),N2))) ).

% div_exp_eq
tff(fact_6134_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: 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))),N2)) = 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),N2))) ).

% div_exp_eq
tff(fact_6135_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: 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))),N2)) = 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),N2))) ).

% div_exp_eq
tff(fact_6136_div__exp__eq,axiom,
    ! [A: code_natural,M: nat,N2: 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))),N2)) = 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),N2))) ).

% div_exp_eq
tff(fact_6137_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_6138_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_6139_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_6140_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_6141_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_6142_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: 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))),N2))),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,N2),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_6143_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: 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))),N2))),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,N2),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_6144_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: 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))),N2))),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,N2),M)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_6145_div__exp__mod__exp__eq,axiom,
    ! [A: code_natural,N2: nat,M: nat] : modulo8411746178871703098atural(divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)),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,N2),M))),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_6146_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_6147_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_6148_set__bit__0,axiom,
    ! [A: code_natural] : bit_se1617098188084679374atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),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_6149_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_6150_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N2)),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(N2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6151_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N2)),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,N2),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_6152_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se7083795435491715335atural(aa_nat_nat(suc,N2),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se7083795435491715335atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6153_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,N2)),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,N2),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_6154_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N2)),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(N2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6155_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N2)),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,N2),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_6156_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se168947363167071951atural(aa_nat_nat(suc,N2),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se168947363167071951atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6157_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,N2)),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,N2),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_6158_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N2)),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(N2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6159_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N2)),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,N2),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_6160_set__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se1617098188084679374atural(aa_nat_nat(suc,N2),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se1617098188084679374atural(N2,divide5121882707175180666atural(A,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6161_set__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,N2)),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,N2),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_6162_signed__take__bit__rec,axiom,
    ! [N2: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N2),A) = $ite(N2 = 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,N2),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_6163_signed__take__bit__rec,axiom,
    ! [N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),A) = $ite(N2 = 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,N2),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_6164_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_6165_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_6166_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% set_bit_nonnegative_int_iff
tff(fact_6167_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_6168_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N2),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_6169_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_6170_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,N2),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_6171_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N2)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_6172_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6173_signed__take__bit__add,axiom,
    ! [N2: nat,K2: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ).

% signed_take_bit_add
tff(fact_6174_unset__bit__less__eq,axiom,
    ! [N2: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,N2),K2)),K2) ).

% unset_bit_less_eq
tff(fact_6175_set__bit__greater__eq,axiom,
    ! [K2: int,N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K2)) ).

% set_bit_greater_eq
tff(fact_6176_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_6177_floor__le__round,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,X)) ).

% floor_le_round
tff(fact_6178_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% signed_take_bit_int_less_exp
tff(fact_6179_round__diff__minimal,axiom,
    ! [Z3: rat,M: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z3),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z3))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Z3),aa_int_rat(ring_1_of_int_rat,M)))) ).

% round_diff_minimal
tff(fact_6180_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_6181_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)),K2)
    <=> 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))),N2)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_6182_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6183_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),K2) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6184_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),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))),N2))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6185_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K2: 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))),N2)),K2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),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,N2)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_6186_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6187_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6188_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),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))),N2)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),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,N2)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_6189_round__def,axiom,
    ! [X: rat] : aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_6190_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N2)),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(N2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_6191_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,N2)),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,N2),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_6192_of__int__round__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6193_of__int__round__ge,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_ge
tff(fact_6194_of__int__round__gt,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_gt
tff(fact_6195_of__int__round__abs__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6196_round__unique_H,axiom,
    ! [X: rat,N2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_int_rat(ring_1_of_int_rat,N2)))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N2 ) ) ).

% round_unique'
tff(fact_6197_round__altdef,axiom,
    ! [X: rat] :
      aa_rat_int(archim7778729529865785530nd_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X),aa_rat_int(archim3151403230148437115or_rat,X)) ).

% round_altdef
tff(fact_6198_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),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,N2),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(8)
tff(fact_6199_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),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,N2)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(8)
tff(fact_6200_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(aa_num_num(bit1,N2)),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(8)
tff(fact_6201_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2),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,N2),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(7)
tff(fact_6202_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2),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,N2)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(7)
tff(fact_6203_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N2),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(aa_num_num(bit1,N2)),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2))))) ).

% divmod_algorithm_code(7)
tff(fact_6204_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_6205_concat__bit__Suc,axiom,
    ! [N2: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,N2)),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_6206_binomial__code,axiom,
    ! [N2: nat,K2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)),one_one_nat)),N2),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2))) ) ).

% binomial_code
tff(fact_6207_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_6208_binomial__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2) ) ).

% binomial_eq_0_iff
tff(fact_6209_binomial__Suc__Suc,axiom,
    ! [N2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_6210_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K2),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_6211_concat__bit__negative__iff,axiom,
    ! [N2: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K2),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_6212_zero__less__binomial__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2) ) ).

% zero_less_binomial_iff
tff(fact_6213_less__numeral__Suc,axiom,
    ! [K2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K2)),N2) ) ).

% less_numeral_Suc
tff(fact_6214_less__Suc__numeral,axiom,
    ! [N2: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_num_nat(pred_numeral,K2)) ) ).

% less_Suc_numeral
tff(fact_6215_le__numeral__Suc,axiom,
    ! [K2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),N2) ) ).

% le_numeral_Suc
tff(fact_6216_le__Suc__numeral,axiom,
    ! [N2: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_num_nat(pred_numeral,K2)) ) ).

% le_Suc_numeral
tff(fact_6217_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_6218_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_6219_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_6220_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6221_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N2)) = 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_6222_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N2)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_6223_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6224_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N2)) = 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_6225_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N2)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_6226_concat__bit__assoc,axiom,
    ! [N2: nat,K2: int,M: nat,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K2),aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),L),R)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K2),L)),R) ).

% concat_bit_assoc
tff(fact_6227_binomial__eq__0,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6228_binomial__symmetric,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)) ) ) ).

% binomial_symmetric
tff(fact_6229_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K2)),K2)),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_6230_binomial__le__pow,axiom,
    ! [R: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),R)) ) ).

% binomial_le_pow
tff(fact_6231_zero__less__binomial,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)) ) ).

% zero_less_binomial
tff(fact_6232_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_6233_choose__mult,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K2))) ) ) ) ).

% choose_mult
tff(fact_6234_binomial__fact__lemma,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,N2) ) ) ).

% binomial_fact_lemma
tff(fact_6235_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6236_choose__reduce__nat,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_6237_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_6238_binomial__le__pow2,axiom,
    ! [N2: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% binomial_le_pow2
tff(fact_6239_binomial__altdef__nat,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_6240_divmod__def,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(M,N2) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2))) ).

% divmod_def
tff(fact_6241_divmod__def,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(M,N2) = 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,N2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ).

% divmod_def
tff(fact_6242_divmod__def,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(M,N2) = 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,N2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))) ).

% divmod_def
tff(fact_6243_binomial__addition__formula,axiom,
    ! [N2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_6244_fact__binomial,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))) ) ) ).

% fact_binomial
tff(fact_6245_binomial__fact,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)))) ) ) ).

% binomial_fact
tff(fact_6246_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      unique5055182867167087721od_nat(M,N2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),unique5026877609467782581ep_nat(N2,unique5055182867167087721od_nat(M,aa_num_num(bit0,N2)))) ).

% divmod_divmod_step
tff(fact_6247_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      unique5052692396658037445od_int(M,N2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),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(N2),unique5052692396658037445od_int(M,aa_num_num(bit0,N2)))) ).

% divmod_divmod_step
tff(fact_6248_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      unique3479559517661332726nteger(M,N2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(N2),unique3479559517661332726nteger(M,aa_num_num(bit0,N2)))) ).

% divmod_divmod_step
tff(fact_6249_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_6250_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6251_even__succ__mod__exp,axiom,
    ! [A: nat,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6252_even__succ__mod__exp,axiom,
    ! [A: int,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6253_even__succ__mod__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_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),N2)
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6254_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: 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),N2)
       => ( 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))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_6255_even__succ__div__exp,axiom,
    ! [A: nat,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_6256_even__succ__div__exp,axiom,
    ! [A: int,N2: 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),N2)
       => ( 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))),N2)) = 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))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_6257_even__succ__div__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_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),N2)
       => ( 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))),N2)) = divide5121882707175180666atural(A,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_6258_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N2)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_io),unique5055182867167087721od_nat(M,N2)) ).

% divmod_algorithm_code(6)
tff(fact_6259_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N2)) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_ip),unique3479559517661332726nteger(M,N2)) ).

% divmod_algorithm_code(6)
tff(fact_6260_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_iq),unique5052692396658037445od_int(M,N2)) ).

% divmod_algorithm_code(6)
tff(fact_6261_mask__numeral,axiom,
    ! [N2: num] : aa_nat_Code_natural(bit_se943457434206027407atural,aa_num_nat(numeral_numeral_nat,N2)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),aa_nat_Code_natural(bit_se943457434206027407atural,aa_num_nat(pred_numeral,N2)))) ).

% mask_numeral
tff(fact_6262_mask__numeral,axiom,
    ! [N2: num] : aa_nat_Code_integer(bit_se2119862282449309892nteger,aa_num_nat(numeral_numeral_nat,N2)) = 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,N2)))) ).

% mask_numeral
tff(fact_6263_mask__numeral,axiom,
    ! [N2: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N2)) = 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,N2)))) ).

% mask_numeral
tff(fact_6264_mask__numeral,axiom,
    ! [N2: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N2)) = 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,N2)))) ).

% mask_numeral
tff(fact_6265_take__bit__rec,axiom,
    ! [N2: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),A) = $ite(N2 = 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,N2),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_6266_take__bit__rec,axiom,
    ! [N2: nat,A: code_natural] :
      bit_se569199155075624693atural(N2,A) = $ite(N2 = zero_zero_nat,zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_6267_take__bit__rec,axiom,
    ! [N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A) = $ite(N2 = 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,N2),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_6268_take__bit__rec,axiom,
    ! [N2: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A) = $ite(N2 = 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,N2),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_6269_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_6270_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_6271_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_6272_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_6273_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_6274_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_6275_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_6276_dvd__1__left,axiom,
    ! [K2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2) ).

% dvd_1_left
tff(fact_6277_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_6278_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% mask_nat_positive_iff
tff(fact_6279_case__prod__conv,axiom,
    ! [Fa: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Fa,A),B) ).

% case_prod_conv
tff(fact_6280_case__prod__conv,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Fa,A),B) ).

% case_prod_conv
tff(fact_6281_case__prod__conv,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,A),B) ).

% case_prod_conv
tff(fact_6282_case__prod__conv,axiom,
    ! [Fa: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(Fa,A),B) ) ).

% case_prod_conv
tff(fact_6283_case__prod__conv,axiom,
    ! [Fa: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(Fa,A),B) ).

% case_prod_conv
tff(fact_6284_curry__case__prod,axiom,
    ! [Fa: fun_na1668100129766719005at_nat] : produc4701859392678344816at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa)) = Fa ).

% curry_case_prod
tff(fact_6285_curry__case__prod,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o] : produc2062876521443793799_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa)) = Fa ).

% curry_case_prod
tff(fact_6286_curry__case__prod,axiom,
    ! [Fa: fun_in1696471567231710611nt_int] : produc8249235968001453780nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa)) = Fa ).

% curry_case_prod
tff(fact_6287_curry__case__prod,axiom,
    ! [Fa: fun_int_fun_int_o] : produc175634133007206835_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa)) = Fa ).

% curry_case_prod
tff(fact_6288_curry__case__prod,axiom,
    ! [Fa: fun_int_fun_int_int] : produc1016772743285680337nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa)) = Fa ).

% curry_case_prod
tff(fact_6289_case__prod__curry,axiom,
    ! [Fa: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,produc4701859392678344816at_nat(Fa)) = Fa ).

% case_prod_curry
tff(fact_6290_case__prod__curry,axiom,
    ! [Fa: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,produc2062876521443793799_nat_o(Fa)) = Fa ).

% case_prod_curry
tff(fact_6291_case__prod__curry,axiom,
    ! [Fa: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,produc8249235968001453780nt_int(Fa)) = Fa ).

% case_prod_curry
tff(fact_6292_case__prod__curry,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,produc175634133007206835_int_o(Fa)) = Fa ).

% case_prod_curry
tff(fact_6293_case__prod__curry,axiom,
    ! [Fa: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,produc1016772743285680337nt_int(Fa)) = Fa ).

% case_prod_curry
tff(fact_6294_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C2: 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,C2),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6295_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C2: 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,C2),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6296_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C2: 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,C2),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6297_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C2: 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,C2),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6298_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C2: 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,C2),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6299_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C2: 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,C2),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6300_div__add,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ) ).

% div_add
tff(fact_6301_div__add,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ) ).

% div_add
tff(fact_6302_div__add,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C2),A)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(C2),B)
       => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C2)),divide5121882707175180666atural(B,C2)) ) ) ) ).

% div_add
tff(fact_6303_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_6304_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_6305_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_6306_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_6307_odd__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))
    <=> ~ ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_6308_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_6309_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_6310_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_6311_even__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))
    <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_6312_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_6313_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_6314_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_6315_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_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(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_6316_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_6317_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_6318_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_6319_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W2: 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,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6320_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: 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,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6321_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: 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,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6322_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6323_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: 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,W2))),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,W2))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6324_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: 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,W2))),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,W2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6325_power__less__zero__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_6326_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),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))),N2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_6327_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),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))),N2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_6328_even__diff__nat,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
        | 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),N2)) ) ) ).

% even_diff_nat
tff(fact_6329_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_6330_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_6331_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_6332_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_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_6333_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_6334_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_6335_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_6336_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_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_6337_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_6338_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_6339_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_6340_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_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_6341_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W2: 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,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( 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,W2))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6342_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: 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,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( 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,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6343_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: 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,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( 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,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6344_even__power,axiom,
    ! [A: nat,N2: 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),N2))
    <=> ( 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),N2) ) ) ).

% even_power
tff(fact_6345_even__power,axiom,
    ! [A: int,N2: 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),N2))
    <=> ( 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),N2) ) ) ).

% even_power
tff(fact_6346_even__power,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_Code_natural(power_7079662738309270450atural(A),N2))
    <=> ( aa_Code_natural_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),N2) ) ) ).

% even_power
tff(fact_6347_even__power,axiom,
    ! [A: code_integer,N2: 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),N2))
    <=> ( 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),N2) ) ) ).

% even_power
tff(fact_6348_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N2)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ir),unique5055182867167087721od_nat(M,N2)) ).

% divmod_algorithm_code(5)
tff(fact_6349_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N2)) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_is),unique3479559517661332726nteger(M,N2)) ).

% divmod_algorithm_code(5)
tff(fact_6350_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_it),unique5052692396658037445od_int(M,N2)) ).

% divmod_algorithm_code(5)
tff(fact_6351_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_6352_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_6353_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_6354_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,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_6355_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W2: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W2))),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & 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,W2))
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6356_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: 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,W2))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & 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,W2))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6357_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: 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,W2))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & 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,W2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6358_case__prod__app,axiom,
    ! [Fa: fun_na1668100129766719005at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa),X),Y) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_iu(Fa),Y)),X) ).

% case_prod_app
tff(fact_6359_case__prod__app,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa),X),Y)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_iv(Fa),Y)),X) ) ).

% case_prod_app
tff(fact_6360_nested__case__prod__simp,axiom,
    ! [Fa: fun_na1668100129766719005at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa),X),Y) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_iu(Fa),Y)),X) ).

% nested_case_prod_simp
tff(fact_6361_nested__case__prod__simp,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa),X),Y)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_iv(Fa),Y)),X) ) ).

% nested_case_prod_simp
tff(fact_6362_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,Fa: 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,Fa),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_iw(Ha),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_6363_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_int,Fa: 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,Fa),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7323606162024495668nt_int(aTP_Lamm_ix(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6364_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_o,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun82482788810740534_int_o(aTP_Lamm_iy(Ha),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_6365_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun6119123525526402106nt_int(aTP_Lamm_iz(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6366_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr3238374608701176162_int_o,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun4545072666309863515_int_o(aTP_Lamm_ja(Ha),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_6367_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr982841511892225912nt_int,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7461304428735181397nt_int(aTP_Lamm_jb(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6368_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_8988054265482332260nt_int,Fa: 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,Fa),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4023131440832735197nt_int(aTP_Lamm_jc(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6369_prod_Ocase__distrib,axiom,
    ! [Ha: fun_in4211310926729683942nt_int,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun1034830265760468567nt_int(aTP_Lamm_jd(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6370_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr2478283820680522243nt_int,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4152872853588681340nt_int(aTP_Lamm_je(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6371_prod_Ocase__distrib,axiom,
    ! [Ha: fun_fu3275390434952296369_nat_o,Fa: 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,Fa),Prod)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun824319262034445606_nat_o(aTP_Lamm_jf(Ha),Fa)),Prod) ).

% prod.case_distrib
tff(fact_6372_take__bit__add,axiom,
    ! [N2: nat,A: int,B: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),B))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_6373_take__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_6374_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B: int,M: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_6375_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).

% take_bit_tightened
tff(fact_6376_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),Q4)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_6377_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_6378_dvd__add,axiom,
    ! [A: rat,B: rat,C2: 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),C2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2)) ) ) ).

% dvd_add
tff(fact_6379_dvd__add,axiom,
    ! [A: nat,B: nat,C2: 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),C2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ) ).

% dvd_add
tff(fact_6380_dvd__add,axiom,
    ! [A: int,B: int,C2: 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),C2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ) ).

% dvd_add
tff(fact_6381_dvd__add__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6382_dvd__add__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6383_dvd__add__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_6384_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C2: 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),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6385_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C2: 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),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6386_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C2: 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),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6387_old_Oprod_Ocase,axiom,
    ! [Fa: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Fa,X1),X2) ).

% old.prod.case
tff(fact_6388_old_Oprod_Ocase,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Fa,X1),X2) ).

% old.prod.case
tff(fact_6389_old_Oprod_Ocase,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,X1),X2) ).

% old.prod.case
tff(fact_6390_old_Oprod_Ocase,axiom,
    ! [Fa: fun_int_fun_int_o,X1: int,X2: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
    <=> aa_int_o(aa_int_fun_int_o(Fa,X1),X2) ) ).

% old.prod.case
tff(fact_6391_old_Oprod_Ocase,axiom,
    ! [Fa: fun_int_fun_int_int,X1: int,X2: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int_int(aa_int_fun_int_int(Fa,X1),X2) ).

% old.prod.case
tff(fact_6392_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,Fa: fun_na1668100129766719005at_nat,Ga: fun_na1668100129766719005at_nat,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) = Q4 )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Fa,X3),Y3) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Ga,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa),P2) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Ga),Q4) ) ) ) ).

% split_cong
tff(fact_6393_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,Fa: fun_na5914473681949881652_nat_o,Ga: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) = Q4 )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Fa,X3),Y3) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Ga,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa),P2) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Ga),Q4) ) ) ) ).

% split_cong
tff(fact_6394_split__cong,axiom,
    ! [Q4: product_prod_int_int,Fa: fun_in1696471567231710611nt_int,Ga: fun_in1696471567231710611nt_int,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,X3),Y3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Ga,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa),P2) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Ga),Q4) ) ) ) ).

% split_cong
tff(fact_6395_split__cong,axiom,
    ! [Q4: product_prod_int_int,Fa: fun_int_fun_int_o,Ga: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int_o(aa_int_fun_int_o(Fa,X3),Y3)
          <=> aa_int_o(aa_int_fun_int_o(Ga,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa),P2)
        <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Ga),Q4) ) ) ) ).

% split_cong
tff(fact_6396_split__cong,axiom,
    ! [Q4: product_prod_int_int,Fa: fun_int_fun_int_int,Ga: fun_int_fun_int_int,P2: product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) = Q4 )
         => ( aa_int_int(aa_int_fun_int_int(Fa,X3),Y3) = aa_int_int(aa_int_fun_int_int(Ga,X3),Y3) ) )
     => ( ( P2 = Q4 )
       => ( aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa),P2) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Ga),Q4) ) ) ) ).

% split_cong
tff(fact_6397_dvd__diff__nat,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),N2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% dvd_diff_nat
tff(fact_6398_uminus__dvd__conv_I1_J,axiom,
    ! [D2: int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),T6)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,D2)),T6) ) ).

% uminus_dvd_conv(1)
tff(fact_6399_uminus__dvd__conv_I2_J,axiom,
    ! [D2: int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),T6)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(uminus_uminus_int,T6)) ) ).

% uminus_dvd_conv(2)
tff(fact_6400_less__eq__mask,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(bit_se2002935070580805687sk_nat,N2)) ).

% less_eq_mask
tff(fact_6401_case__prod__Pair__iden,axiom,
    ! [P2: produc8799533983703829792et_nat] : aa_Pro5931322043027775230et_nat(produc1949718844479134567et_nat(produc6017135503889078682et_nat),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6402_case__prod__Pair__iden,axiom,
    ! [P2: produc3112576180687729086et_nat] : produc2255458887868536227et_nat(produc7291855603632349880et_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6403_case__prod__Pair__iden,axiom,
    ! [P2: produc7388388658123137530it_nat] : aa_Pro88785284265212210it_nat(produc7293086598567236123it_nat(produc4082563078715348724it_nat),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6404_case__prod__Pair__iden,axiom,
    ! [P2: produc3260487557148687353it_nat] : produc2183267410226956569it_nat(produc9178034014595674355it_nat,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6405_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,product_Pair_int_int),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6406_nat__take__bit__eq,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_int_nat(nat2,K2)) ) ) ).

% nat_take_bit_eq
tff(fact_6407_take__bit__nat__eq,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) ) ) ).

% take_bit_nat_eq
tff(fact_6408_case__prodE2,axiom,
    ! [Qa: fun_fu8391177546919232046_nat_o,Pa: fun_na1668100129766719005at_nat,Z3: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Qa,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Pa),Z3))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z3 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_fun3538214411680374423_nat_o(Qa,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6409_case__prodE2,axiom,
    ! [Qa: fun_fu6891447142836604759at_o_o,Pa: fun_na5914473681949881652_nat_o,Z3: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Qa,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Pa),Z3))
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z3 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_fun8408151826566693824at_o_o(Qa,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6410_case__prodE2,axiom,
    ! [Qa: fun_Pr3238374608701176162_int_o,Pa: fun_in1696471567231710611nt_int,Z3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Qa,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Pa),Z3))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z3 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(Qa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6411_case__prodE2,axiom,
    ! [Qa: fun_o_o,Pa: fun_int_fun_int_o,Z3: product_prod_int_int] :
      ( aa_o_o(Qa,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Pa),Z3))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z3 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_o_o(Qa,aa_int_o(aa_int_fun_int_o(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6412_case__prodE2,axiom,
    ! [Qa: fun_int_o,Pa: fun_int_fun_int_int,Z3: product_prod_int_int] :
      ( aa_int_o(Qa,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Pa),Z3))
     => ~ ! [X3: int,Y3: int] :
            ( ( Z3 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(Pa,X3),Y3)) ) ) ).

% case_prodE2
tff(fact_6413_case__prod__eta,axiom,
    ! [Fa: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_jg(Fa)) = Fa ).

% case_prod_eta
tff(fact_6414_case__prod__eta,axiom,
    ! [Fa: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_jh(Fa)) = Fa ).

% case_prod_eta
tff(fact_6415_case__prod__eta,axiom,
    ! [Fa: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_ji(Fa)) = Fa ).

% case_prod_eta
tff(fact_6416_case__prod__eta,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_jj(Fa)) = Fa ).

% case_prod_eta
tff(fact_6417_case__prod__eta,axiom,
    ! [Fa: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_jk(Fa)) = Fa ).

% case_prod_eta
tff(fact_6418_cond__case__prod__eta,axiom,
    ! [Fa: fun_na1668100129766719005at_nat,Ga: fun_Pr2477239776880934807at_nat] :
      ( ! [X3: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Fa,X3),Y3) = aa_Pro8023792847356852296at_nat(Ga,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3))
     => ( aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa) = Ga ) ) ).

% cond_case_prod_eta
tff(fact_6419_cond__case__prod__eta,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o,Ga: fun_Pr2028404973800837934_nat_o] :
      ( ! [X3: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Fa,X3),Y3) = aa_Pro989595756709598047_nat_o(Ga,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3))
     => ( aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa) = Ga ) ) ).

% cond_case_prod_eta
tff(fact_6420_cond__case__prod__eta,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,Ga: fun_Pr2478283820680522243nt_int] :
      ( ! [X3: int,Y3: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,X3),Y3) = aa_Pro6970504127699319548nt_int(Ga,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3))
     => ( aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa) = Ga ) ) ).

% cond_case_prod_eta
tff(fact_6421_cond__case__prod__eta,axiom,
    ! [Fa: fun_int_fun_int_o,Ga: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: int,Y3: int] :
          ( aa_int_o(aa_int_fun_int_o(Fa,X3),Y3)
        <=> aa_Pro4592068669816069595_int_o(Ga,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3)) )
     => ( aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa) = Ga ) ) ).

% cond_case_prod_eta
tff(fact_6422_cond__case__prod__eta,axiom,
    ! [Fa: fun_int_fun_int_int,Ga: fun_Pr982841511892225912nt_int] :
      ( ! [X3: int,Y3: int] : aa_int_int(aa_int_fun_int_int(Fa,X3),Y3) = aa_Pro6799258434465214121nt_int(Ga,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3))
     => ( aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa) = Ga ) ) ).

% cond_case_prod_eta
tff(fact_6423_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jl,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jl,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_6424_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_int_fun_int_o(aTP_Lamm_jm,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_jm,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_6425_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2) = aa_nat_int(bit_se2000444600071755411sk_int,N2) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_6426_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jl,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jl,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_6427_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,collect_int(aa_int_fun_int_o(aTP_Lamm_jm,A))),collect_int(aa_int_fun_int_o(aTP_Lamm_jm,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_6428_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2) = aa_nat_int(bit_se2000444600071755411sk_int,N2) )
    <=> 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))),N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_6429_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K2)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_6430_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_6431_take__bit__nonnegative,axiom,
    ! [N2: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) ).

% take_bit_nonnegative
tff(fact_6432_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_6433_not__take__bit__negative,axiom,
    ! [N2: nat,K2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_6434_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% pinf(9)
tff(fact_6435_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% pinf(9)
tff(fact_6436_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% pinf(9)
tff(fact_6437_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X5)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% pinf(10)
tff(fact_6438_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X5)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% pinf(10)
tff(fact_6439_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X5)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% pinf(10)
tff(fact_6440_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% minf(9)
tff(fact_6441_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% minf(9)
tff(fact_6442_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% minf(9)
tff(fact_6443_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),S2)) ) ) ).

% minf(10)
tff(fact_6444_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),S2)) ) ) ).

% minf(10)
tff(fact_6445_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),S2)) ) ) ).

% minf(10)
tff(fact_6446_signed__take__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_nat_fun_int_int(bit_ri631733984087533419it_int,M)),A) ).

% signed_take_bit_take_bit
tff(fact_6447_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6448_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6449_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C2),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C2)),divide5121882707175180666atural(B,C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6450_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6451_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),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)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6452_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C2),B)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),C2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,C2)),divide5121882707175180666atural(B,C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6453_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A))) ).

% take_bit_unset_bit_eq
tff(fact_6454_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A))) ).

% take_bit_unset_bit_eq
tff(fact_6455_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A))) ).

% take_bit_set_bit_eq
tff(fact_6456_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A))) ).

% take_bit_set_bit_eq
tff(fact_6457_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A))) ).

% take_bit_flip_bit_eq
tff(fact_6458_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A))) ).

% take_bit_flip_bit_eq
tff(fact_6459_nat__dvd__not__less,axiom,
    ! [M: nat,N2: 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),N2)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M) ) ) ).

% nat_dvd_not_less
tff(fact_6460_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_6461_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ).

% le_imp_power_dvd
tff(fact_6462_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_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),N2)) ) ).

% le_imp_power_dvd
tff(fact_6463_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_6464_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),M)),B) ) ) ).

% power_le_dvd
tff(fact_6465_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_6466_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),M)) ) ) ).

% dvd_power_le
tff(fact_6467_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2) ) ) ).

% dvd_minus_self
tff(fact_6468_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: 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),N2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N2)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N2),M)
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_6469_zdvd__not__zless,axiom,
    ! [M: int,N2: 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),N2)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N2),M) ) ) ).

% zdvd_not_zless
tff(fact_6470_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_6471_dvd__diffD1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),N2) ) ) ) ).

% dvd_diffD1
tff(fact_6472_dvd__diffD,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),M) ) ) ) ).

% dvd_diffD
tff(fact_6473_zdvd__mono,axiom,
    ! [K2: int,M: int,T6: int] :
      ( ( K2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),T6)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),T6)) ) ) ).

% zdvd_mono
tff(fact_6474_bezout__lemma__nat,axiom,
    ! [D2: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D2) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D2) ) )
         => ? [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y3)),D2) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_6475_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D: nat,X3: 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),X3) = 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ).

% bezout_add_nat
tff(fact_6476_zdvd__reduce,axiom,
    ! [K2: int,N2: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,N2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),M)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),N2) ) ).

% zdvd_reduce
tff(fact_6477_zdvd__period,axiom,
    ! [A: int,D2: int,X: int,T6: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T6))
      <=> 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,X),aa_int_int(aa_int_fun_int_int(times_times_int,C2),D2))),T6)) ) ) ).

% zdvd_period
tff(fact_6478_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,M)) ) ).

% fact_dvd
tff(fact_6479_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) ) ).

% fact_dvd
tff(fact_6480_uncurry__def,axiom,
    ! [Fa: fun_na1668100129766719005at_nat] : uncurr9212869592684144700at_nat(Fa) = aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Fa) ).

% uncurry_def
tff(fact_6481_uncurry__def,axiom,
    ! [Fa: fun_na5914473681949881652_nat_o] : uncurr8479020417594943315_nat_o(Fa) = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Fa) ).

% uncurry_def
tff(fact_6482_uncurry__def,axiom,
    ! [Fa: fun_in1696471567231710611nt_int] : uncurr7650761721940715016nt_int(Fa) = aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Fa) ).

% uncurry_def
tff(fact_6483_uncurry__def,axiom,
    ! [Fa: fun_int_fun_int_o] : uncurry_int_int_o(Fa) = aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa) ).

% uncurry_def
tff(fact_6484_uncurry__def,axiom,
    ! [Fa: fun_int_fun_int_int] : uncurry_int_int_int(Fa) = aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Fa) ).

% uncurry_def
tff(fact_6485_internal__case__prod__def,axiom,
    produc8104577556401574439at_nat = produc4528391901023682385at_nat ).

% internal_case_prod_def
tff(fact_6486_internal__case__prod__def,axiom,
    produc9189008374811844670_nat_o = produc6395135316500328296_nat_o ).

% internal_case_prod_def
tff(fact_6487_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
tff(fact_6488_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
tff(fact_6489_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
tff(fact_6490_mask__nonnegative__int,axiom,
    ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N2)) ).

% mask_nonnegative_int
tff(fact_6491_not__mask__negative__int,axiom,
    ! [N2: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N2)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_6492_nat__dvd__iff,axiom,
    ! [Z3: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z3)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z3),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_6493_take__bit__signed__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_6494_unity__coeff__ex,axiom,
    ! [Pa: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_6495_unity__coeff__ex,axiom,
    ! [Pa: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat))
          & aa_rat_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_6496_unity__coeff__ex,axiom,
    ! [Pa: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_6497_unity__coeff__ex,axiom,
    ! [Pa: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),zero_zero_int))
          & aa_int_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_6498_inf__period_I3_J,axiom,
    ! [D2: rat,D4: rat,T6: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D4)
     => ! [X5: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),T6))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))),T6)) ) ) ).

% inf_period(3)
tff(fact_6499_inf__period_I3_J,axiom,
    ! [D2: int,D4: int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D4)
     => ! [X5: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T6))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),T6)) ) ) ).

% inf_period(3)
tff(fact_6500_inf__period_I4_J,axiom,
    ! [D2: rat,D4: rat,T6: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D4)
     => ! [X5: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X5),T6))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X5),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))),T6)) ) ) ).

% inf_period(4)
tff(fact_6501_inf__period_I4_J,axiom,
    ! [D2: int,D4: int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D4)
     => ! [X5: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T6))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),T6)) ) ) ).

% inf_period(4)
tff(fact_6502_dvd__imp__le,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2) ) ) ).

% dvd_imp_le
tff(fact_6503_dvd__mult__cancel,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2) ) ) ).

% dvd_mult_cancel
tff(fact_6504_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_6505_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D: nat,X3: 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),X3) = 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) ) ) ) ).

% bezout_add_strong_nat
tff(fact_6506_zdvd__imp__le,axiom,
    ! [Z3: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z3),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),N2) ) ) ).

% zdvd_imp_le
tff(fact_6507_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N2: 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,K2)),aa_nat_rat(semiri773545260158071498ct_rat,N2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2))) ).

% fact_fact_dvd_fact
tff(fact_6508_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N2: 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,K2)),aa_nat_int(semiri1406184849735516958ct_int,N2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2))) ).

% fact_fact_dvd_fact
tff(fact_6509_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N2: 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,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N2))) ).

% fact_fact_dvd_fact
tff(fact_6510_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_6511_dvd__imp__le__int,axiom,
    ! [I2: int,D2: int] :
      ( ( I2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),I2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D2)),aa_int_int(abs_abs_int,I2)) ) ) ).

% dvd_imp_le_int
tff(fact_6512_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_6513_dvd__fact,axiom,
    ! [M: nat,N2: 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),N2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ) ).

% dvd_fact
tff(fact_6514_less__mask,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(bit_se2002935070580805687sk_nat,N2)) ) ).

% less_mask
tff(fact_6515_even__nat__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ) ).

% even_nat_iff
tff(fact_6516_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_jn),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_6517_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_6518_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_6519_odd__even__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),B)
       => aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% odd_even_add
tff(fact_6520_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_6521_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N2: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6522_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: nat] :
      ( ( X != 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,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6523_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: nat] :
      ( ( X != 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,X),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6524_dvd__power,axiom,
    ! [N2: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) ) ).

% dvd_power
tff(fact_6525_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N2)) ) ).

% dvd_power
tff(fact_6526_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)) ) ).

% dvd_power
tff(fact_6527_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)) ) ).

% dvd_power
tff(fact_6528_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: 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),N2)),M)
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_6529_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: 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,N2),M)),M)
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_6530_choose__dvd,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => 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,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)))),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ) ).

% choose_dvd
tff(fact_6531_choose__dvd,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => 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,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)))),aa_nat_int(semiri1406184849735516958ct_int,N2)) ) ).

% choose_dvd
tff(fact_6532_choose__dvd,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => 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,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2)))),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ) ).

% choose_dvd
tff(fact_6533_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_6534_dvd__minus__add,axiom,
    ! [Q4: nat,N2: nat,R: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),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,N2),Q4))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q4))) ) ) ) ).

% dvd_minus_add
tff(fact_6535_mod__nat__eqI,axiom,
    ! [R: nat,N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_6536_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_6537_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% power_dvd_imp_le
tff(fact_6538_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_6539_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% take_bit_nat_less_exp
tff(fact_6540_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_6541_take__bit__int__less__exp,axiom,
    ! [N2: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% take_bit_int_less_exp
tff(fact_6542_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ).

% power_mono_odd
tff(fact_6543_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ).

% power_mono_odd
tff(fact_6544_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ).

% odd_pos
tff(fact_6545_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_diff_iff
tff(fact_6546_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_add_abs_iff
tff(fact_6547_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K2)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_abs_add_iff
tff(fact_6548_dvd__power__iff__le,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% dvd_power_iff_le
tff(fact_6549_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_6550_mask__nat__less__exp,axiom,
    ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% mask_nat_less_exp
tff(fact_6551_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_6552_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)),K2)
    <=> 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))),N2)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_6553_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)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_6554_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)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_6555_oddE,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_natural] : A != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),B3)),one_one_Code_natural) ) ).

% oddE
tff(fact_6556_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)
     => ~ ! [B3: 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))),B3)),one_one_Code_integer) ) ).

% oddE
tff(fact_6557_zero__le__power__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_le_power_eq
tff(fact_6558_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_6559_zero__le__power__eq,axiom,
    ! [A: int,N2: 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),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_6560_zero__le__odd__power,axiom,
    ! [N2: 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))),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ) ).

% zero_le_odd_power
tff(fact_6561_zero__le__odd__power,axiom,
    ! [N2: 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))),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_6562_zero__le__odd__power,axiom,
    ! [N2: 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))),N2)
     => ( 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),N2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_6563_zero__le__even__power,axiom,
    ! [N2: 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))),N2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ) ).

% zero_le_even_power
tff(fact_6564_zero__le__even__power,axiom,
    ! [N2: 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))),N2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)) ) ).

% zero_le_even_power
tff(fact_6565_zero__le__even__power,axiom,
    ! [N2: 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))),N2)
     => 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),N2)) ) ).

% zero_le_even_power
tff(fact_6566_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2)) ) ) ).

% power_mono_even
tff(fact_6567_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)) ) ) ).

% power_mono_even
tff(fact_6568_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),N2)) ) ) ).

% power_mono_even
tff(fact_6569_take__bit__int__eq__self,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_6570_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_6571_take__bit__incr__eq,axiom,
    ! [N2: nat,K2: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2) != 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))),N2)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_6572_zero__less__power__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2))
    <=> ( ( N2 = 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))),N2)
          & ( 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))),N2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ) ) ).

% zero_less_power_eq
tff(fact_6573_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))
    <=> ( ( N2 = 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))),N2)
          & ( 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))),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_6574_zero__less__power__eq,axiom,
    ! [A: int,N2: 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),N2))
    <=> ( ( N2 = 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))),N2)
          & ( 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))),N2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_6575_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] : bit_se569199155075624693atural(aa_nat_nat(suc,N2),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,K2))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(bit_se569199155075624693atural(N2,aa_num_Code_natural(numera5444537566228673987atural,K2)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),one_one_Code_natural) ).

% take_bit_Suc_bit1
tff(fact_6576_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N2)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),aa_num_Code_integer(numera6620942414471956472nteger,K2))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ).

% take_bit_Suc_bit1
tff(fact_6577_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_6578_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_6579_take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N2)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),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_6580_take__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] : bit_se569199155075624693atural(aa_nat_nat(suc,N2),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(bit_se569199155075624693atural(N2,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_6581_take__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),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,N2),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_6582_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N2)),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,N2),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_6583_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jo(L)),Qr) ).

% divmod_step_nat_def
tff(fact_6584_take__bit__int__less__eq,axiom,
    ! [N2: nat,K2: 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))),N2)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ) ).

% take_bit_int_less_eq
tff(fact_6585_take__bit__int__greater__eq,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),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,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_6586_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_6587_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_6588_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_6589_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_natural_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))),N2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% even_mask_div_iff'
tff(fact_6590_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N2: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_6591_power__le__zero__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & 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))),N2)
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6592_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & 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))),N2)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6593_power__le__zero__eq,axiom,
    ! [A: int,N2: 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),N2)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & 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))),N2)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6594_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),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,N2),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_6595_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,L),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,K2))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(bit_se569199155075624693atural(aa_num_nat(pred_numeral,L),aa_num_Code_natural(numera5444537566228673987atural,K2)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)))),one_one_Code_natural) ).

% take_bit_numeral_bit1
tff(fact_6596_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K2))) = 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,K2))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ).

% take_bit_numeral_bit1
tff(fact_6597_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_6598_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_6599_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_jp(L)),Qr) ).

% divmod_step_int_def
tff(fact_6600_take__bit__minus__small__eq,axiom,
    ! [K2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(uminus_uminus_int,K2)) = 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))),N2)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_6601_even__mask__div__iff,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_6602_even__mask__div__iff,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_6603_even__mask__div__iff,axiom,
    ! [M: nat,N2: 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))),N2)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_6604_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_Code_natural_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))),N2)))
    <=> ( ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2) = zero_z2226904508553997617atural )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ) ).

% even_mask_div_iff
tff(fact_6605_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),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,N2),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_6606_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K2: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_6607_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N2: 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))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & 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,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6608_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: 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))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & 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,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6609_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: 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))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & 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,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6610_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),divide5121882707175180666atural(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))),N2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        | ( aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2) = zero_z2226904508553997617atural )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & aa_Code_natural_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,N2),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6611_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jo(L)),Qr) ).

% divmod_step_def
tff(fact_6612_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_jp(L)),Qr) ).

% divmod_step_def
tff(fact_6613_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_jq(L)),Qr) ).

% divmod_step_def
tff(fact_6614_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: 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,N2))) = aa_option_num_int(case_option_int_num(zero_zero_int,aTP_Lamm_jr(M)),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_6615_divmod__nat__if,axiom,
    ! [M: nat,N2: nat] :
      divmod_nat(M,N2) = $ite(
        ( ( N2 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_js),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2),N2)) ) ).

% divmod_nat_if
tff(fact_6616_flip__bit__0,axiom,
    ! [A: code_natural] : bit_se168947363167071951atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))),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_6617_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_6618_flip__bit__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(zero_zero_nat),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))),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_6619_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_6620_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_6621_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_6622_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K2 = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_6623_case__prodI2,axiom,
    ! [P2: produc8799533983703829792et_nat,C2: fun_fu8419763957888848259_nat_o] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
          ( ( P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A3),B3) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(C2,A3),B3) )
     => aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(C2),P2) ) ).

% case_prodI2
tff(fact_6624_case__prodI2,axiom,
    ! [P2: produc3112576180687729086et_nat,C2: fun_fu2611741329380808165_nat_o] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
          ( ( P2 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A3),B3) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(C2,A3),B3) )
     => aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(C2),P2) ) ).

% case_prodI2
tff(fact_6625_case__prodI2,axiom,
    ! [P2: produc7388388658123137530it_nat,C2: fun_b_540864168722188329_nat_o] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3) )
         => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(C2,A3),B3) )
     => aa_Pro1537530206668979808_nat_o(produc3971132677695353399_nat_o(C2),P2) ) ).

% case_prodI2
tff(fact_6626_case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,C2: fun_a_2467521563176467050_nat_o] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3) )
         => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(C2,A3),B3) )
     => aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(C2),P2) ) ).

% case_prodI2
tff(fact_6627_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C2: fun_int_fun_int_o] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(C2,A3),B3) )
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C2),P2) ) ).

% case_prodI2
tff(fact_6628_case__prodI,axiom,
    ! [Fa: fun_fu8419763957888848259_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Fa,A),B)
     => aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(Fa),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B)) ) ).

% case_prodI
tff(fact_6629_case__prodI,axiom,
    ! [Fa: fun_fu2611741329380808165_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Fa,A),B)
     => aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(Fa),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B)) ) ).

% case_prodI
tff(fact_6630_case__prodI,axiom,
    ! [Fa: fun_b_540864168722188329_nat_o,A: b,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Fa,A),B)
     => aa_Pro1537530206668979808_nat_o(produc3971132677695353399_nat_o(Fa),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B)) ) ).

% case_prodI
tff(fact_6631_case__prodI,axiom,
    ! [Fa: fun_a_2467521563176467050_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Fa,A),B)
     => aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(Fa),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B)) ) ).

% case_prodI
tff(fact_6632_case__prodI,axiom,
    ! [Fa: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(Fa,A),B)
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_6633_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z3: $o,C2: fun_in4068701459849370962_set_o] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_o_o(member_o((Z3)),aa_int_set_o(aa_int_fun_int_set_o(C2,A3),B3)) )
     => aa_set_o_o(member_o((Z3)),produc4257766111578684402_set_o(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6634_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z3: nat,C2: fun_in2630332840903238488et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_nat_o(member_nat(Z3),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,A3),B3)) )
     => aa_set_nat_o(member_nat(Z3),produc4251311855443802252et_nat(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6635_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z3: int,C2: fun_in7040156458970095540et_int] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_int_o(member_int(Z3),aa_int_set_int(aa_int8639817059373041269et_int(C2,A3),B3)) )
     => aa_set_int_o(member_int(Z3),produc73460835934605544et_int(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6636_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z3: set_nat,C2: fun_in7207726656141197838et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_set_nat_o(member_set_nat(Z3),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,A3),B3)) )
     => aa_set_set_nat_o(member_set_nat(Z3),produc5233655623923918146et_nat(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6637_mem__case__prodI2,axiom,
    ! [P2: product_prod_int_int,Z3: set_Pr958786334691620121nt_int,C2: fun_in217785400498748499nt_int] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z3),aa_int6638694790789732239nt_int(aa_int76085875543787452nt_int(C2,A3),B3)) )
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z3),produc7771776839612048883nt_int(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6638_mem__case__prodI2,axiom,
    ! [P2: produc7388388658123137530it_nat,Z3: $o,C2: fun_b_9192071537261429513_set_o] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3) )
         => aa_set_o_o(member_o((Z3)),aa_Pro1465782317883297390_set_o(aa_b_f1386041640509420034_set_o(C2,A3),B3)) )
     => aa_set_o_o(member_o((Z3)),produc8188766211626073367_set_o(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6639_mem__case__prodI2,axiom,
    ! [P2: produc7388388658123137530it_nat,Z3: nat,C2: fun_b_7146180689985673569et_nat] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3) )
         => aa_set_nat_o(member_nat(Z3),aa_Pro6139939891222248720et_nat(aa_b_f6129190992908776978et_nat(C2,A3),B3)) )
     => aa_set_nat_o(member_nat(Z3),produc8058763169506871975et_nat(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6640_mem__case__prodI2,axiom,
    ! [P2: produc7388388658123137530it_nat,Z3: int,C2: fun_b_2332632271197754813et_int] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A3),B3) )
         => aa_set_int_o(member_int(Z3),aa_Pro1962088871713052012et_int(aa_b_f3639107847977050222et_int(C2,A3),B3)) )
     => aa_set_int_o(member_int(Z3),produc3880912149997675267et_int(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6641_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z3: $o,C2: fun_a_6228866161775505738_set_o] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3) )
         => aa_set_o_o(member_o((Z3)),aa_Pro1465782317883297390_set_o(aa_a_f7646208301878272067_set_o(C2,A3),B3)) )
     => aa_set_o_o(member_o((Z3)),produc5350075035600711000_set_o(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6642_mem__case__prodI2,axiom,
    ! [P2: produc3260487557148687353it_nat,Z3: nat,C2: fun_a_8173838432492502368et_nat] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A3),B3) )
         => aa_set_nat_o(member_nat(Z3),aa_Pro6139939891222248720et_nat(aa_a_f7156848735415605777et_nat(C2,A3),B3)) )
     => aa_set_nat_o(member_nat(Z3),produc7315013382793309350et_nat(C2,P2)) ) ).

% mem_case_prodI2
tff(fact_6643_mem__case__prodI,axiom,
    ! [Z3: $o,C2: fun_in4068701459849370962_set_o,A: int,B: int] :
      ( aa_set_o_o(member_o((Z3)),aa_int_set_o(aa_int_fun_int_set_o(C2,A),B))
     => aa_set_o_o(member_o((Z3)),produc4257766111578684402_set_o(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6644_mem__case__prodI,axiom,
    ! [Z3: nat,C2: fun_in2630332840903238488et_nat,A: int,B: int] :
      ( aa_set_nat_o(member_nat(Z3),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,A),B))
     => aa_set_nat_o(member_nat(Z3),produc4251311855443802252et_nat(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6645_mem__case__prodI,axiom,
    ! [Z3: int,C2: fun_in7040156458970095540et_int,A: int,B: int] :
      ( aa_set_int_o(member_int(Z3),aa_int_set_int(aa_int8639817059373041269et_int(C2,A),B))
     => aa_set_int_o(member_int(Z3),produc73460835934605544et_int(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6646_mem__case__prodI,axiom,
    ! [Z3: set_nat,C2: fun_in7207726656141197838et_nat,A: int,B: int] :
      ( aa_set_set_nat_o(member_set_nat(Z3),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,A),B))
     => aa_set_set_nat_o(member_set_nat(Z3),produc5233655623923918146et_nat(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6647_mem__case__prodI,axiom,
    ! [Z3: set_Pr958786334691620121nt_int,C2: fun_in217785400498748499nt_int,A: int,B: int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z3),aa_int6638694790789732239nt_int(aa_int76085875543787452nt_int(C2,A),B))
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z3),produc7771776839612048883nt_int(C2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_6648_mem__case__prodI,axiom,
    ! [Z3: $o,C2: fun_b_9192071537261429513_set_o,A: b,B: produc6653097349344004940it_nat] :
      ( aa_set_o_o(member_o((Z3)),aa_Pro1465782317883297390_set_o(aa_b_f1386041640509420034_set_o(C2,A),B))
     => aa_set_o_o(member_o((Z3)),produc8188766211626073367_set_o(C2,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_6649_mem__case__prodI,axiom,
    ! [Z3: nat,C2: fun_b_7146180689985673569et_nat,A: b,B: produc6653097349344004940it_nat] :
      ( aa_set_nat_o(member_nat(Z3),aa_Pro6139939891222248720et_nat(aa_b_f6129190992908776978et_nat(C2,A),B))
     => aa_set_nat_o(member_nat(Z3),produc8058763169506871975et_nat(C2,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_6650_mem__case__prodI,axiom,
    ! [Z3: int,C2: fun_b_2332632271197754813et_int,A: b,B: produc6653097349344004940it_nat] :
      ( aa_set_int_o(member_int(Z3),aa_Pro1962088871713052012et_int(aa_b_f3639107847977050222et_int(C2,A),B))
     => aa_set_int_o(member_int(Z3),produc3880912149997675267et_int(C2,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_6651_mem__case__prodI,axiom,
    ! [Z3: $o,C2: fun_a_6228866161775505738_set_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_set_o_o(member_o((Z3)),aa_Pro1465782317883297390_set_o(aa_a_f7646208301878272067_set_o(C2,A),B))
     => aa_set_o_o(member_o((Z3)),produc5350075035600711000_set_o(C2,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_6652_mem__case__prodI,axiom,
    ! [Z3: nat,C2: fun_a_8173838432492502368et_nat,A: a,B: produc6653097349344004940it_nat] :
      ( aa_set_nat_o(member_nat(Z3),aa_Pro6139939891222248720et_nat(aa_a_f7156848735415605777et_nat(C2,A),B))
     => aa_set_nat_o(member_nat(Z3),produc7315013382793309350et_nat(C2,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_6653_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C2: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C2,A3),B3),X) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C2),P2),X) ) ).

% case_prodI2'
tff(fact_6654_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),aa_o_rat(zero_n2052037380579107095ol_rat,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_6655_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $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,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_6656_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $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,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_6657_of__bool__less__eq__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),aa_o_Code_integer(zero_n356916108424825756nteger,(Qa)))
    <=> ( (Pa)
       => (Qa) ) ) ).

% of_bool_less_eq_iff
tff(fact_6658_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),aa_o_rat(zero_n2052037380579107095ol_rat,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_6659_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $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,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_6660_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $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,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_6661_of__bool__less__iff,axiom,
    ! [Pa: $o,Qa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),aa_o_Code_integer(zero_n356916108424825756nteger,(Qa)))
    <=> ( ~ (Pa)
        & (Qa) ) ) ).

% of_bool_less_iff
tff(fact_6662_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_6663_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_6664_of__nat__of__bool,axiom,
    ! [Pa: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(Pa))) = aa_o_Code_integer(zero_n356916108424825756nteger,(Pa)) ).

% of_nat_of_bool
tff(fact_6665_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_6666_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_6667_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_6668_zero__less__of__bool__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(Pa)))
    <=> (Pa) ) ).

% zero_less_of_bool_iff
tff(fact_6669_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),one_one_rat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_6670_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_6671_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_6672_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),one_one_Code_integer)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_6673_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_6674_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_6675_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N2)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_jt),aa_num_option_num(bit_take_bit_num(N2),M)) ).

% take_bit_num_simps(3)
tff(fact_6676_take__bit__of__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),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),N2)) ).

% take_bit_of_Suc_0
tff(fact_6677_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N2)),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(N2),M))) ).

% take_bit_num_simps(4)
tff(fact_6678_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_jt),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,R)),M)) ).

% take_bit_num_simps(6)
tff(fact_6679_take__bit__of__1,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),one_one_Code_integer) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_1
tff(fact_6680_take__bit__of__1,axiom,
    ! [N2: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_1
tff(fact_6681_take__bit__of__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_1
tff(fact_6682_sgn__of__nat,axiom,
    ! [N2: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)) = aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% sgn_of_nat
tff(fact_6683_sgn__of__nat,axiom,
    ! [N2: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% sgn_of_nat
tff(fact_6684_sgn__of__nat,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% sgn_of_nat
tff(fact_6685_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_6686_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] : bit_se569199155075624693atural(aa_num_nat(numeral_numeral_nat,M),aa_num_Code_natural(numera5444537566228673987atural,N2)) = case_o5621594795226839503al_num(zero_z2226904508553997617atural,numera5444537566228673987atural,aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_numeral
tff(fact_6687_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,M)),aa_num_Code_integer(numera6620942414471956472nteger,N2)) = case_o356765784539232260er_num(zero_z3403309356797280102nteger,numera6620942414471956472nteger,aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N2)) ).

% take_bit_numeral_numeral
tff(fact_6688_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: 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,N2)) = 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)),N2)) ).

% take_bit_numeral_numeral
tff(fact_6689_take__bit__numeral__numeral,axiom,
    ! [M: num,N2: 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,N2)) = 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)),N2)) ).

% take_bit_numeral_numeral
tff(fact_6690_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2773287842338716102atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% and_numerals(7)
tff(fact_6691_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = 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,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% and_numerals(7)
tff(fact_6692_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_6693_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_6694_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : bit_se569199155075624693atural(M,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_6695_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_6696_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_6697_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% take_bit_of_exp
tff(fact_6698_take__bit__of__2,axiom,
    ! [N2: nat] : bit_se569199155075624693atural(N2,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6699_take__bit__of__2,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6700_take__bit__of__2,axiom,
    ! [N2: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6701_take__bit__of__2,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6702_one__mod__2__pow__eq,axiom,
    ! [N2: nat] : modulo8411746178871703098atural(one_one_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2)) = aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_6703_one__mod__2__pow__eq,axiom,
    ! [N2: 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))),N2)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_6704_one__mod__2__pow__eq,axiom,
    ! [N2: 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))),N2)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_6705_one__mod__2__pow__eq,axiom,
    ! [N2: 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))),N2)) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)) ).

% one_mod_2_pow_eq
tff(fact_6706_same__fst__def,axiom,
    ! [Pa: fun_int_o,Ra: fun_in7209865251430457670nt_int] : same_fst_int_int(Pa,Ra) = collec7447955092554649554nt_int(produc1676490119946744748_int_o(produc66813930598470504_int_o(aa_fun8358116108987704894_int_o(aTP_Lamm_jv(Pa),Ra)))) ).

% same_fst_def
tff(fact_6707_same__fst__def,axiom,
    ! [Pa: fun_nat_o,Ra: fun_na7044673412300895210at_nat] : same_fst_nat_nat(Pa,Ra) = collec7088162979684241874at_nat(produc6590410687421337004_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun1888629816020924642_nat_o(aTP_Lamm_jx(Pa),Ra)))) ).

% same_fst_def
tff(fact_6708_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M)
       => ( M = N2 ) ) ) ).

% dvd_antisym
tff(fact_6709_lex__prod__def,axiom,
    ! [Ra2: set_Pr958786334691620121nt_int,Rb: set_Pr958786334691620121nt_int] : lex_prod_int_int(Ra2,Rb) = collec7447955092554649554nt_int(produc1676490119946744748_int_o(produc66813930598470504_int_o(aa_set7124137317651323025_int_o(aTP_Lamm_jz(Ra2),Rb)))) ).

% lex_prod_def
tff(fact_6710_lex__prod__def,axiom,
    ! [Ra2: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] : lex_prod_nat_nat(Ra2,Rb) = collec7088162979684241874at_nat(produc6590410687421337004_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_set1501693898512841689_nat_o(aTP_Lamm_kb(Ra2),Rb)))) ).

% lex_prod_def
tff(fact_6711_mem__case__prodE,axiom,
    ! [Z3: $o,C2: fun_in4068701459849370962_set_o,P2: product_prod_int_int] :
      ( aa_set_o_o(member_o((Z3)),produc4257766111578684402_set_o(C2,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_o_o(member_o((Z3)),aa_int_set_o(aa_int_fun_int_set_o(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6712_mem__case__prodE,axiom,
    ! [Z3: nat,C2: fun_in2630332840903238488et_nat,P2: product_prod_int_int] :
      ( aa_set_nat_o(member_nat(Z3),produc4251311855443802252et_nat(C2,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z3),aa_int_set_nat(aa_int1906528167449992217et_nat(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6713_mem__case__prodE,axiom,
    ! [Z3: int,C2: fun_in7040156458970095540et_int,P2: product_prod_int_int] :
      ( aa_set_int_o(member_int(Z3),produc73460835934605544et_int(C2,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_int_o(member_int(Z3),aa_int_set_int(aa_int8639817059373041269et_int(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6714_mem__case__prodE,axiom,
    ! [Z3: set_nat,C2: fun_in7207726656141197838et_nat,P2: product_prod_int_int] :
      ( aa_set_set_nat_o(member_set_nat(Z3),produc5233655623923918146et_nat(C2,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set_set_nat_o(member_set_nat(Z3),aa_int_set_set_nat(aa_int1843896668418082255et_nat(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6715_mem__case__prodE,axiom,
    ! [Z3: set_Pr958786334691620121nt_int,C2: fun_in217785400498748499nt_int,P2: product_prod_int_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z3),produc7771776839612048883nt_int(C2,P2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Z3),aa_int6638694790789732239nt_int(aa_int76085875543787452nt_int(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6716_mem__case__prodE,axiom,
    ! [Z3: $o,C2: fun_b_9192071537261429513_set_o,P2: produc7388388658123137530it_nat] :
      ( aa_set_o_o(member_o((Z3)),produc8188766211626073367_set_o(C2,P2))
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3) )
           => ~ aa_set_o_o(member_o((Z3)),aa_Pro1465782317883297390_set_o(aa_b_f1386041640509420034_set_o(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6717_mem__case__prodE,axiom,
    ! [Z3: nat,C2: fun_b_7146180689985673569et_nat,P2: produc7388388658123137530it_nat] :
      ( aa_set_nat_o(member_nat(Z3),produc8058763169506871975et_nat(C2,P2))
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z3),aa_Pro6139939891222248720et_nat(aa_b_f6129190992908776978et_nat(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6718_mem__case__prodE,axiom,
    ! [Z3: int,C2: fun_b_2332632271197754813et_int,P2: produc7388388658123137530it_nat] :
      ( aa_set_int_o(member_int(Z3),produc3880912149997675267et_int(C2,P2))
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3) )
           => ~ aa_set_int_o(member_int(Z3),aa_Pro1962088871713052012et_int(aa_b_f3639107847977050222et_int(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6719_mem__case__prodE,axiom,
    ! [Z3: $o,C2: fun_a_6228866161775505738_set_o,P2: produc3260487557148687353it_nat] :
      ( aa_set_o_o(member_o((Z3)),produc5350075035600711000_set_o(C2,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_set_o_o(member_o((Z3)),aa_Pro1465782317883297390_set_o(aa_a_f7646208301878272067_set_o(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6720_mem__case__prodE,axiom,
    ! [Z3: nat,C2: fun_a_8173838432492502368et_nat,P2: produc3260487557148687353it_nat] :
      ( aa_set_nat_o(member_nat(Z3),produc7315013382793309350et_nat(C2,P2))
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_set_nat_o(member_nat(Z3),aa_Pro6139939891222248720et_nat(aa_a_f7156848735415605777et_nat(C2,X3),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6721_case__prodE,axiom,
    ! [C2: fun_fu8419763957888848259_nat_o,P2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(C2),P2)
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X3),Y3) )
           => ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(C2,X3),Y3) ) ) ).

% case_prodE
tff(fact_6722_case__prodE,axiom,
    ! [C2: fun_fu2611741329380808165_nat_o,P2: produc3112576180687729086et_nat] :
      ( aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(C2),P2)
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] :
            ( ( P2 = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X3),Y3) )
           => ~ aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(C2,X3),Y3) ) ) ).

% case_prodE
tff(fact_6723_case__prodE,axiom,
    ! [C2: fun_b_540864168722188329_nat_o,P2: produc7388388658123137530it_nat] :
      ( aa_Pro1537530206668979808_nat_o(produc3971132677695353399_nat_o(C2),P2)
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X3),Y3) )
           => ~ aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(C2,X3),Y3) ) ) ).

% case_prodE
tff(fact_6724_case__prodE,axiom,
    ! [C2: fun_a_2467521563176467050_nat_o,P2: produc3260487557148687353it_nat] :
      ( aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(C2),P2)
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P2 = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X3),Y3) )
           => ~ aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(C2,X3),Y3) ) ) ).

% case_prodE
tff(fact_6725_case__prodE,axiom,
    ! [C2: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C2),P2)
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y3) )
           => ~ aa_int_o(aa_int_fun_int_o(C2,X3),Y3) ) ) ).

% case_prodE
tff(fact_6726_case__prodD,axiom,
    ! [Fa: fun_fu8419763957888848259_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro5643009357358181178_nat_o(produc2915205735004773393_nat_o(Fa),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,A),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(Fa,A),B) ) ).

% case_prodD
tff(fact_6727_case__prodD,axiom,
    ! [Fa: fun_fu2611741329380808165_nat_o,A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro2671238023309943196_nat_o(produc4465912260561112435_nat_o(Fa),aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,A),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(Fa,A),B) ) ).

% case_prodD
tff(fact_6728_case__prodD,axiom,
    ! [Fa: fun_b_540864168722188329_nat_o,A: b,B: produc6653097349344004940it_nat] :
      ( aa_Pro1537530206668979808_nat_o(produc3971132677695353399_nat_o(Fa),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,A),B))
     => aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(Fa,A),B) ) ).

% case_prodD
tff(fact_6729_case__prodD,axiom,
    ! [Fa: fun_a_2467521563176467050_nat_o,A: a,B: produc6653097349344004940it_nat] :
      ( aa_Pro6426884464393220769_nat_o(produc8860486935419594360_nat_o(Fa),aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,A),B))
     => aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(Fa,A),B) ) ).

% case_prodD
tff(fact_6730_case__prodD,axiom,
    ! [Fa: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(Fa,A),B) ) ).

% case_prodD
tff(fact_6731_case__prodE_H,axiom,
    ! [C2: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C2),P2),Z3)
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C2,X3),Y3),Z3) ) ) ).

% case_prodE'
tff(fact_6732_case__prodD_H,axiom,
    ! [Ra: fun_na5914473681949881652_nat_o,A: nat,B: nat,C2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Ra),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C2)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Ra,A),B),C2) ) ).

% case_prodD'
tff(fact_6733_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_6734_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_6735_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_6736_zero__less__eq__of__bool,axiom,
    ! [Pa: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))) ).

% zero_less_eq_of_bool
tff(fact_6737_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(zero_n2052037380579107095ol_rat,(Pa))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_6738_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_6739_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_6740_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(Pa))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_6741_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)) ) ).

% AND_lower
tff(fact_6742_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),X) ) ).

% AND_upper1
tff(fact_6743_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Y) ) ).

% AND_upper2
tff(fact_6744_AND__upper1_H,axiom,
    ! [Y: int,Z3: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z3) ) ) ).

% AND_upper1'
tff(fact_6745_AND__upper2_H,axiom,
    ! [Y: int,Z3: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z3) ) ) ).

% AND_upper2'
tff(fact_6746_Collect__case__prod__mono,axiom,
    ! [A4: fun_int_fun_int_o,B4: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A4,B4)
     => aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A4))),collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,B4))) ) ).

% Collect_case_prod_mono
tff(fact_6747_execute__bind__case,axiom,
    ! [Fa: heap_Time_Heap_b,Ga: fun_b_8426044978055006235Heap_b,Ha: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_b_b(Fa,Ga)),Ha) = case_o5901184372446974929it_nat(none_P6779099274072355161it_nat,produc3931674313442123243it_nat(aTP_Lamm_kd(Ga)),aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha)) ).

% execute_bind_case
tff(fact_6748_execute__bind__case,axiom,
    ! [Fa: heap_Time_Heap_a,Ga: fun_a_6164325696594227994Heap_b,Ha: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(heap_Time_execute_b(heap_Time_bind_a_b(Fa,Ga)),Ha) = case_o1773283271472524752it_nat(none_P6779099274072355161it_nat,produc8352694711413002346it_nat(aTP_Lamm_kf(Ga)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha)) ).

% execute_bind_case
tff(fact_6749_execute__bind__case,axiom,
    ! [Fa: heap_Time_Heap_b,Ga: fun_b_8355010938008660250Heap_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_b_a(Fa,Ga)),Ha) = case_o965627003169710416it_nat(none_P2651198173097904984it_nat,produc3428986626803312746it_nat(aTP_Lamm_kh(Ga)),aa_hea5227063804513972443it_nat(heap_Time_execute_b(Fa),Ha)) ).

% execute_bind_case
tff(fact_6750_execute__bind__case,axiom,
    ! [Fa: heap_Time_Heap_a,Ga: fun_a_6093291656547882009Heap_a,Ha: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(heap_Time_execute_a(heap_Time_bind_a_a(Fa,Ga)),Ha) = case_o6061097939050036047it_nat(none_P2651198173097904984it_nat,produc7850007024774191849it_nat(aTP_Lamm_kj(Ga)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Fa),Ha)) ).

% execute_bind_case
tff(fact_6751_map__to__set__def,axiom,
    ! [M: fun_int_option_int] : map_to_set_int_int(M) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_kk(M))) ).

% map_to_set_def
tff(fact_6752_and__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),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))),K2)
            & ~ 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,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_6753_AND__upper2_H_H,axiom,
    ! [Y: int,Z3: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z3) ) ) ).

% AND_upper2''
tff(fact_6754_AND__upper1_H_H,axiom,
    ! [Y: int,Z3: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z3) ) ) ).

% AND_upper1''
tff(fact_6755_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),K2) ) ).

% and_less_eq
tff(fact_6756_Heap__Time__Monad_Obind__def,axiom,
    ! [Fa: heap_Time_Heap_b,Ga: fun_b_8426044978055006235Heap_b] : heap_Time_bind_b_b(Fa,Ga) = heap_Time_Heap_b2(aa_fun5043953227945495435it_nat(aTP_Lamm_kl(Fa),Ga)) ).

% Heap_Time_Monad.bind_def
tff(fact_6757_Heap__Time__Monad_Obind__def,axiom,
    ! [Fa: heap_Time_Heap_a,Ga: fun_a_6164325696594227994Heap_b] : heap_Time_bind_a_b(Fa,Ga) = heap_Time_Heap_b2(aa_fun7255524789075031756it_nat(aTP_Lamm_km(Fa),Ga)) ).

% Heap_Time_Monad.bind_def
tff(fact_6758_Heap__Time__Monad_Obind__def,axiom,
    ! [Fa: heap_Time_Heap_b,Ga: fun_b_8355010938008660250Heap_a] : heap_Time_bind_b_a(Fa,Ga) = heap_Time_Heap_a2(aa_fun3907649205913156555it_nat(aTP_Lamm_kn(Fa),Ga)) ).

% Heap_Time_Monad.bind_def
tff(fact_6759_Heap__Time__Monad_Obind__def,axiom,
    ! [Fa: heap_Time_Heap_a,Ga: fun_a_6093291656547882009Heap_a] : heap_Time_bind_a_a(Fa,Ga) = heap_Time_Heap_a2(aa_fun6119220767042692876it_nat(aTP_Lamm_ko(Fa),Ga)) ).

% Heap_Time_Monad.bind_def
tff(fact_6760_Heap__Time__Monad_Obind__def,axiom,
    ! [Fa: heap_Time_Heap_b,Ga: fun_b_6677255003012645673t_unit] : heap_T3413130826733729493t_unit(Fa,Ga) = heap_T6183433275982383450t_unit(aa_fun3115769655970763019it_nat(aTP_Lamm_kr(Fa),Ga)) ).

% Heap_Time_Monad.bind_def
tff(fact_6761_Heap__Time__Monad_Obind__def,axiom,
    ! [Fa: heap_Time_Heap_a,Ga: fun_a_2990225006182978344t_unit] : heap_T8685611227969916822t_unit(Fa,Ga) = heap_T6183433275982383450t_unit(aa_fun372765568861188684it_nat(aTP_Lamm_ku(Fa),Ga)) ).

% Heap_Time_Monad.bind_def
tff(fact_6762_rel__of__def,axiom,
    ! [M: fun_fu2352890940381120596et_nat,Pa: fun_Pr2620521676128474833_nat_o] : rel_of7233497478102412281et_nat(M,Pa) = collec6193113176214168693et_nat(produc2915205735004773393_nat_o(aa_fun1045796833200626456_nat_o(aTP_Lamm_kv(M),Pa))) ).

% rel_of_def
tff(fact_6763_rel__of__def,axiom,
    ! [M: fun_fu1905361174868762994et_nat,Pa: fun_Pr1227177680593602867_nat_o] : rel_of4573248530116435351et_nat(M,Pa) = collec5916191906451531411et_nat(produc4465912260561112435_nat_o(aa_fun4422562838209026524_nat_o(aTP_Lamm_kw(M),Pa))) ).

% rel_of_def
tff(fact_6764_rel__of__def,axiom,
    ! [M: fun_b_7832390750199427630it_nat,Pa: fun_Pr856651894144386551_nat_o] : rel_of6889123638293104083it_nat(M,Pa) = collec8276579174399660751it_nat(produc3971132677695353399_nat_o(aa_fun5445639589913049316_nat_o(aTP_Lamm_kx(M),Pa))) ).

% rel_of_def
tff(fact_6765_rel__of__def,axiom,
    ! [M: fun_a_1313742400928293293it_nat,Pa: fun_Pr5746006151868627512_nat_o] : rel_of2761222537318653906it_nat(M,Pa) = collec4148678073425210574it_nat(produc8860486935419594360_nat_o(aa_fun7773039478287727974_nat_o(aTP_Lamm_ky(M),Pa))) ).

% rel_of_def
tff(fact_6766_rel__of__def,axiom,
    ! [M: fun_int_option_int,Pa: fun_Pr3238374608701176162_int_o] : rel_of_int_int(M,Pa) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun5140975492127840594_int_o(aTP_Lamm_kz(M),Pa))) ).

% rel_of_def
tff(fact_6767_bits__induct,axiom,
    ! [Pa: fun_Code_natural_o,A: code_natural] :
      ( ! [A3: code_natural] :
          ( ( divide5121882707175180666atural(A3,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A3 )
         => aa_Code_natural_o(Pa,A3) )
     => ( ! [A3: code_natural,B3: $o] :
            ( aa_Code_natural_o(Pa,A3)
           => ( ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_o_Code_natural(zero_n8403883297036319079atural,(B3))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A3)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) = A3 )
             => aa_Code_natural_o(Pa,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_o_Code_natural(zero_n8403883297036319079atural,(B3))),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A3))) ) )
       => aa_Code_natural_o(Pa,A) ) ) ).

% bits_induct
tff(fact_6768_bits__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => aa_nat_o(Pa,A3) )
     => ( ! [A3: nat,B3: $o] :
            ( aa_nat_o(Pa,A3)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))) ) )
       => aa_nat_o(Pa,A) ) ) ).

% bits_induct
tff(fact_6769_bits__induct,axiom,
    ! [Pa: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => aa_int_o(Pa,A3) )
     => ( ! [A3: int,B3: $o] :
            ( aa_int_o(Pa,A3)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))) ) )
       => aa_int_o(Pa,A) ) ) ).

% bits_induct
tff(fact_6770_bits__induct,axiom,
    ! [Pa: fun_Code_integer_o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,A3),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A3 )
         => aa_Code_integer_o(Pa,A3) )
     => ( ! [A3: code_integer,B3: $o] :
            ( aa_Code_integer_o(Pa,A3)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B3))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A3))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A3 )
             => aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B3))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A3))) ) )
       => aa_Code_integer_o(Pa,A) ) ) ).

% bits_induct
tff(fact_6771_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] : modulo8411746178871703098atural(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))),N2)) = times_2397367101498566445atural(aa_o_Code_natural(zero_n8403883297036319079atural,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)),aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6772_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6773_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6774_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,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))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6775_exp__div__exp__eq,axiom,
    ! [M: nat,N2: 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))),N2)) = times_2397367101498566445atural(aa_o_Code_natural(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,N2),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),N2))) ).

% exp_div_exp_eq
tff(fact_6776_exp__div__exp__eq,axiom,
    ! [M: nat,N2: 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))),N2)) = 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,N2),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),N2))) ).

% exp_div_exp_eq
tff(fact_6777_exp__div__exp__eq,axiom,
    ! [M: nat,N2: 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))),N2)) = 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,N2),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),N2))) ).

% exp_div_exp_eq
tff(fact_6778_exp__div__exp__eq,axiom,
    ! [M: nat,N2: 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))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          aa_o_Code_integer(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,N2),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),N2))) ).

% exp_div_exp_eq
tff(fact_6779_divide__int__unfold,axiom,
    ! [K2: int,M: nat,L: int,N2: 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,K2)),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,N2))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( N2 = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),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),N2)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M)))))) ) ).

% divide_int_unfold
tff(fact_6780_divide__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = 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,K2))),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,K2))),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),K2)))))) ) ).

% divide_int_def
tff(fact_6781_mlex__eq,axiom,
    ! [Fa: fun_int_nat,Ra: set_Pr958786334691620121nt_int] : mlex_prod_int(Fa,Ra) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_set742265667465749333_int_o(aTP_Lamm_la(Fa),Ra))) ).

% mlex_eq
tff(fact_6782_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: 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,N2)))) = 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,N2))) ).

% and_minus_numerals(4)
tff(fact_6783_and__minus__numerals_I8_J,axiom,
    ! [N2: 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,N2)))),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,N2))) ).

% and_minus_numerals(8)
tff(fact_6784_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(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))),K2)
            & ~ 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))),K2)
              & ~ 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,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_6785_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xa),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))),X)
                & ~ 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_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))),X)
                  & ~ 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_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,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_6786_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_o_int(zero_n2684676970156552555ol_int,R != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_6787_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(X,A4)) = insert9069300056098147895at_nat(X,A4) ).

% insert_absorb2
tff(fact_6788_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(X,A4)) = insert8211810215607154385at_nat(X,A4) ).

% insert_absorb2
tff(fact_6789_insert__absorb2,axiom,
    ! [X: $o,A4: set_o] : insert_o((X),insert_o((X),A4)) = insert_o((X),A4) ).

% insert_absorb2
tff(fact_6790_insert__absorb2,axiom,
    ! [X: nat,A4: set_nat] : insert_nat(X,insert_nat(X,A4)) = insert_nat(X,A4) ).

% insert_absorb2
tff(fact_6791_insert__absorb2,axiom,
    ! [X: int,A4: set_int] : insert_int(X,insert_int(X,A4)) = insert_int(X,A4) ).

% insert_absorb2
tff(fact_6792_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A4))
    <=> ( ( A = B )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4) ) ) ).

% insert_iff
tff(fact_6793_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A4))
    <=> ( ( A = B )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4) ) ) ).

% insert_iff
tff(fact_6794_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A4))
    <=> ( ( (A)
        <=> (B) )
        | aa_set_o_o(member_o((A)),A4) ) ) ).

% insert_iff
tff(fact_6795_insert__iff,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(B,A4))
    <=> ( ( A = B )
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4) ) ) ).

% insert_iff
tff(fact_6796_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,A4))
    <=> ( ( A = B )
        | aa_set_set_nat_o(member_set_nat(A),A4) ) ) ).

% insert_iff
tff(fact_6797_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A4))
    <=> ( ( A = B )
        | aa_set_nat_o(member_nat(A),A4) ) ) ).

% insert_iff
tff(fact_6798_insert__iff,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A4))
    <=> ( ( A = B )
        | aa_set_int_o(member_int(A),A4) ) ) ).

% insert_iff
tff(fact_6799_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
       => ( A = B ) )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B4)) ) ).

% insertCI
tff(fact_6800_insertCI,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
       => ( A = B ) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B4)) ) ).

% insertCI
tff(fact_6801_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ aa_set_o_o(member_o((A)),B4)
       => ( (A)
        <=> (B) ) )
     => aa_set_o_o(member_o((A)),insert_o((B),B4)) ) ).

% insertCI
tff(fact_6802_insertCI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B4)
       => ( A = B ) )
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(B,B4)) ) ).

% insertCI
tff(fact_6803_insertCI,axiom,
    ! [A: set_nat,B4: set_set_nat,B: set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(A),B4)
       => ( A = B ) )
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,B4)) ) ).

% insertCI
tff(fact_6804_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B4)
       => ( A = B ) )
     => aa_set_nat_o(member_nat(A),insert_nat(B,B4)) ) ).

% insertCI
tff(fact_6805_insertCI,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A),B4)
       => ( A = B ) )
     => aa_set_int_o(member_int(A),insert_int(B,B4)) ) ).

% insertCI
tff(fact_6806_split__part,axiom,
    ! [Pa: $o,Qa: fun_int_fun_int_o,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_lb((Pa)),Qa)),X5)
    <=> ( (Pa)
        & aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Qa),X5) ) ) ).

% split_part
tff(fact_6807_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% singletonI
tff(fact_6808_singletonI,axiom,
    ! [A: set_Pr958786334691620121nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int)) ).

% singletonI
tff(fact_6809_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,bot_bot_set_set_nat)) ).

% singletonI
tff(fact_6810_singletonI,axiom,
    ! [A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% singletonI
tff(fact_6811_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),insert_o((A),bot_bot_set_o)) ).

% singletonI
tff(fact_6812_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),insert_nat(A,bot_bot_set_nat)) ).

% singletonI
tff(fact_6813_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),insert_int(A,bot_bot_set_int)) ).

% singletonI
tff(fact_6814_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,A4)),B4)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B4)
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4) ) ) ).

% insert_subset
tff(fact_6815_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(X,A4)),B4)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B4)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ) ).

% insert_subset
tff(fact_6816_insert__subset,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o((X),A4)),B4)
    <=> ( aa_set_o_o(member_o((X)),B4)
        & aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ) ).

% insert_subset
tff(fact_6817_insert__subset,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(insert8897473484851387113nt_int(X,A4)),B4)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B4)
        & aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4) ) ) ).

% insert_subset
tff(fact_6818_insert__subset,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(X,A4)),B4)
    <=> ( aa_set_set_nat_o(member_set_nat(X),B4)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4) ) ) ).

% insert_subset
tff(fact_6819_insert__subset,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(X,A4)),B4)
    <=> ( aa_set_nat_o(member_nat(X),B4)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ) ).

% insert_subset
tff(fact_6820_insert__subset,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,insert_int(X,A4)),B4)
    <=> ( aa_set_int_o(member_int(X),B4)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4) ) ) ).

% insert_subset
tff(fact_6821_Int__insert__right__if1,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(A,B4)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6822_Int__insert__right__if1,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( inf_inf_set_o(A4,insert_o((A),B4)) = insert_o((A),inf_inf_set_o(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6823_Int__insert__right__if1,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
     => ( inf_in8396524679539076455nt_int(A4,insert8897473484851387113nt_int(A,B4)) = insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6824_Int__insert__right__if1,axiom,
    ! [A: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ( inf_inf_set_set_nat(A4,insert_set_nat(A,B4)) = insert_set_nat(A,inf_inf_set_set_nat(A4,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6825_Int__insert__right__if1,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),insert_int(A,B4)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6826_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(A,B4)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6827_Int__insert__right__if1,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(A,B4)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_6828_Int__insert__right__if0,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(A,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_6829_Int__insert__right__if0,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A4)
     => ( inf_inf_set_o(A4,insert_o((A),B4)) = inf_inf_set_o(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_6830_Int__insert__right__if0,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
     => ( inf_in8396524679539076455nt_int(A4,insert8897473484851387113nt_int(A,B4)) = inf_in8396524679539076455nt_int(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_6831_Int__insert__right__if0,axiom,
    ! [A: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A4)
     => ( inf_inf_set_set_nat(A4,insert_set_nat(A,B4)) = inf_inf_set_set_nat(A4,B4) ) ) ).

% Int_insert_right_if0
tff(fact_6832_Int__insert__right__if0,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),A4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),insert_int(A,B4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_6833_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(A,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_6834_Int__insert__right__if0,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ) ) ).

% Int_insert_right_if0
tff(fact_6835_insert__inter__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A4)),insert9069300056098147895at_nat(A,B4)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) ).

% insert_inter_insert
tff(fact_6836_insert__inter__insert,axiom,
    ! [A: $o,A4: set_o,B4: set_o] : inf_inf_set_o(insert_o((A),A4),insert_o((A),B4)) = insert_o((A),inf_inf_set_o(A4,B4)) ).

% insert_inter_insert
tff(fact_6837_insert__inter__insert,axiom,
    ! [A: int,A4: set_int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A4)),insert_int(A,B4)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) ).

% insert_inter_insert
tff(fact_6838_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A4)),insert8211810215607154385at_nat(A,B4)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ).

% insert_inter_insert
tff(fact_6839_insert__inter__insert,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A4)),insert_nat(A,B4)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ).

% insert_inter_insert
tff(fact_6840_Int__insert__left__if1,axiom,
    ! [A: produc3843707927480180839at_nat,C3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B4)),C3) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6841_Int__insert__left__if1,axiom,
    ! [A: $o,C3: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A)),C3)
     => ( inf_inf_set_o(insert_o((A),B4),C3) = insert_o((A),inf_inf_set_o(B4,C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6842_Int__insert__left__if1,axiom,
    ! [A: set_Pr958786334691620121nt_int,C3: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C3)
     => ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,B4),C3) = insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(B4,C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6843_Int__insert__left__if1,axiom,
    ! [A: set_nat,C3: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),C3)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B4),C3) = insert_set_nat(A,inf_inf_set_set_nat(B4,C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6844_Int__insert__left__if1,axiom,
    ! [A: int,C3: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A),C3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B4)),C3) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6845_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B4)),C3) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6846_Int__insert__left__if1,axiom,
    ! [A: nat,C3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A),C3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B4)),C3) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_6847_Int__insert__left__if0,axiom,
    ! [A: produc3843707927480180839at_nat,C3: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B4)),C3) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3) ) ) ).

% Int_insert_left_if0
tff(fact_6848_Int__insert__left__if0,axiom,
    ! [A: $o,C3: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),C3)
     => ( inf_inf_set_o(insert_o((A),B4),C3) = inf_inf_set_o(B4,C3) ) ) ).

% Int_insert_left_if0
tff(fact_6849_Int__insert__left__if0,axiom,
    ! [A: set_Pr958786334691620121nt_int,C3: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C3)
     => ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,B4),C3) = inf_in8396524679539076455nt_int(B4,C3) ) ) ).

% Int_insert_left_if0
tff(fact_6850_Int__insert__left__if0,axiom,
    ! [A: set_nat,C3: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),C3)
     => ( inf_inf_set_set_nat(insert_set_nat(A,B4),C3) = inf_inf_set_set_nat(B4,C3) ) ) ).

% Int_insert_left_if0
tff(fact_6851_Int__insert__left__if0,axiom,
    ! [A: int,C3: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),C3)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B4)),C3) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),C3) ) ) ).

% Int_insert_left_if0
tff(fact_6852_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C3)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B4)),C3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3) ) ) ).

% Int_insert_left_if0
tff(fact_6853_Int__insert__left__if0,axiom,
    ! [A: nat,C3: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),C3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B4)),C3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3) ) ) ).

% Int_insert_left_if0
tff(fact_6854_Un__insert__left,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A,B4)),C3) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B4),C3)) ).

% Un_insert_left
tff(fact_6855_Un__insert__left,axiom,
    ! [A: $o,B4: set_o,C3: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,insert_o((A),B4)),C3) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B4),C3)) ).

% Un_insert_left
tff(fact_6856_Un__insert__left,axiom,
    ! [A: int,B4: set_int,C3: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A,B4)),C3) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B4),C3)) ).

% Un_insert_left
tff(fact_6857_Un__insert__left,axiom,
    ! [A: produc4166570645942440679at_nat,B4: set_Pr8551490117392284871at_nat,C3: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,insert6337962749363155127at_nat(A,B4)),C3) = insert6337962749363155127at_nat(A,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,B4),C3)) ).

% Un_insert_left
tff(fact_6858_Un__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A,B4)),C3) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B4),C3)) ).

% Un_insert_left
tff(fact_6859_Un__insert__left,axiom,
    ! [A: nat,B4: set_nat,C3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,B4)),C3) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C3)) ).

% Un_insert_left
tff(fact_6860_Un__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),insert8211810215607154385at_nat(A,B4)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) ).

% Un_insert_right
tff(fact_6861_Un__insert__right,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),insert_o((A),B4)) = insert_o((A),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) ).

% Un_insert_right
tff(fact_6862_Un__insert__right,axiom,
    ! [A4: set_int,A: int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),insert_int(A,B4)) = insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) ).

% Un_insert_right
tff(fact_6863_Un__insert__right,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,A: produc4166570645942440679at_nat,B4: set_Pr8551490117392284871at_nat] : aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),insert6337962749363155127at_nat(A,B4)) = insert6337962749363155127at_nat(A,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) ).

% Un_insert_right
tff(fact_6864_Un__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),insert9069300056098147895at_nat(A,B4)) = insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) ).

% Un_insert_right
tff(fact_6865_Un__insert__right,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),insert_nat(A,B4)) = insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) ).

% Un_insert_right
tff(fact_6866_insert__Diff1,axiom,
    ! [X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A4)),B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4) ) ) ).

% insert_Diff1
tff(fact_6867_insert__Diff1,axiom,
    ! [X: $o,B4: set_o,A4: set_o] :
      ( aa_set_o_o(member_o((X)),B4)
     => ( minus_minus_set_o(insert_o((X),A4),B4) = minus_minus_set_o(A4,B4) ) ) ).

% insert_Diff1
tff(fact_6868_insert__Diff1,axiom,
    ! [X: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B4)
     => ( minus_2612819937483484256nt_int(insert8897473484851387113nt_int(X,A4),B4) = minus_2612819937483484256nt_int(A4,B4) ) ) ).

% insert_Diff1
tff(fact_6869_insert__Diff1,axiom,
    ! [X: set_nat,B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),B4)
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A4),B4) = minus_2163939370556025621et_nat(A4,B4) ) ) ).

% insert_Diff1
tff(fact_6870_insert__Diff1,axiom,
    ! [X: nat,B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(X),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A4)),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) ) ) ).

% insert_Diff1
tff(fact_6871_insert__Diff1,axiom,
    ! [X: int,B4: set_int,A4: set_int] :
      ( aa_set_int_o(member_int(X),B4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A4)),B4) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) ) ) ).

% insert_Diff1
tff(fact_6872_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A4)),B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) ) ) ).

% insert_Diff1
tff(fact_6873_Diff__insert0,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4) ) ) ).

% Diff_insert0
tff(fact_6874_Diff__insert0,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A4)
     => ( minus_minus_set_o(A4,insert_o((X),B4)) = minus_minus_set_o(A4,B4) ) ) ).

% Diff_insert0
tff(fact_6875_Diff__insert0,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
     => ( minus_2612819937483484256nt_int(A4,insert8897473484851387113nt_int(X,B4)) = minus_2612819937483484256nt_int(A4,B4) ) ) ).

% Diff_insert0
tff(fact_6876_Diff__insert0,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A4)
     => ( minus_2163939370556025621et_nat(A4,insert_set_nat(X,B4)) = minus_2163939370556025621et_nat(A4,B4) ) ) ).

% Diff_insert0
tff(fact_6877_Diff__insert0,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4) ) ) ).

% Diff_insert0
tff(fact_6878_Diff__insert0,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,B4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4) ) ) ).

% Diff_insert0
tff(fact_6879_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4) ) ) ).

% Diff_insert0
tff(fact_6880_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(aTP_Lamm_lc(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv
tff(fact_6881_singleton__conv,axiom,
    ! [A: list_nat] : collect_list_nat(aTP_Lamm_ld(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv
tff(fact_6882_singleton__conv,axiom,
    ! [A: set_Pr958786334691620121nt_int] : collec5210948495886036740nt_int(aTP_Lamm_le(A)) = insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int) ).

% singleton_conv
tff(fact_6883_singleton__conv,axiom,
    ! [A: set_nat] : collect_set_nat(aTP_Lamm_lf(A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv
tff(fact_6884_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_lg,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv
tff(fact_6885_singleton__conv,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(aTP_Lamm_lh,(A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv
tff(fact_6886_singleton__conv,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_li,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv
tff(fact_6887_singleton__conv,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_lj,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv
tff(fact_6888_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] : collec6321179662152712658at_nat(fequal7949547063978738364at_nat(A)) = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ).

% singleton_conv2
tff(fact_6889_singleton__conv2,axiom,
    ! [A: list_nat] : collect_list_nat(fequal_list_nat(A)) = insert_list_nat(A,bot_bot_set_list_nat) ).

% singleton_conv2
tff(fact_6890_singleton__conv2,axiom,
    ! [A: set_Pr958786334691620121nt_int] : collec5210948495886036740nt_int(fequal1127456680179375854nt_int(A)) = insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int) ).

% singleton_conv2
tff(fact_6891_singleton__conv2,axiom,
    ! [A: set_nat] : collect_set_nat(fequal_set_nat(A)) = insert_set_nat(A,bot_bot_set_set_nat) ).

% singleton_conv2
tff(fact_6892_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] : collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,A)) = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ).

% singleton_conv2
tff(fact_6893_singleton__conv2,axiom,
    ! [A: $o] : collect_o(aa_o_fun_o_o(fequal_o,(A))) = insert_o((A),bot_bot_set_o) ).

% singleton_conv2
tff(fact_6894_singleton__conv2,axiom,
    ! [A: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = insert_nat(A,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_6895_singleton__conv2,axiom,
    ! [A: int] : collect_int(aa_int_fun_int_o(fequal_int,A)) = insert_int(A,bot_bot_set_int) ).

% singleton_conv2
tff(fact_6896_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,A4) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6897_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,A4) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6898_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( insert_o((A),A4) = insert_o((B),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A4),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6899_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( insert_nat(A,A4) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6900_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( insert_int(A,A4) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_6901_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A,A4) )
    <=> ( ( A = B )
        & aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6902_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A,A4) )
    <=> ( ( A = B )
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6903_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( insert_o((B),bot_bot_set_o) = insert_o((A),A4) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A4),insert_o((B),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6904_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A4) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6905_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A4) )
    <=> ( ( A = B )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_6906_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(B,B4)) )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A4)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6907_disjoint__insert_I2_J,axiom,
    ! [A4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] :
      ( ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(A4,insert8897473484851387113nt_int(B,B4)) )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),A4)
        & ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6908_disjoint__insert_I2_J,axiom,
    ! [A4: set_set_nat,B: set_nat,B4: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(A4,insert_set_nat(B,B4)) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(B),A4)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6909_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(B,B4)) )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A4)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6910_disjoint__insert_I2_J,axiom,
    ! [A4: set_o,B: $o,B4: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(A4,insert_o((B),B4)) )
    <=> ( ~ aa_set_o_o(member_o((B)),A4)
        & ( bot_bot_set_o = inf_inf_set_o(A4,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6911_disjoint__insert_I2_J,axiom,
    ! [A4: set_nat,B: nat,B4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(B,B4)) )
    <=> ( ~ aa_set_nat_o(member_nat(B),A4)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6912_disjoint__insert_I2_J,axiom,
    ! [A4: set_int,B: int,B4: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),insert_int(B,B4)) )
    <=> ( ~ aa_set_int_o(member_int(B),A4)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_6913_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),insert9069300056098147895at_nat(A,A4)) = bot_bo228742789529271731at_nat )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),A4) = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6914_disjoint__insert_I1_J,axiom,
    ! [B4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( inf_in8396524679539076455nt_int(B4,insert8897473484851387113nt_int(A,A4)) = bot_bo1488462491386950373nt_int )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B4)
        & ( inf_in8396524679539076455nt_int(B4,A4) = bot_bo1488462491386950373nt_int ) ) ) ).

% disjoint_insert(1)
tff(fact_6915_disjoint__insert_I1_J,axiom,
    ! [B4: set_set_nat,A: set_nat,A4: set_set_nat] :
      ( ( inf_inf_set_set_nat(B4,insert_set_nat(A,A4)) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B4)
        & ( inf_inf_set_set_nat(B4,A4) = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6916_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),insert8211810215607154385at_nat(A,A4)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),A4) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6917_disjoint__insert_I1_J,axiom,
    ! [B4: set_o,A: $o,A4: set_o] :
      ( ( inf_inf_set_o(B4,insert_o((A),A4)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B4)
        & ( inf_inf_set_o(B4,A4) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_6918_disjoint__insert_I1_J,axiom,
    ! [B4: set_nat,A: nat,A4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),insert_nat(A,A4)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A4) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_6919_disjoint__insert_I1_J,axiom,
    ! [B4: set_int,A: int,A4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),insert_int(A,A4)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B4)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),A4) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_6920_insert__disjoint_I2_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A4)),B4) )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6921_insert__disjoint_I2_J,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,A4),B4) )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B4)
        & ( bot_bo1488462491386950373nt_int = inf_in8396524679539076455nt_int(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6922_insert__disjoint_I2_J,axiom,
    ! [A: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( bot_bot_set_set_nat = inf_inf_set_set_nat(insert_set_nat(A,A4),B4) )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B4)
        & ( bot_bot_set_set_nat = inf_inf_set_set_nat(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6923_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A4)),B4) )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6924_insert__disjoint_I2_J,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(insert_o((A),A4),B4) )
    <=> ( ~ aa_set_o_o(member_o((A)),B4)
        & ( bot_bot_set_o = inf_inf_set_o(A4,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6925_insert__disjoint_I2_J,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A4)),B4) )
    <=> ( ~ aa_set_nat_o(member_nat(A),B4)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6926_insert__disjoint_I2_J,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A4)),B4) )
    <=> ( ~ aa_set_int_o(member_int(A),B4)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_6927_insert__disjoint_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,A4)),B4) = bot_bo228742789529271731at_nat )
    <=> ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4) = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6928_insert__disjoint_I1_J,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ( inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,A4),B4) = bot_bo1488462491386950373nt_int )
    <=> ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B4)
        & ( inf_in8396524679539076455nt_int(A4,B4) = bot_bo1488462491386950373nt_int ) ) ) ).

% insert_disjoint(1)
tff(fact_6929_insert__disjoint_I1_J,axiom,
    ! [A: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ( inf_inf_set_set_nat(insert_set_nat(A,A4),B4) = bot_bot_set_set_nat )
    <=> ( ~ aa_set_set_nat_o(member_set_nat(A),B4)
        & ( inf_inf_set_set_nat(A4,B4) = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6930_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A4)),B4) = bot_bo2099793752762293965at_nat )
    <=> ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6931_insert__disjoint_I1_J,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o(insert_o((A),A4),B4) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A)),B4)
        & ( inf_inf_set_o(A4,B4) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_6932_insert__disjoint_I1_J,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,A4)),B4) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_6933_insert__disjoint_I1_J,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,A4)),B4) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A),B4)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_6934_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A,A4) ).

% insert_Diff_single
tff(fact_6935_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] : insert_o((A),minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = insert_o((A),A4) ).

% insert_Diff_single
tff(fact_6936_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A4) ).

% insert_Diff_single
tff(fact_6937_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] : insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(A,bot_bot_set_int))) = insert_int(A,A4) ).

% insert_Diff_single
tff(fact_6938_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A4) ).

% insert_Diff_single
tff(fact_6939_subset__Compl__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),uminus935396558254630718at_nat(insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)))
    <=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_6940_subset__Compl__singleton,axiom,
    ! [A4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),uminus6423885277529793776nt_int(insert8897473484851387113nt_int(B,bot_bo1488462491386950373nt_int)))
    <=> ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),A4) ) ).

% subset_Compl_singleton
tff(fact_6941_subset__Compl__singleton,axiom,
    ! [A4: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_6942_subset__Compl__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),uminus6524753893492686040at_nat(insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)))
    <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_6943_subset__Compl__singleton,axiom,
    ! [A4: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(insert_o((B),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B)),A4) ) ).

% subset_Compl_singleton
tff(fact_6944_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_6945_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A4) ) ).

% subset_Compl_singleton
tff(fact_6946_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_in),Prod) ).

% prod.disc_eq_case
tff(fact_6947_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ? [B7: set_Pr4329608150637261639at_nat] :
          ( ( A4 = insert9069300056098147895at_nat(A,B7) )
          & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6948_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ? [B7: set_Pr1261947904930325089at_nat] :
          ( ( A4 = insert8211810215607154385at_nat(A,B7) )
          & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6949_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ? [B7: set_o] :
          ( ( A4 = insert_o((A),B7) )
          & ~ aa_set_o_o(member_o((A)),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6950_mk__disjoint__insert,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
     => ? [B7: set_se6260736226359567993nt_int] :
          ( ( A4 = insert8897473484851387113nt_int(A,B7) )
          & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6951_mk__disjoint__insert,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ? [B7: set_set_nat] :
          ( ( A4 = insert_set_nat(A,B7) )
          & ~ aa_set_set_nat_o(member_set_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6952_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ? [B7: set_nat] :
          ( ( A4 = insert_nat(A,B7) )
          & ~ aa_set_nat_o(member_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6953_mk__disjoint__insert,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ? [B7: set_int] :
          ( ( A4 = insert_int(A,B7) )
          & ~ aa_set_int_o(member_int(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_6954_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(Y,A4)) = insert9069300056098147895at_nat(Y,insert9069300056098147895at_nat(X,A4)) ).

% insert_commute
tff(fact_6955_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(Y,A4)) = insert8211810215607154385at_nat(Y,insert8211810215607154385at_nat(X,A4)) ).

% insert_commute
tff(fact_6956_insert__commute,axiom,
    ! [X: $o,Y: $o,A4: set_o] : insert_o((X),insert_o((Y),A4)) = insert_o((Y),insert_o((X),A4)) ).

% insert_commute
tff(fact_6957_insert__commute,axiom,
    ! [X: nat,Y: nat,A4: set_nat] : insert_nat(X,insert_nat(Y,A4)) = insert_nat(Y,insert_nat(X,A4)) ).

% insert_commute
tff(fact_6958_insert__commute,axiom,
    ! [X: int,Y: int,A4: set_int] : insert_int(X,insert_int(Y,A4)) = insert_int(Y,insert_int(X,A4)) ).

% insert_commute
tff(fact_6959_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),B4)
       => ( ( insert9069300056098147895at_nat(A,A4) = insert9069300056098147895at_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C5: set_Pr4329608150637261639at_nat] :
                ( ( A4 = insert9069300056098147895at_nat(B,C5) )
                & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),C5)
                & ( B4 = insert9069300056098147895at_nat(A,C5) )
                & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6960_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),B4)
       => ( ( insert8211810215607154385at_nat(A,A4) = insert8211810215607154385at_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C5: set_Pr1261947904930325089at_nat] :
                ( ( A4 = insert8211810215607154385at_nat(B,C5) )
                & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),C5)
                & ( B4 = insert8211810215607154385at_nat(A,C5) )
                & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6961_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A4)
     => ( ~ aa_set_o_o(member_o((B)),B4)
       => ( ( insert_o((A),A4) = insert_o((B),B4) )
        <=> $ite(
              (A) = (B),
              A4 = B4,
              ? [C5: set_o] :
                ( ( A4 = insert_o((B),C5) )
                & ~ aa_set_o_o(member_o((B)),C5)
                & ( B4 = insert_o((A),C5) )
                & ~ aa_set_o_o(member_o((A)),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6962_insert__eq__iff,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),B4)
       => ( ( insert8897473484851387113nt_int(A,A4) = insert8897473484851387113nt_int(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C5: set_se6260736226359567993nt_int] :
                ( ( A4 = insert8897473484851387113nt_int(B,C5) )
                & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),C5)
                & ( B4 = insert8897473484851387113nt_int(A,C5) )
                & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6963_insert__eq__iff,axiom,
    ! [A: set_nat,A4: set_set_nat,B: set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(B),B4)
       => ( ( insert_set_nat(A,A4) = insert_set_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C5: set_set_nat] :
                ( ( A4 = insert_set_nat(B,C5) )
                & ~ aa_set_set_nat_o(member_set_nat(B),C5)
                & ( B4 = insert_set_nat(A,C5) )
                & ~ aa_set_set_nat_o(member_set_nat(A),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6964_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A4)
     => ( ~ aa_set_nat_o(member_nat(B),B4)
       => ( ( insert_nat(A,A4) = insert_nat(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C5: set_nat] :
                ( ( A4 = insert_nat(B,C5) )
                & ~ aa_set_nat_o(member_nat(B),C5)
                & ( B4 = insert_nat(A,C5) )
                & ~ aa_set_nat_o(member_nat(A),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6965_insert__eq__iff,axiom,
    ! [A: int,A4: set_int,B: int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),A4)
     => ( ~ aa_set_int_o(member_int(B),B4)
       => ( ( insert_int(A,A4) = insert_int(B,B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C5: set_int] :
                ( ( A4 = insert_int(B,C5) )
                & ~ aa_set_int_o(member_int(B),C5)
                & ( B4 = insert_int(A,C5) )
                & ~ aa_set_int_o(member_int(A),C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_6966_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( insert9069300056098147895at_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_6967_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( insert8211810215607154385at_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_6968_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( insert_o((A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_6969_insert__absorb,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
     => ( insert8897473484851387113nt_int(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_6970_insert__absorb,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ( insert_set_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_6971_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( insert_nat(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_6972_insert__absorb,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( insert_int(A,A4) = A4 ) ) ).

% insert_absorb
tff(fact_6973_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B4)
       => ( ( insert9069300056098147895at_nat(X,A4) = insert9069300056098147895at_nat(X,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6974_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B4)
       => ( ( insert8211810215607154385at_nat(X,A4) = insert8211810215607154385at_nat(X,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6975_insert__ident,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A4)
     => ( ~ aa_set_o_o(member_o((X)),B4)
       => ( ( insert_o((X),A4) = insert_o((X),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6976_insert__ident,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
     => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B4)
       => ( ( insert8897473484851387113nt_int(X,A4) = insert8897473484851387113nt_int(X,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6977_insert__ident,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(X),B4)
       => ( ( insert_set_nat(X,A4) = insert_set_nat(X,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6978_insert__ident,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A4)
     => ( ~ aa_set_nat_o(member_nat(X),B4)
       => ( ( insert_nat(X,A4) = insert_nat(X,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6979_insert__ident,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A4)
     => ( ~ aa_set_int_o(member_int(X),B4)
       => ( ( insert_int(X,A4) = insert_int(X,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_6980_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => ~ ! [B7: set_Pr4329608150637261639at_nat] :
            ( ( A4 = insert9069300056098147895at_nat(X,B7) )
           => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B7) ) ) ).

% Set.set_insert
tff(fact_6981_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => ~ ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( A4 = insert8211810215607154385at_nat(X,B7) )
           => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B7) ) ) ).

% Set.set_insert
tff(fact_6982_Set_Oset__insert,axiom,
    ! [X: $o,A4: set_o] :
      ( aa_set_o_o(member_o((X)),A4)
     => ~ ! [B7: set_o] :
            ( ( A4 = insert_o((X),B7) )
           => aa_set_o_o(member_o((X)),B7) ) ) ).

% Set.set_insert
tff(fact_6983_Set_Oset__insert,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
     => ~ ! [B7: set_se6260736226359567993nt_int] :
            ( ( A4 = insert8897473484851387113nt_int(X,B7) )
           => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B7) ) ) ).

% Set.set_insert
tff(fact_6984_Set_Oset__insert,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),A4)
     => ~ ! [B7: set_set_nat] :
            ( ( A4 = insert_set_nat(X,B7) )
           => aa_set_set_nat_o(member_set_nat(X),B7) ) ) ).

% Set.set_insert
tff(fact_6985_Set_Oset__insert,axiom,
    ! [X: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(X),A4)
     => ~ ! [B7: set_nat] :
            ( ( A4 = insert_nat(X,B7) )
           => aa_set_nat_o(member_nat(X),B7) ) ) ).

% Set.set_insert
tff(fact_6986_Set_Oset__insert,axiom,
    ! [X: int,A4: set_int] :
      ( aa_set_int_o(member_int(X),A4)
     => ~ ! [B7: set_int] :
            ( ( A4 = insert_int(X,B7) )
           => aa_set_int_o(member_int(X),B7) ) ) ).

% Set.set_insert
tff(fact_6987_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),B4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,B4)) ) ).

% insertI2
tff(fact_6988_insertI2,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),B4)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,B4)) ) ).

% insertI2
tff(fact_6989_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( aa_set_o_o(member_o((A)),B4)
     => aa_set_o_o(member_o((A)),insert_o((B),B4)) ) ).

% insertI2
tff(fact_6990_insertI2,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),B4)
     => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(B,B4)) ) ).

% insertI2
tff(fact_6991_insertI2,axiom,
    ! [A: set_nat,B4: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),B4)
     => aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,B4)) ) ).

% insertI2
tff(fact_6992_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A),B4)
     => aa_set_nat_o(member_nat(A),insert_nat(B,B4)) ) ).

% insertI2
tff(fact_6993_insertI2,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( aa_set_int_o(member_int(A),B4)
     => aa_set_int_o(member_int(A),insert_int(B,B4)) ) ).

% insertI2
tff(fact_6994_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(A,B4)) ).

% insertI1
tff(fact_6995_insertI1,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(A,B4)) ).

% insertI1
tff(fact_6996_insertI1,axiom,
    ! [A: $o,B4: set_o] : aa_set_o_o(member_o((A)),insert_o((A),B4)) ).

% insertI1
tff(fact_6997_insertI1,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(A,B4)) ).

% insertI1
tff(fact_6998_insertI1,axiom,
    ! [A: set_nat,B4: set_set_nat] : aa_set_set_nat_o(member_set_nat(A),insert_set_nat(A,B4)) ).

% insertI1
tff(fact_6999_insertI1,axiom,
    ! [A: nat,B4: set_nat] : aa_set_nat_o(member_nat(A),insert_nat(A,B4)) ).

% insertI1
tff(fact_7000_insertI1,axiom,
    ! [A: int,B4: set_int] : aa_set_int_o(member_int(A),insert_int(A,B4)) ).

% insertI1
tff(fact_7001_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),insert9069300056098147895at_nat(B,A4))
     => ( ( A != B )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4) ) ) ).

% insertE
tff(fact_7002_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),insert8211810215607154385at_nat(B,A4))
     => ( ( A != B )
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4) ) ) ).

% insertE
tff(fact_7003_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),insert_o((B),A4))
     => ( ( (A)
        <=> ~ (B) )
       => aa_set_o_o(member_o((A)),A4) ) ) ).

% insertE
tff(fact_7004_insertE,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),insert8897473484851387113nt_int(B,A4))
     => ( ( A != B )
       => aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4) ) ) ).

% insertE
tff(fact_7005_insertE,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),insert_set_nat(B,A4))
     => ( ( A != B )
       => aa_set_set_nat_o(member_set_nat(A),A4) ) ) ).

% insertE
tff(fact_7006_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),insert_nat(B,A4))
     => ( ( A != B )
       => aa_set_nat_o(member_nat(A),A4) ) ) ).

% insertE
tff(fact_7007_insertE,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),insert_int(B,A4))
     => ( ( A != B )
       => aa_set_int_o(member_int(A),A4) ) ) ).

% insertE
tff(fact_7008_insert__compr,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B4) = collec6321179662152712658at_nat(aa_set6353399069419443519_nat_o(aTP_Lamm_lk(A),B4)) ).

% insert_compr
tff(fact_7009_insert__compr,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_ll(A),B4)) ).

% insert_compr
tff(fact_7010_insert__compr,axiom,
    ! [A: $o,B4: set_o] : insert_o((A),B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_lm((A)),B4)) ).

% insert_compr
tff(fact_7011_insert__compr,axiom,
    ! [A: list_nat,B4: set_list_nat] : insert_list_nat(A,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ln(A),B4)) ).

% insert_compr
tff(fact_7012_insert__compr,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] : insert8897473484851387113nt_int(A,B4) = collec5210948495886036740nt_int(aa_set4062574822058255039_int_o(aTP_Lamm_lo(A),B4)) ).

% insert_compr
tff(fact_7013_insert__compr,axiom,
    ! [A: set_nat,B4: set_set_nat] : insert_set_nat(A,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_lp(A),B4)) ).

% insert_compr
tff(fact_7014_insert__compr,axiom,
    ! [A: nat,B4: set_nat] : insert_nat(A,B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_lq(A),B4)) ).

% insert_compr
tff(fact_7015_insert__compr,axiom,
    ! [A: int,B4: set_int] : insert_int(A,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_lr(A),B4)) ).

% insert_compr
tff(fact_7016_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_ls(A),Pa)) ).

% insert_Collect
tff(fact_7017_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_lt(A),Pa)) ).

% insert_Collect
tff(fact_7018_insert__Collect,axiom,
    ! [A: $o,Pa: fun_o_o] : insert_o((A),collect_o(Pa)) = collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_lu((A)),Pa)) ).

% insert_Collect
tff(fact_7019_insert__Collect,axiom,
    ! [A: list_nat,Pa: fun_list_nat_o] : insert_list_nat(A,collect_list_nat(Pa)) = collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_lv(A),Pa)) ).

% insert_Collect
tff(fact_7020_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_lw(A),Pa)) ).

% insert_Collect
tff(fact_7021_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_lx(A),Pa)) ).

% insert_Collect
tff(fact_7022_insert__Collect,axiom,
    ! [A: nat,Pa: fun_nat_o] : insert_nat(A,collect_nat(Pa)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ly(A),Pa)) ).

% insert_Collect
tff(fact_7023_insert__Collect,axiom,
    ! [A: int,Pa: fun_int_o] : insert_int(A,collect_int(Pa)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_lz(A),Pa)) ).

% insert_Collect
tff(fact_7024_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_7025_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_7026_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( insert_o((A),bot_bot_set_o) = insert_o((B),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_7027_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( insert_nat(A,bot_bot_set_nat) = insert_nat(B,bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_7028_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( insert_int(A,bot_bot_set_int) = insert_int(B,bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_7029_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A4) != bot_bo228742789529271731at_nat ).

% insert_not_empty
tff(fact_7030_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A4) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_7031_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] : insert_o((A),A4) != bot_bot_set_o ).

% insert_not_empty
tff(fact_7032_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,A4) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_7033_insert__not__empty,axiom,
    ! [A: int,A4: set_int] : insert_int(A,A4) != bot_bot_set_int ).

% insert_not_empty
tff(fact_7034_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C2: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(C2,insert9069300056098147895at_nat(D2,bot_bo228742789529271731at_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_7035_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C2: product_prod_nat_nat,D2: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C2,insert8211810215607154385at_nat(D2,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_7036_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C2: $o,D2: $o] :
      ( ( insert_o((A),insert_o((B),bot_bot_set_o)) = insert_o((C2),insert_o((D2),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C2) )
          & ( (B)
          <=> (D2) ) )
        | ( ( (A)
          <=> (D2) )
          & ( (B)
          <=> (C2) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_7037_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( insert_nat(A,insert_nat(B,bot_bot_set_nat)) = insert_nat(C2,insert_nat(D2,bot_bot_set_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_7038_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( insert_int(A,insert_int(B,bot_bot_set_int)) = insert_int(C2,insert_int(D2,bot_bot_set_int)) )
    <=> ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_7039_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_7040_singleton__iff,axiom,
    ! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_7041_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_7042_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_7043_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_7044_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_7045_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_7046_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_7047_singletonD,axiom,
    ! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(B),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_7048_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),insert_set_nat(A,bot_bot_set_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_7049_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_7050_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),insert_o((A),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_7051_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),insert_nat(A,bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_7052_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),insert_int(A,bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_7053_insert__mono,axiom,
    ! [C3: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(C3),D4)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(A,C3)),insert9069300056098147895at_nat(A,D4)) ) ).

% insert_mono
tff(fact_7054_insert__mono,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),D4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(A,C3)),insert8211810215607154385at_nat(A,D4)) ) ).

% insert_mono
tff(fact_7055_insert__mono,axiom,
    ! [C3: set_o,D4: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C3),D4)
     => aa_set_o_o(ord_less_eq_set_o(insert_o((A),C3)),insert_o((A),D4)) ) ).

% insert_mono
tff(fact_7056_insert__mono,axiom,
    ! [C3: set_nat,D4: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),D4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(A,C3)),insert_nat(A,D4)) ) ).

% insert_mono
tff(fact_7057_insert__mono,axiom,
    ! [C3: set_int,D4: set_int,A: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C3),D4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,insert_int(A,C3)),insert_int(A,D4)) ) ).

% insert_mono
tff(fact_7058_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X,B4))
      <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4) ) ) ).

% subset_insert
tff(fact_7059_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,B4))
      <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4) ) ) ).

% subset_insert
tff(fact_7060_subset__insert,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X),B4))
      <=> aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ) ).

% subset_insert
tff(fact_7061_subset__insert,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),insert8897473484851387113nt_int(X,B4))
      <=> aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4) ) ) ).

% subset_insert
tff(fact_7062_subset__insert,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),insert_set_nat(X,B4))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4) ) ) ).

% subset_insert
tff(fact_7063_subset__insert,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(X,B4))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ) ).

% subset_insert
tff(fact_7064_subset__insert,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(X,B4))
      <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4) ) ) ).

% subset_insert
tff(fact_7065_subset__insertI,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(B4),insert9069300056098147895at_nat(A,B4)) ).

% subset_insertI
tff(fact_7066_subset__insertI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),insert8211810215607154385at_nat(A,B4)) ).

% subset_insertI
tff(fact_7067_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B4),insert_o((A),B4)) ).

% subset_insertI
tff(fact_7068_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),insert_nat(A,B4)) ).

% subset_insertI
tff(fact_7069_subset__insertI,axiom,
    ! [B4: set_int,A: int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),insert_int(A,B4)) ).

% subset_insertI
tff(fact_7070_subset__insertI2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_7071_subset__insertI2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_7072_subset__insertI2,axiom,
    ! [A4: set_o,B4: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => aa_set_o_o(ord_less_eq_set_o(A4),insert_o((B),B4)) ) ).

% subset_insertI2
tff(fact_7073_subset__insertI2,axiom,
    ! [A4: set_nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_7074_subset__insertI2,axiom,
    ! [A4: set_int,B4: set_int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(B,B4)) ) ).

% subset_insertI2
tff(fact_7075_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,X6: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X6),A4)
       => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7076_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X6),A4)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(X,X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7077_insert__subsetI,axiom,
    ! [X: $o,A4: set_o,X6: set_o] :
      ( aa_set_o_o(member_o((X)),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(X6),A4)
       => aa_set_o_o(ord_less_eq_set_o(insert_o((X),X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7078_insert__subsetI,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,X6: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(X6),A4)
       => aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(insert8897473484851387113nt_int(X,X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7079_insert__subsetI,axiom,
    ! [X: set_nat,A4: set_set_nat,X6: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),A4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(X6),A4)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(insert_set_nat(X,X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7080_insert__subsetI,axiom,
    ! [X: nat,A4: set_nat,X6: set_nat] :
      ( aa_set_nat_o(member_nat(X),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X6),A4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,insert_nat(X,X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7081_insert__subsetI,axiom,
    ! [X: int,A4: set_int,X6: set_int] :
      ( aa_set_int_o(member_int(X),A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X6),A4)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,insert_int(X,X6)),A4) ) ) ).

% insert_subsetI
tff(fact_7082_Int__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),insert9069300056098147895at_nat(A,B4)) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4),insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4)) ).

% Int_insert_right
tff(fact_7083_Int__insert__right,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      inf_inf_set_o(A4,insert_o((A),B4)) = $ite(aa_set_o_o(member_o((A)),A4),insert_o((A),inf_inf_set_o(A4,B4)),inf_inf_set_o(A4,B4)) ).

% Int_insert_right
tff(fact_7084_Int__insert__right,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] :
      inf_in8396524679539076455nt_int(A4,insert8897473484851387113nt_int(A,B4)) = $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4),insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(A4,B4)),inf_in8396524679539076455nt_int(A4,B4)) ).

% Int_insert_right
tff(fact_7085_Int__insert__right,axiom,
    ! [A4: set_set_nat,A: set_nat,B4: set_set_nat] :
      inf_inf_set_set_nat(A4,insert_set_nat(A,B4)) = $ite(aa_set_set_nat_o(member_set_nat(A),A4),insert_set_nat(A,inf_inf_set_set_nat(A4,B4)),inf_inf_set_set_nat(A4,B4)) ).

% Int_insert_right
tff(fact_7086_Int__insert__right,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),insert_int(A,B4)) = $ite(aa_set_int_o(member_int(A),A4),insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) ).

% Int_insert_right
tff(fact_7087_Int__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),insert8211810215607154385at_nat(A,B4)) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4)) ).

% Int_insert_right
tff(fact_7088_Int__insert__right,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),insert_nat(A,B4)) = $ite(aa_set_nat_o(member_nat(A),A4),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4)) ).

% Int_insert_right
tff(fact_7089_Int__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A,B4)),C3) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),C3),insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B4),C3)) ).

% Int_insert_left
tff(fact_7090_Int__insert__left,axiom,
    ! [A: $o,B4: set_o,C3: set_o] :
      inf_inf_set_o(insert_o((A),B4),C3) = $ite(aa_set_o_o(member_o((A)),C3),insert_o((A),inf_inf_set_o(B4,C3)),inf_inf_set_o(B4,C3)) ).

% Int_insert_left
tff(fact_7091_Int__insert__left,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int,C3: set_se6260736226359567993nt_int] :
      inf_in8396524679539076455nt_int(insert8897473484851387113nt_int(A,B4),C3) = $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),C3),insert8897473484851387113nt_int(A,inf_in8396524679539076455nt_int(B4,C3)),inf_in8396524679539076455nt_int(B4,C3)) ).

% Int_insert_left
tff(fact_7092_Int__insert__left,axiom,
    ! [A: set_nat,B4: set_set_nat,C3: set_set_nat] :
      inf_inf_set_set_nat(insert_set_nat(A,B4),C3) = $ite(aa_set_set_nat_o(member_set_nat(A),C3),insert_set_nat(A,inf_inf_set_set_nat(B4,C3)),inf_inf_set_set_nat(B4,C3)) ).

% Int_insert_left
tff(fact_7093_Int__insert__left,axiom,
    ! [A: int,B4: set_int,C3: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int(A,B4)),C3) = $ite(aa_set_int_o(member_int(A),C3),insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),C3)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B4),C3)) ).

% Int_insert_left
tff(fact_7094_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B4)),C3) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),C3),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B4),C3)) ).

% Int_insert_left
tff(fact_7095_Int__insert__left,axiom,
    ! [A: nat,B4: set_nat,C3: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat(A,B4)),C3) = $ite(aa_set_nat_o(member_nat(A),C3),insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C3)) ).

% Int_insert_left
tff(fact_7096_insert__Diff__if,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A4)),B4) = $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4),insert9069300056098147895at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4))) ).

% insert_Diff_if
tff(fact_7097_insert__Diff__if,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      minus_minus_set_o(insert_o((X),A4),B4) = $ite(aa_set_o_o(member_o((X)),B4),minus_minus_set_o(A4,B4),insert_o((X),minus_minus_set_o(A4,B4))) ).

% insert_Diff_if
tff(fact_7098_insert__Diff__if,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
      minus_2612819937483484256nt_int(insert8897473484851387113nt_int(X,A4),B4) = $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B4),minus_2612819937483484256nt_int(A4,B4),insert8897473484851387113nt_int(X,minus_2612819937483484256nt_int(A4,B4))) ).

% insert_Diff_if
tff(fact_7099_insert__Diff__if,axiom,
    ! [X: set_nat,A4: set_set_nat,B4: set_set_nat] :
      minus_2163939370556025621et_nat(insert_set_nat(X,A4),B4) = $ite(aa_set_set_nat_o(member_set_nat(X),B4),minus_2163939370556025621et_nat(A4,B4),insert_set_nat(X,minus_2163939370556025621et_nat(A4,B4))) ).

% insert_Diff_if
tff(fact_7100_insert__Diff__if,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A4)),B4) = $ite(aa_set_nat_o(member_nat(X),B4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4),insert_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ).

% insert_Diff_if
tff(fact_7101_insert__Diff__if,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A4)),B4) = $ite(aa_set_int_o(member_int(X),B4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4),insert_int(X,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))) ).

% insert_Diff_if
tff(fact_7102_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A4)),B4) = $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4),insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))) ).

% insert_Diff_if
tff(fact_7103_Collect__conv__if,axiom,
    ! [A: produc3843707927480180839at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_ma(A),Pa)) = $ite(aa_Pro5947980084102686253_nat_o(Pa,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).

% Collect_conv_if
tff(fact_7104_Collect__conv__if,axiom,
    ! [A: list_nat,Pa: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_mb(A),Pa)) = $ite(aa_list_nat_o(Pa,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ).

% Collect_conv_if
tff(fact_7105_Collect__conv__if,axiom,
    ! [A: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_mc(A),Pa)) = $ite(aa_set6170482117830564987_int_o(Pa,A),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int),bot_bo1488462491386950373nt_int) ).

% Collect_conv_if
tff(fact_7106_Collect__conv__if,axiom,
    ! [A: set_nat,Pa: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_md(A),Pa)) = $ite(aa_set_nat_o(Pa,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if
tff(fact_7107_Collect__conv__if,axiom,
    ! [A: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_me(A),Pa)) = $ite(aa_Pro6150637604445964947_nat_o(Pa,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).

% Collect_conv_if
tff(fact_7108_Collect__conv__if,axiom,
    ! [A: $o,Pa: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_mf((A)),Pa)) = $ite(aa_o_o(Pa,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if
tff(fact_7109_Collect__conv__if,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_mg(A),Pa)) = $ite(aa_nat_o(Pa,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_7110_Collect__conv__if,axiom,
    ! [A: int,Pa: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_mh(A),Pa)) = $ite(aa_int_o(Pa,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_7111_Collect__conv__if2,axiom,
    ! [A: produc3843707927480180839at_nat,Pa: fun_Pr3059359313727237556_nat_o] :
      collec6321179662152712658at_nat(aa_fun2566793329686429606_nat_o(aTP_Lamm_mi(A),Pa)) = $ite(aa_Pro5947980084102686253_nat_o(Pa,A),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat),bot_bo228742789529271731at_nat) ).

% Collect_conv_if2
tff(fact_7112_Collect__conv__if2,axiom,
    ! [A: list_nat,Pa: fun_list_nat_o] :
      collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_mj(A),Pa)) = $ite(aa_list_nat_o(Pa,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ).

% Collect_conv_if2
tff(fact_7113_Collect__conv__if2,axiom,
    ! [A: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_mk(A),Pa)) = $ite(aa_set6170482117830564987_int_o(Pa,A),insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int),bot_bo1488462491386950373nt_int) ).

% Collect_conv_if2
tff(fact_7114_Collect__conv__if2,axiom,
    ! [A: set_nat,Pa: fun_set_nat_o] :
      collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ml(A),Pa)) = $ite(aa_set_nat_o(Pa,A),insert_set_nat(A,bot_bot_set_set_nat),bot_bot_set_set_nat) ).

% Collect_conv_if2
tff(fact_7115_Collect__conv__if2,axiom,
    ! [A: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_mm(A),Pa)) = $ite(aa_Pro6150637604445964947_nat_o(Pa,A),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat) ).

% Collect_conv_if2
tff(fact_7116_Collect__conv__if2,axiom,
    ! [A: $o,Pa: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_mn((A)),Pa)) = $ite(aa_o_o(Pa,(A)),insert_o((A),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if2
tff(fact_7117_Collect__conv__if2,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_mo(A),Pa)) = $ite(aa_nat_o(Pa,A),insert_nat(A,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_7118_Collect__conv__if2,axiom,
    ! [A: int,Pa: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_mp(A),Pa)) = $ite(aa_int_o(Pa,A),insert_int(A,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_7119_insert__def,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,B4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_lg,A))),B4) ).

% insert_def
tff(fact_7120_insert__def,axiom,
    ! [A: $o,B4: set_o] : insert_o((A),B4) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,collect_o(aa_o_fun_o_o(aTP_Lamm_lh,(A)))),B4) ).

% insert_def
tff(fact_7121_insert__def,axiom,
    ! [A: list_nat,B4: set_list_nat] : insert_list_nat(A,B4) = sup_sup_set_list_nat(collect_list_nat(aTP_Lamm_ld(A)),B4) ).

% insert_def
tff(fact_7122_insert__def,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] : insert8897473484851387113nt_int(A,B4) = sup_su2047564715030645325nt_int(collec5210948495886036740nt_int(aTP_Lamm_le(A)),B4) ).

% insert_def
tff(fact_7123_insert__def,axiom,
    ! [A: set_nat,B4: set_set_nat] : insert_set_nat(A,B4) = sup_sup_set_set_nat(collect_set_nat(aTP_Lamm_lf(A)),B4) ).

% insert_def
tff(fact_7124_insert__def,axiom,
    ! [A: int,B4: set_int] : insert_int(A,B4) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,collect_int(aa_int_fun_int_o(aTP_Lamm_lj,A))),B4) ).

% insert_def
tff(fact_7125_insert__def,axiom,
    ! [A: produc4166570645942440679at_nat,B4: set_Pr8551490117392284871at_nat] : insert6337962749363155127at_nat(A,B4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,collec5204685387357076818at_nat(aTP_Lamm_mq(A))),B4) ).

% insert_def
tff(fact_7126_insert__def,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,B4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,collec6321179662152712658at_nat(aTP_Lamm_lc(A))),B4) ).

% insert_def
tff(fact_7127_insert__def,axiom,
    ! [A: nat,B4: set_nat] : insert_nat(A,B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_li,A))),B4) ).

% insert_def
tff(fact_7128_subset__singleton__iff,axiom,
    ! [X6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(X6),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))
    <=> ( ( X6 = bot_bo228742789529271731at_nat )
        | ( X6 = insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_7129_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X6),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))
    <=> ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6 = insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_7130_subset__singleton__iff,axiom,
    ! [X6: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X6),insert_o((A),bot_bot_set_o))
    <=> ( ( X6 = bot_bot_set_o )
        | ( X6 = insert_o((A),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_7131_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X6),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_7132_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X6),insert_int(A,bot_bot_set_int))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_7133_subset__singletonD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))
     => ( ( A4 = bot_bo228742789529271731at_nat )
        | ( A4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_7134_subset__singletonD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))
     => ( ( A4 = bot_bo2099793752762293965at_nat )
        | ( A4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_7135_subset__singletonD,axiom,
    ! [A4: set_o,X: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X),bot_bot_set_o))
     => ( ( A4 = bot_bot_set_o )
        | ( A4 = insert_o((X),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_7136_subset__singletonD,axiom,
    ! [A4: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(X,bot_bot_set_nat))
     => ( ( A4 = bot_bot_set_nat )
        | ( A4 = insert_nat(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_7137_subset__singletonD,axiom,
    ! [A4: set_int,X: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(X,bot_bot_set_int))
     => ( ( A4 = bot_bot_set_int )
        | ( A4 = insert_int(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_7138_insert__is__Un,axiom,
    ! [A: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat] : insert6337962749363155127at_nat(A,A4) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,insert6337962749363155127at_nat(A,bot_bo8422036546324065075at_nat)),A4) ).

% insert_is_Un
tff(fact_7139_insert__is__Un,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A,A4) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)),A4) ).

% insert_is_Un
tff(fact_7140_insert__is__Un,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,A4) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),A4) ).

% insert_is_Un
tff(fact_7141_insert__is__Un,axiom,
    ! [A: $o,A4: set_o] : insert_o((A),A4) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,insert_o((A),bot_bot_set_o)),A4) ).

% insert_is_Un
tff(fact_7142_insert__is__Un,axiom,
    ! [A: nat,A4: set_nat] : insert_nat(A,A4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat(A,bot_bot_set_nat)),A4) ).

% insert_is_Un
tff(fact_7143_insert__is__Un,axiom,
    ! [A: int,A4: set_int] : insert_int(A,A4) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int(A,bot_bot_set_int)),A4) ).

% insert_is_Un
tff(fact_7144_Un__singleton__iff,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,X: produc4166570645942440679at_nat] :
      ( ( aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
    <=> ( ( ( A4 = bot_bo8422036546324065075at_nat )
          & ( B4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) )
        | ( ( A4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B4 = bot_bo8422036546324065075at_nat ) )
        | ( ( A4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_7145_Un__singleton__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
    <=> ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_7146_Un__singleton__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
    <=> ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_7147_Un__singleton__iff,axiom,
    ! [A4: set_o,B4: set_o,X: $o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4) = insert_o((X),bot_bot_set_o) )
    <=> ( ( ( A4 = bot_bot_set_o )
          & ( B4 = insert_o((X),bot_bot_set_o) ) )
        | ( ( A4 = insert_o((X),bot_bot_set_o) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A4 = insert_o((X),bot_bot_set_o) )
          & ( B4 = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_7148_Un__singleton__iff,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) = insert_nat(X,bot_bot_set_nat) )
    <=> ( ( ( A4 = bot_bot_set_nat )
          & ( B4 = insert_nat(X,bot_bot_set_nat) ) )
        | ( ( A4 = insert_nat(X,bot_bot_set_nat) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A4 = insert_nat(X,bot_bot_set_nat) )
          & ( B4 = insert_nat(X,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_7149_Un__singleton__iff,axiom,
    ! [A4: set_int,B4: set_int,X: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) = insert_int(X,bot_bot_set_int) )
    <=> ( ( ( A4 = bot_bot_set_int )
          & ( B4 = insert_int(X,bot_bot_set_int) ) )
        | ( ( A4 = insert_int(X,bot_bot_set_int) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A4 = insert_int(X,bot_bot_set_int) )
          & ( B4 = insert_int(X,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_7150_singleton__Un__iff,axiom,
    ! [X: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
      ( ( insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bo8422036546324065075at_nat )
          & ( B4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) )
        | ( ( A4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B4 = bot_bo8422036546324065075at_nat ) )
        | ( ( A4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) )
          & ( B4 = insert6337962749363155127at_nat(X,bot_bo8422036546324065075at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_7151_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B4 = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_7152_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B4 = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_7153_singleton__Un__iff,axiom,
    ! [X: $o,A4: set_o,B4: set_o] :
      ( ( insert_o((X),bot_bot_set_o) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4) )
    <=> ( ( ( A4 = bot_bot_set_o )
          & ( B4 = insert_o((X),bot_bot_set_o) ) )
        | ( ( A4 = insert_o((X),bot_bot_set_o) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A4 = insert_o((X),bot_bot_set_o) )
          & ( B4 = insert_o((X),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_7154_singleton__Un__iff,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ( insert_nat(X,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4) )
    <=> ( ( ( A4 = bot_bot_set_nat )
          & ( B4 = insert_nat(X,bot_bot_set_nat) ) )
        | ( ( A4 = insert_nat(X,bot_bot_set_nat) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A4 = insert_nat(X,bot_bot_set_nat) )
          & ( B4 = insert_nat(X,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_7155_singleton__Un__iff,axiom,
    ! [X: int,A4: set_int,B4: set_int] :
      ( ( insert_int(X,bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4) )
    <=> ( ( ( A4 = bot_bot_set_int )
          & ( B4 = insert_int(X,bot_bot_set_int) ) )
        | ( ( A4 = insert_int(X,bot_bot_set_int) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A4 = insert_int(X,bot_bot_set_int) )
          & ( B4 = insert_int(X,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_7156_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X6: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),X6)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,X6),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7157_set__minus__singleton__eq,axiom,
    ! [X: set_Pr958786334691620121nt_int,X6: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),X6)
     => ( minus_2612819937483484256nt_int(X6,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7158_set__minus__singleton__eq,axiom,
    ! [X: set_nat,X6: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),X6)
     => ( minus_2163939370556025621et_nat(X6,insert_set_nat(X,bot_bot_set_set_nat)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7159_set__minus__singleton__eq,axiom,
    ! [X: $o,X6: set_o] :
      ( ~ aa_set_o_o(member_o((X)),X6)
     => ( minus_minus_set_o(X6,insert_o((X),bot_bot_set_o)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7160_set__minus__singleton__eq,axiom,
    ! [X: nat,X6: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),X6)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X6),insert_nat(X,bot_bot_set_nat)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7161_set__minus__singleton__eq,axiom,
    ! [X: int,X6: set_int] :
      ( ~ aa_set_int_o(member_int(X),X6)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,X6),insert_int(X,bot_bot_set_int)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7162_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),X6)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X6),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_7163_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A4)),insert9069300056098147895at_nat(Y,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(Y,bot_bo228742789529271731at_nat))) ) ) ).

% insert_minus_eq
tff(fact_7164_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A4: set_o] :
      ( ~ ( (X)
        <=> (Y) )
     => ( minus_minus_set_o(insert_o((X),A4),insert_o((Y),bot_bot_set_o)) = insert_o((X),minus_minus_set_o(A4,insert_o((Y),bot_bot_set_o))) ) ) ).

% insert_minus_eq
tff(fact_7165_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( X != Y )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A4)),insert_nat(Y,bot_bot_set_nat)) = insert_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(Y,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_7166_insert__minus__eq,axiom,
    ! [X: int,Y: int,A4: set_int] :
      ( ( X != Y )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A4)),insert_int(Y,bot_bot_set_int)) = insert_int(X,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(Y,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_7167_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A4)),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat))) ) ) ).

% insert_minus_eq
tff(fact_7168_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A4)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7169_Diff__insert__absorb,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4)
     => ( minus_2612819937483484256nt_int(insert8897473484851387113nt_int(X,A4),insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7170_Diff__insert__absorb,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(X),A4)
     => ( minus_2163939370556025621et_nat(insert_set_nat(X,A4),insert_set_nat(X,bot_bot_set_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7171_Diff__insert__absorb,axiom,
    ! [X: $o,A4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A4)
     => ( minus_minus_set_o(insert_o((X),A4),insert_o((X),bot_bot_set_o)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7172_Diff__insert__absorb,axiom,
    ! [X: nat,A4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat(X,A4)),insert_nat(X,bot_bot_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7173_Diff__insert__absorb,axiom,
    ! [X: int,A4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A4)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,insert_int(X,A4)),insert_int(X,bot_bot_set_int)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7174_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A4)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_7175_Diff__insert2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))),B4) ).

% Diff_insert2
tff(fact_7176_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : minus_minus_set_o(A4,insert_o((A),B4)) = minus_minus_set_o(minus_minus_set_o(A4,insert_o((A),bot_bot_set_o)),B4) ).

% Diff_insert2
tff(fact_7177_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))),B4) ).

% Diff_insert2
tff(fact_7178_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(A,B4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(A,bot_bot_set_int))),B4) ).

% Diff_insert2
tff(fact_7179_Diff__insert2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))),B4) ).

% Diff_insert2
tff(fact_7180_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( insert9069300056098147895at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_7181_insert__Diff,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
     => ( insert8897473484851387113nt_int(A,minus_2612819937483484256nt_int(A4,insert8897473484851387113nt_int(A,bot_bo1488462491386950373nt_int))) = A4 ) ) ).

% insert_Diff
tff(fact_7182_insert__Diff,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ( insert_set_nat(A,minus_2163939370556025621et_nat(A4,insert_set_nat(A,bot_bot_set_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_7183_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( insert_o((A),minus_minus_set_o(A4,insert_o((A),bot_bot_set_o))) = A4 ) ) ).

% insert_Diff
tff(fact_7184_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( insert_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,bot_bot_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_7185_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( insert_int(A,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(A,bot_bot_set_int))) = A4 ) ) ).

% insert_Diff
tff(fact_7186_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_7187_Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(A,B4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),B4)),insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% Diff_insert
tff(fact_7188_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : minus_minus_set_o(A4,insert_o((A),B4)) = minus_minus_set_o(minus_minus_set_o(A4,B4),insert_o((A),bot_bot_set_o)) ).

% Diff_insert
tff(fact_7189_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)),insert_nat(A,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_7190_Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(A,B4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)),insert_int(A,bot_bot_set_int)) ).

% Diff_insert
tff(fact_7191_Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(A,B4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_7192_subset__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),insert9069300056098147895at_nat(X,C3)))
    <=> ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B4),C3))
        & ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4) ) ) ).

% subset_Diff_insert
tff(fact_7193_subset__Diff__insert,axiom,
    ! [A4: set_o,B4: set_o,X: $o,C3: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),minus_minus_set_o(B4,insert_o((X),C3)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A4),minus_minus_set_o(B4,C3))
        & ~ aa_set_o_o(member_o((X)),A4) ) ) ).

% subset_Diff_insert
tff(fact_7194_subset__Diff__insert,axiom,
    ! [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,C3: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),minus_2612819937483484256nt_int(B4,insert8897473484851387113nt_int(X,C3)))
    <=> ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),minus_2612819937483484256nt_int(B4,C3))
        & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4) ) ) ).

% subset_Diff_insert
tff(fact_7195_subset__Diff__insert,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,X: set_nat,C3: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),minus_2163939370556025621et_nat(B4,insert_set_nat(X,C3)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),minus_2163939370556025621et_nat(B4,C3))
        & ~ aa_set_set_nat_o(member_set_nat(X),A4) ) ) ).

% subset_Diff_insert
tff(fact_7196_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),insert_nat(X,C3)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),C3))
        & ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% subset_Diff_insert
tff(fact_7197_subset__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),insert8211810215607154385at_nat(X,C3)))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),C3))
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4) ) ) ).

% subset_Diff_insert
tff(fact_7198_subset__Diff__insert,axiom,
    ! [A4: set_int,B4: set_int,X: int,C3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),insert_int(X,C3)))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),C3))
        & ~ aa_set_int_o(member_int(X),A4) ) ) ).

% subset_Diff_insert
tff(fact_7199_subset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X,B4))
    <=> $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B4),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)) ) ).

% subset_insert_iff
tff(fact_7200_subset__insert__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),insert8897473484851387113nt_int(X,B4))
    <=> $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4),aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(minus_2612819937483484256nt_int(A4,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))),B4),aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)) ) ).

% subset_insert_iff
tff(fact_7201_subset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),insert_set_nat(X,B4))
    <=> $ite(aa_set_set_nat_o(member_set_nat(X),A4),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat))),B4),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)) ) ).

% subset_insert_iff
tff(fact_7202_subset__insert__iff,axiom,
    ! [A4: set_o,X: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X),B4))
    <=> $ite(aa_set_o_o(member_o((X)),A4),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A4,insert_o((X),bot_bot_set_o))),B4),aa_set_o_o(ord_less_eq_set_o(A4),B4)) ) ).

% subset_insert_iff
tff(fact_7203_subset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(X,B4))
    <=> $ite(aa_set_nat_o(member_nat(X),A4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat))),B4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)) ) ).

% subset_insert_iff
tff(fact_7204_subset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,B4))
    <=> $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)) ) ).

% subset_insert_iff
tff(fact_7205_subset__insert__iff,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(X,B4))
    <=> $ite(aa_set_int_o(member_int(X),A4),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int))),B4),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)) ) ).

% subset_insert_iff
tff(fact_7206_Diff__single__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B4)
     => aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),insert9069300056098147895at_nat(X,B4)) ) ).

% Diff_single_insert
tff(fact_7207_Diff__single__insert,axiom,
    ! [A4: set_o,X: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A4,insert_o((X),bot_bot_set_o))),B4)
     => aa_set_o_o(ord_less_eq_set_o(A4),insert_o((X),B4)) ) ).

% Diff_single_insert
tff(fact_7208_Diff__single__insert,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat))),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),insert_nat(X,B4)) ) ).

% Diff_single_insert
tff(fact_7209_Diff__single__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),insert8211810215607154385at_nat(X,B4)) ) ).

% Diff_single_insert
tff(fact_7210_Diff__single__insert,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int))),B4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),insert_int(X,B4)) ) ).

% Diff_single_insert
tff(fact_7211_remove__subset,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),S)
     => aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),S) ) ).

% remove_subset
tff(fact_7212_remove__subset,axiom,
    ! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),S)
     => aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(minus_2612819937483484256nt_int(S,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))),S) ) ).

% remove_subset
tff(fact_7213_remove__subset,axiom,
    ! [X: set_nat,S: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(X),S)
     => aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(S,insert_set_nat(X,bot_bot_set_set_nat))),S) ) ).

% remove_subset
tff(fact_7214_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => aa_set_o_o(ord_less_set_o(minus_minus_set_o(S,insert_o((X),bot_bot_set_o))),S) ) ).

% remove_subset
tff(fact_7215_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(X),S)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(X,bot_bot_set_nat))),S) ) ).

% remove_subset
tff(fact_7216_remove__subset,axiom,
    ! [X: int,S: set_int] :
      ( aa_set_int_o(member_int(X),S)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(X,bot_bot_set_int))),S) ) ).

% remove_subset
tff(fact_7217_remove__subset,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),S) ) ).

% remove_subset
tff(fact_7218_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : uminus935396558254630718at_nat(insert9069300056098147895at_nat(X,A4)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,uminus935396558254630718at_nat(A4)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% Compl_insert
tff(fact_7219_Compl__insert,axiom,
    ! [X: $o,A4: set_o] : uminus_uminus_set_o(insert_o((X),A4)) = minus_minus_set_o(uminus_uminus_set_o(A4),insert_o((X),bot_bot_set_o)) ).

% Compl_insert
tff(fact_7220_Compl__insert,axiom,
    ! [X: nat,A4: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,uminus5710092332889474511et_nat(A4)),insert_nat(X,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_7221_Compl__insert,axiom,
    ! [X: int,A4: set_int] : uminus1532241313380277803et_int(insert_int(X,A4)) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,uminus1532241313380277803et_int(A4)),insert_int(X,bot_bot_set_int)) ).

% Compl_insert
tff(fact_7222_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X,A4)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,uminus6524753893492686040at_nat(A4)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% Compl_insert
tff(fact_7223_psubset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A4),insert9069300056098147895at_nat(X,B4))
    <=> $ite(
          aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B4),
          aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(A4),B4),
          $ite(aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4),aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A4),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))),B4),aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7224_psubset__insert__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A4),insert8897473484851387113nt_int(X,B4))
    <=> $ite(
          aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),B4),
          aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(A4),B4),
          $ite(aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),A4),aa_set6820988702716878619_int_o(ord_le1924305788584680229nt_int(minus_2612819937483484256nt_int(A4,insert8897473484851387113nt_int(X,bot_bo1488462491386950373nt_int))),B4),aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7225_psubset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),insert_set_nat(X,B4))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(X),B4),
          aa_set_set_nat_o(ord_less_set_set_nat(A4),B4),
          $ite(aa_set_set_nat_o(member_set_nat(X),A4),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A4,insert_set_nat(X,bot_bot_set_set_nat))),B4),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7226_psubset__insert__iff,axiom,
    ! [A4: set_o,X: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),insert_o((X),B4))
    <=> $ite(
          aa_set_o_o(member_o((X)),B4),
          aa_set_o_o(ord_less_set_o(A4),B4),
          $ite(aa_set_o_o(member_o((X)),A4),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A4,insert_o((X),bot_bot_set_o))),B4),aa_set_o_o(ord_less_eq_set_o(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7227_psubset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),insert_nat(X,B4))
    <=> $ite(
          aa_set_nat_o(member_nat(X),B4),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4),
          $ite(aa_set_nat_o(member_nat(X),A4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat))),B4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7228_psubset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),insert8211810215607154385at_nat(X,B4))
    <=> $ite(
          aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B4),
          aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),B4),
          $ite(aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B4),aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7229_psubset__insert__iff,axiom,
    ! [A4: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),insert_int(X,B4))
    <=> $ite(
          aa_set_int_o(member_int(X),B4),
          aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4),
          $ite(aa_set_int_o(member_int(X),A4),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int))),B4),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7230_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),aa_num_num(bit0,N2)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_mr),aa_num_option_num(bit_and_not_num(M),N2)) ).

% and_not_num.simps(8)
tff(fact_7231_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_ms),Qr) ).

% Divides.adjust_div_def
tff(fact_7232_and__nat__rec,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N2) = 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))),N2) ))),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_7233_and__nat__unfold,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N2) = $ite(
        ( ( M = zero_zero_nat )
        | ( N2 = 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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_7234_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xa),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))),X)
                      & ~ 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_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))),X)
                        & ~ 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_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,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),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,X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_7235_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
            ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(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))),K2)
                & ~ 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))),K2)
                  & ~ 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,K2),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_7236_and__minus__numerals_I7_J,axiom,
    ! [N2: 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,N2)))),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,N2))) ).

% and_minus_numerals(7)
tff(fact_7237_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: 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,N2)))) = 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,N2))) ).

% and_minus_numerals(3)
tff(fact_7238_int__ge__less__than2__def,axiom,
    ! [D2: int] : aa_int8519581412571446831nt_int(int_ge_less_than2,D2) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_mt(D2))) ).

% int_ge_less_than2_def
tff(fact_7239_trancl__single,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : transi2703068831062848130at_nat(insert9069300056098147895at_nat(produc2922128104949294807at_nat(A,B),bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(produc2922128104949294807at_nat(A,B),bot_bo228742789529271731at_nat) ).

% trancl_single
tff(fact_7240_trancl__single,axiom,
    ! [A: int,B: int] : transi6261509568448316235cl_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),bot_bo1796632182523588997nt_int)) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),bot_bo1796632182523588997nt_int) ).

% trancl_single
tff(fact_7241_trancl__single,axiom,
    ! [A: nat,B: nat] : transi6264000038957366511cl_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),bot_bo2099793752762293965at_nat) ).

% trancl_single
tff(fact_7242_trancl__sub__insert__trancl,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(transi2703068831062848130at_nat(Ra)),transi2703068831062848130at_nat(insert9069300056098147895at_nat(X,Ra))) ).

% trancl_sub_insert_trancl
tff(fact_7243_trancl__sub__insert__trancl,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,transi6264000038957366511cl_nat(Ra)),transi6264000038957366511cl_nat(insert8211810215607154385at_nat(X,Ra))) ).

% trancl_sub_insert_trancl
tff(fact_7244_one__plus__BitM,axiom,
    ! [N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,N2)) = aa_num_num(bit0,N2) ).

% one_plus_BitM
tff(fact_7245_BitM__plus__one,axiom,
    ! [N2: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,N2)),one) = aa_num_num(bit0,N2) ).

% BitM_plus_one
tff(fact_7246_trancl__insert2,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,R: set_Pr8551490117392284871at_nat] : transi5749863603491437800at_nat(insert6337962749363155127at_nat(produc4348348721325984599at_nat(A,B),R)) = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,transi5749863603491437800at_nat(R)),collec5204685387357076818at_nat(produc5410597840814988588_nat_o(aa_set6182894787541909731_nat_o(aa_mul6910776717834277237_nat_o(aTP_Lamm_mu(A),B),R)))) ).

% trancl_insert2
tff(fact_7247_trancl__insert2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] : transi2703068831062848130at_nat(insert9069300056098147895at_nat(produc2922128104949294807at_nat(A,B),R)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,transi2703068831062848130at_nat(R)),collec6321179662152712658at_nat(produc410239310623530412_nat_o(aa_set5605188905385065315_nat_o(aa_set3198043947577432463_nat_o(aTP_Lamm_mv(A),B),R)))) ).

% trancl_insert2
tff(fact_7248_trancl__insert2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] : transi6264000038957366511cl_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),R)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,transi6264000038957366511cl_nat(R)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aa_set2634951030311766213_nat_o(aa_nat5178545565753023578_nat_o(aTP_Lamm_mw(A),B),R)))) ).

% trancl_insert2
tff(fact_7249_trancl__insert2,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] : transi6261509568448316235cl_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),R)) = sup_su6024340866399070445nt_int(transi6261509568448316235cl_int(R),collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_set742265667465749333_int_o(aa_int2891257317550437190_int_o(aTP_Lamm_mx(A),B),R)))) ).

% trancl_insert2
tff(fact_7250_int__ge__less__than__def,axiom,
    ! [D2: int] : aa_int8519581412571446831nt_int(int_ge_less_than,D2) = collec213857154873943460nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_my(D2))) ).

% int_ge_less_than_def
tff(fact_7251_log_Osimps,axiom,
    ! [B: code_natural,I2: code_natural] :
      log(B,I2) = $ite(
        ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),one_one_Code_natural)
        | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,I2),B) ),
        one_one_Code_natural,
        aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(B,divide5121882707175180666atural(I2,B))) ) ).

% log.simps
tff(fact_7252_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( Y = $ite(
            ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),one_one_Code_natural)
            | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa),X) ),
            one_one_Code_natural,
            aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,divide5121882707175180666atural(Xa,X))) ) ) ) ).

% log.elims
tff(fact_7253_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa))
       => ~ ( ( Y = $ite(
                  ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),one_one_Code_natural)
                  | aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,Xa),X) ),
                  one_one_Code_natural,
                  aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,divide5121882707175180666atural(Xa,X))) ) )
           => ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa)) ) ) ) ).

% log.pelims
tff(fact_7254_minus__shift__def,axiom,
    ! [R: code_natural,K2: code_natural,L: code_natural] :
      minus_shift(R,K2,L) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,K2),L),minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,R),K2),L),minus_7197305767214868737atural(K2,L)) ).

% minus_shift_def
tff(fact_7255_next_Osimps,axiom,
    ! [V: code_natural,W2: code_natural] :
      aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W2)) = $let(
        v: code_natural,
        v:= minus_shift(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(V,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(V,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),
        $let(
          w2: code_natural,
          w2:= minus_shift(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(W2,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(W2,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))),
          aa_Pro7165575656943818355atural(aa_Cod6352003912080677724atural(produc6639722614265839536atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,minus_shift(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),v,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,w2),one_one_Code_natural))),one_one_Code_natural)),produc3574140220909816553atural(v,w2)) ) ) ).

% next.simps
tff(fact_7256_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mz,X)) ).

% set_decode_def
tff(fact_7257_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z3: nat] :
      ( ~ aa_set_nat_o(member_nat(N2),nat_set_decode(Z3))
     => ( 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))),N2)),Z3)) = insert_nat(N2,nat_set_decode(Z3)) ) ) ).

% set_decode_plus_power_2
tff(fact_7258_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,nat_set_decode(M)),nat_set_decode(N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% subset_decode_imp_le
tff(fact_7259_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)) ).

% take_bit_Suc_from_most
tff(fact_7260_Random_Orange__def,axiom,
    ! [K2: code_natural] : aa_Cod6352003912080677724atural(range,K2) = produc5538323210962509403atural(aa_Cod6352003912080677724atural(iterat8892046348760725948atural(log(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),K2),aTP_Lamm_nb),one_one_Code_natural),aTP_Lamm_nc(K2)) ).

% Random.range_def
tff(fact_7261_xor__Suc__0__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N2),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,N2),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))),N2)))),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))),N2))) ).

% xor_Suc_0_eq
tff(fact_7262_Suc__0__xor__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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))),N2)))),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))),N2))) ).

% Suc_0_xor_eq
tff(fact_7263_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_7264_scomp__apply,axiom,
    ! [Fa: fun_Pr7769407664385210434atural,Ga: fun_Co1105491363547889651atural,X: produc7822875418678951345atural] : aa_Pro7165575656943818355atural(produc5538323210962509403atural(Fa,Ga),X) = produc8638916746724166107atural(Ga,aa_Pro7165575656943818355atural(Fa,X)) ).

% scomp_apply
tff(fact_7265_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_7266_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),K2)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2) ) ).

% signed_take_bit_negative_iff
tff(fact_7267_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% xor_numerals(4)
tff(fact_7268_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = 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,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% xor_numerals(4)
tff(fact_7269_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_7270_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(4)
tff(fact_7271_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se2046307713759805098atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% xor_numerals(6)
tff(fact_7272_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = 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,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% xor_numerals(6)
tff(fact_7273_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_7274_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(6)
tff(fact_7275_scomp__scomp,axiom,
    ! [Fa: fun_Pr7769407664385210434atural,Ga: fun_Co1105491363547889651atural,Ha: fun_Co1105491363547889651atural] : produc5538323210962509403atural(produc5538323210962509403atural(Fa,Ga),Ha) = produc5538323210962509403atural(Fa,aa_fun5568814847722588540atural(aTP_Lamm_nd(Ga),Ha)) ).

% scomp_scomp
tff(fact_7276_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [N5: nat] :
          ( ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N5)
          | ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),N5) )
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N2)
      <=> ( 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) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_7277_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [N5: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N5)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N2)
      <=> ( 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) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_7278_scomp__Pair,axiom,
    ! [X: fun_Pr7769407664385210434atural] : produc5538323210962509403atural(X,produc6639722614265839536atural) = X ).

% scomp_Pair
tff(fact_7279_Pair__scomp,axiom,
    ! [X: code_natural,Fa: fun_Co1105491363547889651atural] : produc5538323210962509403atural(aa_Cod6352003912080677724atural(produc6639722614265839536atural,X),Fa) = aa_Cod6352003912080677724atural(Fa,X) ).

% Pair_scomp
tff(fact_7280_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),A)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N2) ) ) ).

% bit_take_bit_iff
tff(fact_7281_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N2) ) ) ).

% bit_take_bit_iff
tff(fact_7282_scomp__def,axiom,
    ! [Fa: fun_Pr7769407664385210434atural,Ga: fun_Co1105491363547889651atural,X5: produc7822875418678951345atural] : aa_Pro7165575656943818355atural(produc5538323210962509403atural(Fa,Ga),X5) = produc8638916746724166107atural(Ga,aa_Pro7165575656943818355atural(Fa,X5)) ).

% scomp_def
tff(fact_7283_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(groups2241214984954800037atural(zero_n8403883297036319079atural,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),Bs)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),N2) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7284_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,groups3417619833198082522nteger(zero_n356916108424825756nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)),Bs)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),N2) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7285_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,groups9119017779487936845_o_nat(zero_n2687167440665602831ol_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)),Bs)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),N2) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7286_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),size_size_list_o(Bs))
        & aa_nat_o(nth_o(Bs),N2) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_7287_iterate_Osimps,axiom,
    ! [K2: code_natural,Fa: fun_in1696471567231710611nt_int,X: int] :
      aa_int2565445551602696444nt_int(iterate_int_int(K2,Fa),X) = $ite(K2 = zero_z2226904508553997617atural,aa_int2565445551602696444nt_int(product_Pair_int_int,X),produc6963641260316735276nt_int(aa_int2565445551602696444nt_int(Fa,X),iterate_int_int(minus_7197305767214868737atural(K2,one_one_Code_natural),Fa))) ).

% iterate.simps
tff(fact_7288_iterate_Osimps,axiom,
    ! [K2: code_natural,Fa: fun_fu2265384587115734689et_nat,X: fun_Pr6460944699146269935_nat_o] :
      aa_fun3926903090414666066et_nat(iterat3566508888001595302et_nat(K2,Fa),X) = $ite(K2 = zero_z2226904508553997617atural,aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),produc8910761370405222282et_nat(aa_fun3926903090414666066et_nat(Fa,X),iterat3566508888001595302et_nat(minus_7197305767214868737atural(K2,one_one_Code_natural),Fa))) ).

% iterate.simps
tff(fact_7289_iterate_Osimps,axiom,
    ! [K2: code_natural,Fa: fun_fu3932451362284101085et_nat,X: fun_Pr6460944699146269935_nat_o] :
      aa_fun615918787490101902et_nat(iterat7598301416969415364et_nat(K2,Fa),X) = $ite(K2 = zero_z2226904508553997617atural,aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),produc8975870419876881960et_nat(aa_fun615918787490101902et_nat(Fa,X),iterat7598301416969415364et_nat(minus_7197305767214868737atural(K2,one_one_Code_natural),Fa))) ).

% iterate.simps
tff(fact_7290_iterate_Osimps,axiom,
    ! [K2: code_natural,Fa: fun_b_6809638951405183189it_nat,X: b] :
      aa_b_f1993928172615730054it_nat(iterat9120928542346325760it_nat(K2,Fa),X) = $ite(K2 = zero_z2226904508553997617atural,aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),produc8824551317683396724it_nat(aa_b_f1993928172615730054it_nat(Fa,X),iterat9120928542346325760it_nat(minus_7197305767214868737atural(K2,one_one_Code_natural),Fa))) ).

% iterate.simps
tff(fact_7291_iterate_Osimps,axiom,
    ! [K2: code_natural,Fa: fun_a_576316402698418899it_nat,X: a] :
      aa_a_f2234720956197727492it_nat(iterat4993027441371875583it_nat(K2,Fa),X) = $ite(K2 = zero_z2226904508553997617atural,aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),produc3714732129343117170it_nat(aa_a_f2234720956197727492it_nat(Fa,X),iterat4993027441371875583it_nat(minus_7197305767214868737atural(K2,one_one_Code_natural),Fa))) ).

% iterate.simps
tff(fact_7292_iterate_Osimps,axiom,
    ! [K2: code_natural,Fa: fun_Co1105491363547889651atural,X: code_natural] :
      aa_Cod6352003912080677724atural(iterat8892046348760725948atural(K2,Fa),X) = $ite(K2 = zero_z2226904508553997617atural,aa_Cod6352003912080677724atural(produc6639722614265839536atural,X),produc5538323210962509403atural(aa_Cod6352003912080677724atural(Fa,X),iterat8892046348760725948atural(minus_7197305767214868737atural(K2,one_one_Code_natural),Fa))) ).

% iterate.simps
tff(fact_7293_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: fun_in1696471567231710611nt_int,Xb: int,Y: fun_in4211310926729683942nt_int] :
      ( ( aa_int2565445551602696444nt_int(iterate_int_int(X,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_z2226904508553997617atural,aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),produc6963641260316735276nt_int(aa_int2565445551602696444nt_int(Xa,Xb),iterate_int_int(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) ) ) ).

% iterate.elims
tff(fact_7294_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: fun_fu2265384587115734689et_nat,Xb: fun_Pr6460944699146269935_nat_o,Y: fun_Pr4885567672785483807et_nat] :
      ( ( aa_fun3926903090414666066et_nat(iterat3566508888001595302et_nat(X,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_z2226904508553997617atural,aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xb),produc8910761370405222282et_nat(aa_fun3926903090414666066et_nat(Xa,Xb),iterat3566508888001595302et_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) ) ) ).

% iterate.elims
tff(fact_7295_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: fun_fu3932451362284101085et_nat,Xb: fun_Pr6460944699146269935_nat_o,Y: fun_Pr3978243888342905947et_nat] :
      ( ( aa_fun615918787490101902et_nat(iterat7598301416969415364et_nat(X,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_z2226904508553997617atural,aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Xb),produc8975870419876881960et_nat(aa_fun615918787490101902et_nat(Xa,Xb),iterat7598301416969415364et_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) ) ) ).

% iterate.elims
tff(fact_7296_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: fun_b_6809638951405183189it_nat,Xb: b,Y: fun_Pr3216343399241953091it_nat] :
      ( ( aa_b_f1993928172615730054it_nat(iterat9120928542346325760it_nat(X,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_z2226904508553997617atural,aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Xb),produc8824551317683396724it_nat(aa_b_f1993928172615730054it_nat(Xa,Xb),iterat9120928542346325760it_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) ) ) ).

% iterate.elims
tff(fact_7297_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: fun_a_576316402698418899it_nat,Xb: a,Y: fun_Pr2713655712603142594it_nat] :
      ( ( aa_a_f2234720956197727492it_nat(iterat4993027441371875583it_nat(X,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_z2226904508553997617atural,aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Xb),produc3714732129343117170it_nat(aa_a_f2234720956197727492it_nat(Xa,Xb),iterat4993027441371875583it_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) ) ) ).

% iterate.elims
tff(fact_7298_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: fun_Co1105491363547889651atural,Xb: code_natural,Y: fun_Pr7769407664385210434atural] :
      ( ( aa_Cod6352003912080677724atural(iterat8892046348760725948atural(X,Xa),Xb) = Y )
     => ( Y = $ite(X = zero_z2226904508553997617atural,aa_Cod6352003912080677724atural(produc6639722614265839536atural,Xb),produc5538323210962509403atural(aa_Cod6352003912080677724atural(Xa,Xb),iterat8892046348760725948atural(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) ) ) ).

% iterate.elims
tff(fact_7299_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K2)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_7300_bit__concat__bit__iff,axiom,
    ! [M: nat,K2: int,L: int,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K2),L)),N2)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_7301_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N5: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),M4)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),M4)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N5) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N5)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N5),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N5) ) ) ) ).

% int_bit_bound
tff(fact_7302_even__bit__succ__iff,axiom,
    ! [A: code_natural,N2: nat] :
      ( aa_Code_natural_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)),N2)
      <=> ( aa_nat_o(bit_se8040316288895769887atural(A),N2)
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7303_even__bit__succ__iff,axiom,
    ! [A: code_integer,N2: 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_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A)),N2)
      <=> ( aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,A),N2)
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7304_even__bit__succ__iff,axiom,
    ! [A: int,N2: 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)),N2)
      <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N2)
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7305_even__bit__succ__iff,axiom,
    ! [A: nat,N2: 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)),N2)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N2)
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_7306_bit__sum__mult__2__cases,axiom,
    ! [A: code_natural,B: code_natural,N2: nat] :
      ( ! [J3: nat] : ~ aa_nat_o(bit_se8040316288895769887atural(A),aa_nat_nat(suc,J3))
     => ( aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),B))),N2)
      <=> $ite(N2 = zero_zero_nat,~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A),aa_nat_o(bit_se8040316288895769887atural(times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),B)),N2)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7307_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ! [J3: nat] : ~ aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,A),aa_nat_nat(suc,J3))
     => ( aa_nat_o(aa_Cod1075288278386115654_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))),N2)
      <=> $ite(N2 = 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(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),N2)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7308_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [J3: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,J3))
     => ( 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))),N2)
      <=> $ite(N2 = 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)),N2)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7309_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [J3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J3))
     => ( 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))),N2)
      <=> $ite(N2 = 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)),N2)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_7310_xor__nat__unfold,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N2) = $ite(
        M = zero_zero_nat,
        N2,
        $ite(N2 = 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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_7311_xor__nat__rec,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N2) = 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))),N2))),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_7312_xor__one__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(A,one_one_Code_natural) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))),aa_o_Code_natural(zero_n8403883297036319079atural,~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_7313_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),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_7314_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_7315_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_7316_one__xor__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(one_one_Code_natural,A) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))),aa_o_Code_natural(zero_n8403883297036319079atural,~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_7317_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),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)))),aa_o_Code_integer(zero_n356916108424825756nteger,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_7318_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_7319_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_7320_set__bit__eq,axiom,
    ! [N2: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,N2),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ).

% set_bit_eq
tff(fact_7321_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_int_rat,A: rat,X: int,Xs: list_int] : groups7852591826665563233nt_rat(Fa,A,cons_int(X,Xs)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Fa,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),groups7852591826665563233nt_rat(Fa,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_7322_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_nat_rat,A: rat,X: nat,Xs: list_nat] : aa_list_nat_rat(groups6853238114764508677at_rat(Fa,A),cons_nat(X,Xs)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Fa,X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_list_nat_rat(groups6853238114764508677at_rat(Fa,A),Xs))) ).

% horner_sum_simps(2)
tff(fact_7323_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_int_nat,A: nat,X: int,Xs: list_int] : groups8487721886752058969nt_nat(Fa,A,cons_int(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Fa,X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),groups8487721886752058969nt_nat(Fa,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_7324_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_nat_nat,A: nat,X: nat,Xs: list_nat] : aa_list_nat_nat(groups7488368174851004413at_nat(Fa,A),cons_nat(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Fa,X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_list_nat_nat(groups7488368174851004413at_nat(Fa,A),Xs))) ).

% horner_sum_simps(2)
tff(fact_7325_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_int_int,A: int,X: int,Xs: list_int] : groups8485231416243008693nt_int(Fa,A,cons_int(X,Xs)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Fa,X)),aa_int_int(aa_int_fun_int_int(times_times_int,A),groups8485231416243008693nt_int(Fa,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_7326_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_nat_int,A: int,X: nat,Xs: list_nat] : groups7485877704341954137at_int(Fa,A,cons_nat(X,Xs)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Fa,X)),aa_int_int(aa_int_fun_int_int(times_times_int,A),groups7485877704341954137at_int(Fa,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_7327_horner__sum__simps_I2_J,axiom,
    ! [Fa: fun_o_int,A: int,X: $o,Xs: list_o] : groups9116527308978886569_o_int(Fa,A,cons_o((X),Xs)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(Fa,(X))),aa_int_int(aa_int_fun_int_int(times_times_int,A),groups9116527308978886569_o_int(Fa,A,Xs))) ).

% horner_sum_simps(2)
tff(fact_7328_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: fun_in1696471567231710611nt_int,Xb: int,Y: fun_in4211310926729683942nt_int] :
      ( ( aa_int2565445551602696444nt_int(iterate_int_int(X,Xa),Xb) = Y )
     => ( accp_P4660419946791220096nt_int(iterate_rel_int_int,produc6271331564418763055nt_int(X,produc8498902281960182888nt_int(Xa,Xb)))
       => ~ ( ( Y = $ite(X = zero_z2226904508553997617atural,aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),produc6963641260316735276nt_int(aa_int2565445551602696444nt_int(Xa,Xb),iterate_int_int(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) )
           => ~ accp_P4660419946791220096nt_int(iterate_rel_int_int,produc6271331564418763055nt_int(X,produc8498902281960182888nt_int(Xa,Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7329_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: fun_fu2265384587115734689et_nat,Xb: fun_Pr6460944699146269935_nat_o,Y: fun_Pr4885567672785483807et_nat] :
      ( ( aa_fun3926903090414666066et_nat(iterat3566508888001595302et_nat(X,Xa),Xb) = Y )
     => ( accp_P6427617220533653561_nat_o(iterat2110091641442184387et_nat,produc5881189497046062388_nat_o(X,produc3883715688632579877_nat_o(Xa,Xb)))
       => ~ ( ( Y = $ite(X = zero_z2226904508553997617atural,aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Xb),produc8910761370405222282et_nat(aa_fun3926903090414666066et_nat(Xa,Xb),iterat3566508888001595302et_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) )
           => ~ accp_P6427617220533653561_nat_o(iterat2110091641442184387et_nat,produc5881189497046062388_nat_o(X,produc3883715688632579877_nat_o(Xa,Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7330_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: fun_fu3932451362284101085et_nat,Xb: fun_Pr6460944699146269935_nat_o,Y: fun_Pr3978243888342905947et_nat] :
      ( ( aa_fun615918787490101902et_nat(iterat7598301416969415364et_nat(X,Xa),Xb) = Y )
     => ( accp_P7384312103228075517_nat_o(iterat9053333248564644193et_nat,produc3491284741032513272_nat_o(X,produc9105880373283336425_nat_o(Xa,Xb)))
       => ~ ( ( Y = $ite(X = zero_z2226904508553997617atural,aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Xb),produc8975870419876881960et_nat(aa_fun615918787490101902et_nat(Xa,Xb),iterat7598301416969415364et_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) )
           => ~ accp_P7384312103228075517_nat_o(iterat9053333248564644193et_nat,produc3491284741032513272_nat_o(X,produc9105880373283336425_nat_o(Xa,Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7331_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: fun_b_6809638951405183189it_nat,Xb: b,Y: fun_Pr3216343399241953091it_nat] :
      ( ( aa_b_f1993928172615730054it_nat(iterat9120928542346325760it_nat(X,Xa),Xb) = Y )
     => ( accp_P5700246227841468181_nat_b(iterat7179843830775474589it_nat,produc2894144817684435216_nat_b(X,produc2559749204529753345_nat_b(Xa,Xb)))
       => ~ ( ( Y = $ite(X = zero_z2226904508553997617atural,aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Xb),produc8824551317683396724it_nat(aa_b_f1993928172615730054it_nat(Xa,Xb),iterat9120928542346325760it_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) )
           => ~ accp_P5700246227841468181_nat_b(iterat7179843830775474589it_nat,produc2894144817684435216_nat_b(X,produc2559749204529753345_nat_b(Xa,Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7332_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: fun_a_576316402698418899it_nat,Xb: a,Y: fun_Pr2713655712603142594it_nat] :
      ( ( aa_a_f2234720956197727492it_nat(iterat4993027441371875583it_nat(X,Xa),Xb) = Y )
     => ( accp_P6489688631332939926_nat_a(iterat3051942729801024412it_nat,produc7847374167541719825_nat_a(X,produc7512978558690266754_nat_a(Xa,Xb)))
       => ~ ( ( Y = $ite(X = zero_z2226904508553997617atural,aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Xb),produc3714732129343117170it_nat(aa_a_f2234720956197727492it_nat(Xa,Xb),iterat4993027441371875583it_nat(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) )
           => ~ accp_P6489688631332939926_nat_a(iterat3051942729801024412it_nat,produc7847374167541719825_nat_a(X,produc7512978558690266754_nat_a(Xa,Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7333_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: fun_Co1105491363547889651atural,Xb: code_natural,Y: fun_Pr7769407664385210434atural] :
      ( ( aa_Cod6352003912080677724atural(iterat8892046348760725948atural(X,Xa),Xb) = Y )
     => ( accp_P4076469161163034844atural(iterat8136814461032266713atural,produc8129436409730272651atural(X,produc231982712012277444atural(Xa,Xb)))
       => ~ ( ( Y = $ite(X = zero_z2226904508553997617atural,aa_Cod6352003912080677724atural(produc6639722614265839536atural,Xb),produc5538323210962509403atural(aa_Cod6352003912080677724atural(Xa,Xb),iterat8892046348760725948atural(minus_7197305767214868737atural(X,one_one_Code_natural),Xa))) )
           => ~ accp_P4076469161163034844atural(iterat8136814461032266713atural,produc8129436409730272651atural(X,produc231982712012277444atural(Xa,Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7334_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] : the_el221668144340439132at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X ).

% the_elem_eq
tff(fact_7335_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] : the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X ).

% the_elem_eq
tff(fact_7336_the__elem__eq,axiom,
    ! [X: $o] :
      ( aa_set_o_o(the_elem_o,insert_o((X),bot_bot_set_o))
    <=> (X) ) ).

% the_elem_eq
tff(fact_7337_the__elem__eq,axiom,
    ! [X: nat] : aa_set_nat_nat(the_elem_nat,insert_nat(X,bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_7338_the__elem__eq,axiom,
    ! [X: int] : the_elem_int(insert_int(X,bot_bot_set_int)) = X ).

% the_elem_eq
tff(fact_7339_inc__shift__def,axiom,
    ! [V: code_natural,K2: code_natural] :
      inc_shift(V,K2) = $ite(V = K2,one_one_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,K2),one_one_Code_natural)) ).

% inc_shift_def
tff(fact_7340_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_jq(L)),Qr) ).

% divmod_step_integer_def
tff(fact_7341_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_7342_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_7343_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_7344_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),L) = L ).

% plus_integer_code(2)
tff(fact_7345_plus__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K2),zero_z3403309356797280102nteger) = K2 ).

% plus_integer_code(1)
tff(fact_7346_sgn__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = $ite(
        K2 = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_7347_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)) ) ) ).

% XOR_lower
tff(fact_7348_bit__nat__iff,axiom,
    ! [K2: int,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K2)),N2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),N2) ) ) ).

% bit_nat_iff
tff(fact_7349_XOR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ) ).

% XOR_upper
tff(fact_7350_xor__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),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))),K2) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_7351_integer__of__int__code,axiom,
    ! [K2: int] :
      aa_int_Code_integer(code_integer_of_int,K2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K2))),
        $ite(
          K2 = 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,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),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_7352_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = $ite(
        K2 = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K2),
          $ite(
            K2 = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_7353_is__singleton__the__elem,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A4)
    <=> ( A4 = insert9069300056098147895at_nat(the_el221668144340439132at_nat(A4),bot_bo228742789529271731at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_7354_is__singleton__the__elem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4)
    <=> ( A4 = insert8211810215607154385at_nat(the_el2281957884133575798at_nat(A4),bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_7355_is__singleton__the__elem,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
    <=> ( A4 = insert_o(aa_set_o_o(the_elem_o,A4),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_7356_is__singleton__the__elem,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ( A4 = insert_nat(aa_set_nat_nat(the_elem_nat,A4),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_7357_is__singleton__the__elem,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
    <=> ( A4 = insert_int(the_elem_int(A4),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_7358_integer__of__num_I3_J,axiom,
    ! [N2: num] :
      code_integer_of_num(aa_num_num(bit1,N2)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N2),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k)),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_7359_take__bit__num__code,axiom,
    ! [N2: nat,M: num] : aa_num_option_num(bit_take_bit_num(N2),M) = produc478579273971653890on_num(aTP_Lamm_nh,product_Pair_nat_num(N2,M)) ).

% take_bit_num_code
tff(fact_7360_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% is_singletonI
tff(fact_7361_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% is_singletonI
tff(fact_7362_is__singletonI,axiom,
    ! [X: $o] : aa_set_o_o(is_singleton_o,insert_o((X),bot_bot_set_o)) ).

% is_singletonI
tff(fact_7363_is__singletonI,axiom,
    ! [X: nat] : aa_set_nat_o(is_singleton_nat,insert_nat(X,bot_bot_set_nat)) ).

% is_singletonI
tff(fact_7364_is__singletonI,axiom,
    ! [X: int] : aa_set_int_o(is_singleton_int,insert_int(X,bot_bot_set_int)) ).

% is_singletonI
tff(fact_7365_case__nat__numeral,axiom,
    ! [A: option_num,Fa: fun_nat_option_num,V: num] : aa_nat_option_num(case_nat_option_num(A,Fa),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_option_num(Fa,aa_num_nat(pred_numeral,V)) ).

% case_nat_numeral
tff(fact_7366_case__nat__numeral,axiom,
    ! [A: nat,Fa: fun_nat_nat,V: num] : aa_nat_nat(case_nat_nat(A,Fa),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(Fa,aa_num_nat(pred_numeral,V)) ).

% case_nat_numeral
tff(fact_7367_case__nat__numeral,axiom,
    ! [A: $o,Fa: fun_nat_o,V: num] :
      ( aa_nat_o(case_nat_o((A),Fa),aa_num_nat(numeral_numeral_nat,V))
    <=> aa_nat_o(Fa,aa_num_nat(pred_numeral,V)) ) ).

% case_nat_numeral
tff(fact_7368_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_7369_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% not_negative_int_iff
tff(fact_7370_case__nat__add__eq__if,axiom,
    ! [A: option_num,Fa: fun_nat_option_num,V: num,N2: nat] : aa_nat_option_num(case_nat_option_num(A,Fa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2)) = aa_nat_option_num(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N2)) ).

% case_nat_add_eq_if
tff(fact_7371_case__nat__add__eq__if,axiom,
    ! [A: nat,Fa: fun_nat_nat,V: num,N2: nat] : aa_nat_nat(case_nat_nat(A,Fa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2)) = aa_nat_nat(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N2)) ).

% case_nat_add_eq_if
tff(fact_7372_case__nat__add__eq__if,axiom,
    ! [A: $o,Fa: fun_nat_o,V: num,N2: nat] :
      ( aa_nat_o(case_nat_o((A),Fa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2))
    <=> aa_nat_o(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(pred_numeral,V)),N2)) ) ).

% case_nat_add_eq_if
tff(fact_7373_abs__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),K2) ).

% abs_integer_code
tff(fact_7374_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_7375_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X) ) ).

% less_integer.abs_eq
tff(fact_7376_nat_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F2: fun_nat_option_num,Nat: nat] : aa_opt5369159329878751634on_num(Ha,aa_nat_option_num(case_nat_option_num(F1,F2),Nat)) = aa_nat_option_num(case_nat_option_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun7716400287237121964on_num(aTP_Lamm_ni(Ha),F2)),Nat) ).

% nat.case_distrib
tff(fact_7377_nat_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_nat,F1: option_num,F2: fun_nat_option_num,Nat: nat] : aa_option_num_nat(Ha,aa_nat_option_num(case_nat_option_num(F1,F2),Nat)) = aa_nat_nat(case_nat_nat(aa_option_num_nat(Ha,F1),aa_fun7387201323300553874at_nat(aTP_Lamm_nj(Ha),F2)),Nat) ).

% nat.case_distrib
tff(fact_7378_nat_Ocase__distrib,axiom,
    ! [Ha: fun_option_num_o,F1: option_num,F2: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Ha,aa_nat_option_num(case_nat_option_num(F1,F2),Nat))
    <=> aa_nat_o(case_nat_o(aa_option_num_o(Ha,F1),aa_fun1675964276245824108_nat_o(aTP_Lamm_nk(Ha),F2)),Nat) ) ).

% nat.case_distrib
tff(fact_7379_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_option_num,F1: nat,F2: fun_nat_nat,Nat: nat] : aa_nat_option_num(Ha,aa_nat_nat(case_nat_nat(F1,F2),Nat)) = aa_nat_option_num(case_nat_option_num(aa_nat_option_num(Ha,F1),aa_fun6517009837249254290on_num(aTP_Lamm_nl(Ha),F2)),Nat) ).

% nat.case_distrib
tff(fact_7380_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_nat,F1: nat,F2: fun_nat_nat,Nat: nat] : aa_nat_nat(Ha,aa_nat_nat(case_nat_nat(F1,F2),Nat)) = aa_nat_nat(case_nat_nat(aa_nat_nat(Ha,F1),aa_fun5791449513213591928at_nat(aTP_Lamm_nm(Ha),F2)),Nat) ).

% nat.case_distrib
tff(fact_7381_nat_Ocase__distrib,axiom,
    ! [Ha: fun_nat_o,F1: nat,F2: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Ha,aa_nat_nat(case_nat_nat(F1,F2),Nat))
    <=> aa_nat_o(case_nat_o(aa_nat_o(Ha,F1),aa_fun8673969331427483654_nat_o(aTP_Lamm_nn(Ha),F2)),Nat) ) ).

% nat.case_distrib
tff(fact_7382_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_option_num,F1: $o,F2: fun_nat_o,Nat: nat] : aa_o_option_num(Ha,aa_nat_o(case_nat_o((F1),F2),Nat)) = aa_nat_option_num(case_nat_option_num(aa_o_option_num(Ha,(F1)),aa_fun6990899178391151900on_num(aTP_Lamm_no(Ha),F2)),Nat) ).

% nat.case_distrib
tff(fact_7383_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_nat,F1: $o,F2: fun_nat_o,Nat: nat] : aa_o_nat(Ha,aa_nat_o(case_nat_o((F1),F2),Nat)) = aa_nat_nat(case_nat_nat(aa_o_nat(Ha,(F1)),aa_fun4204860950123276802at_nat(aTP_Lamm_np(Ha),F2)),Nat) ).

% nat.case_distrib
tff(fact_7384_nat_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,F1: $o,F2: fun_nat_o,Nat: nat] :
      ( aa_o_o(Ha,aa_nat_o(case_nat_o((F1),F2),Nat))
    <=> aa_nat_o(case_nat_o(aa_o_o(Ha,(F1)),aa_fun5937193402907943164_nat_o(aTP_Lamm_nq(Ha),F2)),Nat) ) ).

% nat.case_distrib
tff(fact_7385_num_Ocase__distrib,axiom,
    ! [Ha: fun_op8989173174992673105on_num,F1: option_num,F2: fun_num_option_num,F3: fun_num_option_num,Num: num] : aa_opt5369159329878751634on_num(Ha,aa_num_option_num(case_num_option_num(F1,F2,F3),Num)) = aa_num_option_num(case_num_option_num(aa_opt5369159329878751634on_num(Ha,F1),aa_fun3636977833914307392on_num(aTP_Lamm_db(Ha),F2),aa_fun3636977833914307392on_num(aTP_Lamm_db(Ha),F3)),Num) ).

% num.case_distrib
tff(fact_7386_old_Onat_Osimps_I4_J,axiom,
    ! [F1: option_num,F2: fun_nat_option_num] : aa_nat_option_num(case_nat_option_num(F1,F2),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_7387_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F2: fun_nat_nat] : aa_nat_nat(case_nat_nat(F1,F2),zero_zero_nat) = F1 ).

% old.nat.simps(4)
tff(fact_7388_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F2: fun_nat_o] :
      ( aa_nat_o(case_nat_o((F1),F2),zero_zero_nat)
    <=> (F1) ) ).

% old.nat.simps(4)
tff(fact_7389_old_Onat_Osimps_I5_J,axiom,
    ! [F1: option_num,F2: fun_nat_option_num,X2: nat] : aa_nat_option_num(case_nat_option_num(F1,F2),aa_nat_nat(suc,X2)) = aa_nat_option_num(F2,X2) ).

% old.nat.simps(5)
tff(fact_7390_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F2: fun_nat_nat,X2: nat] : aa_nat_nat(case_nat_nat(F1,F2),aa_nat_nat(suc,X2)) = aa_nat_nat(F2,X2) ).

% old.nat.simps(5)
tff(fact_7391_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F2: fun_nat_o,X2: nat] :
      ( aa_nat_o(case_nat_o((F1),F2),aa_nat_nat(suc,X2))
    <=> aa_nat_o(F2,X2) ) ).

% old.nat.simps(5)
tff(fact_7392_not__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B) ).

% not_add_distrib
tff(fact_7393_not__diff__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_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(bit_ri7919022796975470100ot_int,A)),B) ).

% not_diff_distrib
tff(fact_7394_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),X)) ).

% plus_integer.abs_eq
tff(fact_7395_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X) ) ).

% less_eq_integer.abs_eq
tff(fact_7396_verit__eq__simplify_I17_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num,F3: fun_num_option_num,X2: num] : aa_num_option_num(case_num_option_num(F1,F2,F3),aa_num_num(bit0,X2)) = aa_num_option_num(F2,X2) ).

% verit_eq_simplify(17)
tff(fact_7397_verit__eq__simplify_I16_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num,F3: fun_num_option_num] : aa_num_option_num(case_num_option_num(F1,F2,F3),one) = F1 ).

% verit_eq_simplify(16)
tff(fact_7398_verit__eq__simplify_I18_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num,F3: fun_num_option_num,X32: num] : aa_num_option_num(case_num_option_num(F1,F2,F3),aa_num_num(bit1,X32)) = aa_num_option_num(F3,X32) ).

% verit_eq_simplify(18)
tff(fact_7399_is__singletonI_H,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ( A4 != bot_bo1488462491386950373nt_int )
     => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
           => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y3),A4)
             => ( X3 = Y3 ) ) )
       => aa_set6820988702716878619_int_o(is_sin6299389887212142093nt_int,A4) ) ) ).

% is_singletonI'
tff(fact_7400_is__singletonI_H,axiom,
    ! [A4: set_set_nat] :
      ( ( A4 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat,Y3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A4)
           => ( aa_set_set_nat_o(member_set_nat(Y3),A4)
             => ( X3 = Y3 ) ) )
       => aa_set_set_nat_o(is_singleton_set_nat,A4) ) ) ).

% is_singletonI'
tff(fact_7401_is__singletonI_H,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
           => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y3),A4)
             => ( X3 = Y3 ) ) )
       => aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4) ) ) ).

% is_singletonI'
tff(fact_7402_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X3: $o,Y3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
           => ( aa_set_o_o(member_o((Y3)),A4)
             => ( (X3)
              <=> (Y3) ) ) )
       => aa_set_o_o(is_singleton_o,A4) ) ) ).

% is_singletonI'
tff(fact_7403_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => ( aa_set_nat_o(member_nat(Y3),A4)
             => ( X3 = Y3 ) ) )
       => aa_set_nat_o(is_singleton_nat,A4) ) ) ).

% is_singletonI'
tff(fact_7404_is__singletonI_H,axiom,
    ! [A4: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X3: int,Y3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => ( aa_set_int_o(member_int(Y3),A4)
             => ( X3 = Y3 ) ) )
       => aa_set_int_o(is_singleton_int,A4) ) ) ).

% is_singletonI'
tff(fact_7405_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A)),one_one_Code_integer) ).

% minus_eq_not_plus_1
tff(fact_7406_minus__eq__not__plus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).

% minus_eq_not_plus_1
tff(fact_7407_nth__Cons,axiom,
    ! [X: int,Xs: list_int,N2: nat] : aa_nat_int(nth_int(cons_int(X,Xs)),N2) = aa_nat_int(case_nat_int(X,nth_int(Xs)),N2) ).

% nth_Cons
tff(fact_7408_nth__Cons,axiom,
    ! [X: option_num,Xs: list_option_num,N2: nat] : aa_nat_option_num(nth_option_num(cons_option_num(X,Xs)),N2) = aa_nat_option_num(case_nat_option_num(X,nth_option_num(Xs)),N2) ).

% nth_Cons
tff(fact_7409_nth__Cons,axiom,
    ! [X: nat,Xs: list_nat,N2: nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),N2) = aa_nat_nat(case_nat_nat(X,nth_nat(Xs)),N2) ).

% nth_Cons
tff(fact_7410_nth__Cons,axiom,
    ! [X: $o,Xs: list_o,N2: nat] :
      ( aa_nat_o(nth_o(cons_o((X),Xs)),N2)
    <=> aa_nat_o(case_nat_o((X),nth_o(Xs)),N2) ) ).

% nth_Cons
tff(fact_7411_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(M),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_nat_Code_integer(bit_se2119862282449309892nteger,N2))) = zero_z3403309356797280102nteger ) ) ).

% take_bit_not_mask_eq_0
tff(fact_7412_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N2))) = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
tff(fact_7413_is__singleton__def,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A4)
    <=> ? [X4: produc3843707927480180839at_nat] : A4 = insert9069300056098147895at_nat(X4,bot_bo228742789529271731at_nat) ) ).

% is_singleton_def
tff(fact_7414_is__singleton__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4)
    <=> ? [X4: product_prod_nat_nat] : A4 = insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat) ) ).

% is_singleton_def
tff(fact_7415_is__singleton__def,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
    <=> ? [X4: $o] : A4 = insert_o((X4),bot_bot_set_o) ) ).

% is_singleton_def
tff(fact_7416_is__singleton__def,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ? [X4: nat] : A4 = insert_nat(X4,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_7417_is__singleton__def,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
    <=> ? [X4: int] : A4 = insert_int(X4,bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_7418_is__singletonE,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(is_sin2937591304547752795at_nat,A4)
     => ~ ! [X3: produc3843707927480180839at_nat] : A4 != insert9069300056098147895at_nat(X3,bot_bo228742789529271731at_nat) ) ).

% is_singletonE
tff(fact_7419_is__singletonE,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(is_sin2850979758926227957at_nat,A4)
     => ~ ! [X3: product_prod_nat_nat] : A4 != insert8211810215607154385at_nat(X3,bot_bo2099793752762293965at_nat) ) ).

% is_singletonE
tff(fact_7420_is__singletonE,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
     => ~ ! [X3: $o] : A4 != insert_o((X3),bot_bot_set_o) ) ).

% is_singletonE
tff(fact_7421_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
     => ~ ! [X3: nat] : A4 != insert_nat(X3,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_7422_is__singletonE,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
     => ~ ! [X3: int] : A4 != insert_int(X3,bot_bot_set_int) ) ).

% is_singletonE
tff(fact_7423_integer__of__num_I2_J,axiom,
    ! [N2: num] :
      code_integer_of_num(aa_num_num(bit0,N2)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N2),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k) ) ).

% integer_of_num(2)
tff(fact_7424_int__numeral__not__and__num,axiom,
    ! [M: num,N2: 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,N2)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(N2),M)) ).

% int_numeral_not_and_num
tff(fact_7425_int__numeral__and__not__num,axiom,
    ! [M: num,N2: 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,N2))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),N2)) ).

% int_numeral_and_not_num
tff(fact_7426_and__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N2))))) ).

% and_not_numerals(8)
tff(fact_7427_not__int__rec,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = 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))),K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_7428_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Code_integer_num(code_num_of_integer,K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_nr,code_divmod_integer(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_7429_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K2) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2))),
        $ite(K2 = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_ns,code_divmod_integer(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_7430_nat_Osplit__sels_I2_J,axiom,
    ! [Pa: fun_option_num_o,F1: option_num,F2: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Pa,aa_nat_option_num(case_nat_option_num(F1,F2),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(F2,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_7431_nat_Osplit__sels_I2_J,axiom,
    ! [Pa: fun_nat_o,F1: nat,F2: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(case_nat_nat(F1,F2),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(F2,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_7432_nat_Osplit__sels_I2_J,axiom,
    ! [Pa: fun_o_o,F1: $o,F2: fun_nat_o,Nat: nat] :
      ( aa_o_o(Pa,aa_nat_o(case_nat_o((F1),F2),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(F2,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_7433_nat_Osplit__sels_I1_J,axiom,
    ! [Pa: fun_option_num_o,F1: option_num,F2: fun_nat_option_num,Nat: nat] :
      ( aa_option_num_o(Pa,aa_nat_option_num(case_nat_option_num(F1,F2),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(F2,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_7434_nat_Osplit__sels_I1_J,axiom,
    ! [Pa: fun_nat_o,F1: nat,F2: fun_nat_nat,Nat: nat] :
      ( aa_nat_o(Pa,aa_nat_nat(case_nat_nat(F1,F2),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(F2,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_7435_nat_Osplit__sels_I1_J,axiom,
    ! [Pa: fun_o_o,F1: $o,F2: fun_nat_o,Nat: nat] :
      ( aa_o_o(Pa,aa_nat_o(case_nat_o((F1),F2),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(F2,aa_nat_nat(pred,Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_7436_mask__eq__sum__exp__nat,axiom,
    ! [N2: 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))),N2)),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_by,N2))) ).

% mask_eq_sum_exp_nat
tff(fact_7437_of__int__sum,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(Fa,A4)) = groups3906332499630173760nt_rat(aTP_Lamm_nt(Fa),A4) ).

% of_int_sum
tff(fact_7438_of__int__sum,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(Fa,A4)) = groups4538972089207619220nt_int(aTP_Lamm_nu(Fa),A4) ).

% of_int_sum
tff(fact_7439_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa)) ).

% plus_integer.rep_eq
tff(fact_7440_of__nat__sum,axiom,
    ! [Fa: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(Fa,A4)) = groups4538972089207619220nt_int(aTP_Lamm_nv(Fa),A4) ).

% of_nat_sum
tff(fact_7441_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_nw(Fa)),A4) ).

% of_nat_sum
tff(fact_7442_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_nx(Fa)),A4) ).

% of_nat_sum
tff(fact_7443_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ny(Fa)),A4) ).

% of_nat_sum
tff(fact_7444_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_fq),Nat) ).

% pred_def
tff(fact_7445_mod__sum__eq,axiom,
    ! [Fa: fun_nat_nat,A: nat,A4: 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_nz(Fa),A)),A4)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),A) ).

% mod_sum_eq
tff(fact_7446_mod__sum__eq,axiom,
    ! [Fa: fun_int_int,A: int,A4: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_oa(Fa),A),A4)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(Fa,A4)),A) ).

% mod_sum_eq
tff(fact_7447_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_ob),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_7448_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_eo),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_7449_integer__less__iff,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,K2)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_7450_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% less_integer.rep_eq
tff(fact_7451_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ).

% int_of_integer_less_iff
tff(fact_7452_integer__less__eq__iff,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K2)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_7453_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% less_eq_integer.rep_eq
tff(fact_7454_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oc(X),M)),I5) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),I5)) ).

% sum_power_add
tff(fact_7455_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_od(X),M)),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),I5)) ).

% sum_power_add
tff(fact_7456_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),N2) ) ).

% less_eq_nat.simps(2)
tff(fact_7457_diff__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_fq),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ).

% diff_Suc
tff(fact_7458_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,W2))),N2)
    <=> aa_nat_o(case_nat_o($false,bit_se8040316288895769887atural(aa_num_Code_natural(numera5444537566228673987atural,W2))),N2) ) ).

% bit_numeral_rec(1)
tff(fact_7459_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,W2))),N2)
    <=> aa_nat_o(case_nat_o($false,aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_num_Code_integer(numera6620942414471956472nteger,W2))),N2) ) ).

% bit_numeral_rec(1)
tff(fact_7460_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))),N2)
    <=> aa_nat_o(case_nat_o($false,aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2))),N2) ) ).

% bit_numeral_rec(1)
tff(fact_7461_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W2))),N2)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W2))),N2) ) ).

% bit_numeral_rec(1)
tff(fact_7462_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,W2))),N2)
    <=> aa_nat_o(case_nat_o($true,bit_se8040316288895769887atural(aa_num_Code_natural(numera5444537566228673987atural,W2))),N2) ) ).

% bit_numeral_rec(2)
tff(fact_7463_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,W2))),N2)
    <=> aa_nat_o(case_nat_o($true,aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_num_Code_integer(numera6620942414471956472nteger,W2))),N2) ) ).

% bit_numeral_rec(2)
tff(fact_7464_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2))),N2)
    <=> aa_nat_o(case_nat_o($true,aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2))),N2) ) ).

% bit_numeral_rec(2)
tff(fact_7465_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W2))),N2)
    <=> aa_nat_o(case_nat_o($true,aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W2))),N2) ) ).

% bit_numeral_rec(2)
tff(fact_7466_mask__eq__sum__exp,axiom,
    ! [N2: 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))),N2)),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_by,N2))) ).

% mask_eq_sum_exp
tff(fact_7467_mask__eq__sum__exp,axiom,
    ! [N2: nat] : minus_7197305767214868737atural(aa_nat_Code_natural(power_7079662738309270450atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),N2),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_by,N2))) ).

% mask_eq_sum_exp
tff(fact_7468_mask__eq__sum__exp,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2)),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_by,N2))) ).

% mask_eq_sum_exp
tff(fact_7469_mask__eq__sum__exp,axiom,
    ! [N2: 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))),N2)),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_by,N2))) ).

% mask_eq_sum_exp
tff(fact_7470_sum__abs__ge__zero,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_oe(Fa),A4)) ).

% sum_abs_ge_zero
tff(fact_7471_sum__abs,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(Fa,A4))),groups4538972089207619220nt_int(aTP_Lamm_oe(Fa),A4)) ).

% sum_abs
tff(fact_7472_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: fun_o_Code_integer,A: fun_o_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(X,(I3))) )
     => ( ( groups4406642042086082107nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_o_Code_integer(A,(I3))),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_of(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7473_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I3)) )
     => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(X),I5) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),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,I3)),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_og(X),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7474_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I3)) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),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,I3)),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_oh(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7475_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: fun_o_rat,A: fun_o_rat,B: rat,Delta: rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(X,(I3))) )
     => ( ( groups7872700643590313910_o_rat(X,I5) = one_one_rat )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_o_rat(A,(I3))),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_oi(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7476_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I3)) )
     => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(X),I5) = one_one_rat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(A,I3)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_oj(X),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7477_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(X,I3)) )
     => ( ( groups3906332499630173760nt_rat(X,I5) = one_one_rat )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_int_rat(A,I3)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ok(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7478_convex__sum__bound__le,axiom,
    ! [I5: set_o,X: fun_o_int,A: fun_o_int,B: int,Delta: int] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(X,(I3))) )
     => ( ( groups8505340233167759370_o_int(X,I5) = one_one_int )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),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_o_int(A,(I3))),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,groups8505340233167759370_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_ol(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7479_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_int,A: fun_nat_int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(X,I3)) )
     => ( ( aa_set_nat_int(groups3539618377306564664at_int(X),I5) = one_one_int )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),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,I3)),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_om(X),A)),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7480_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_int,A: fun_int_int,B: int,Delta: int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(X,I3)) )
     => ( ( groups4538972089207619220nt_int(X,I5) = one_one_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),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,I3)),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_on(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7481_convex__sum__bound__le,axiom,
    ! [I5: set_set_nat,X: fun_se7586849273513198688nteger,A: fun_se7586849273513198688nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I3),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(X,I3)) )
     => ( ( groups9190459664516455967nteger(X,I5) = one_one_Code_integer )
       => ( ! [I3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(I3),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,I3)),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_oo(X),A),I5)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7482_abs__sum__abs,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_oe(Fa),A4)) = groups4538972089207619220nt_int(aTP_Lamm_oe(Fa),A4) ).

% abs_sum_abs
tff(fact_7483_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fl),A4) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7484_sum_Oneutral__const,axiom,
    ! [A4: set_int] : groups4538972089207619220nt_int(aTP_Lamm_fm,A4) = zero_zero_int ).

% sum.neutral_const
tff(fact_7485_int__sum,axiom,
    ! [Fa: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(Fa,A4)) = groups4538972089207619220nt_int(aTP_Lamm_nv(Fa),A4) ).

% int_sum
tff(fact_7486_int__sum,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_nw(Fa)),A4) ).

% int_sum
tff(fact_7487_sum_Oswap,axiom,
    ! [Ga: fun_nat_fun_nat_nat,B4: set_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_op(Ga),B4)),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_or(Ga),A4)),B4) ).

% sum.swap
tff(fact_7488_sum_Oswap,axiom,
    ! [Ga: fun_int_fun_int_int,B4: set_int,A4: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_os(Ga),B4),A4) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_ou(Ga),A4),B4) ).

% sum.swap
tff(fact_7489_sum__mono,axiom,
    ! [K5: set_o,Fa: fun_o_rat,Ga: fun_o_rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(I3))),aa_o_rat(Ga,(I3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(Fa,K5)),groups7872700643590313910_o_rat(Ga,K5)) ) ).

% sum_mono
tff(fact_7490_sum__mono,axiom,
    ! [K5: set_nat,Fa: fun_nat_rat,Ga: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I3)),aa_nat_rat(Ga,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),K5)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),K5)) ) ).

% sum_mono
tff(fact_7491_sum__mono,axiom,
    ! [K5: set_int,Fa: fun_int_rat,Ga: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I3)),aa_int_rat(Ga,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(Fa,K5)),groups3906332499630173760nt_rat(Ga,K5)) ) ).

% sum_mono
tff(fact_7492_sum__mono,axiom,
    ! [K5: set_o,Fa: fun_o_nat,Ga: fun_o_nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(I3))),aa_o_nat(Ga,(I3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(Fa,K5)),groups8507830703676809646_o_nat(Ga,K5)) ) ).

% sum_mono
tff(fact_7493_sum__mono,axiom,
    ! [K5: set_int,Fa: fun_int_nat,Ga: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,I3)),aa_int_nat(Ga,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(Fa,K5)),groups4541462559716669496nt_nat(Ga,K5)) ) ).

% sum_mono
tff(fact_7494_sum__mono,axiom,
    ! [K5: set_o,Fa: fun_o_int,Ga: fun_o_int] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(Fa,(I3))),aa_o_int(Ga,(I3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(Fa,K5)),groups8505340233167759370_o_int(Ga,K5)) ) ).

% sum_mono
tff(fact_7495_sum__mono,axiom,
    ! [K5: set_nat,Fa: fun_nat_int,Ga: fun_nat_int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,I3)),aa_nat_int(Ga,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),K5)),aa_set_nat_int(groups3539618377306564664at_int(Ga),K5)) ) ).

% sum_mono
tff(fact_7496_sum__mono,axiom,
    ! [K5: set_nat,Fa: fun_nat_nat,Ga: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,I3)),aa_nat_nat(Ga,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),K5)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),K5)) ) ).

% sum_mono
tff(fact_7497_sum__mono,axiom,
    ! [K5: set_int,Fa: fun_int_int,Ga: fun_int_int] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(Fa,I3)),aa_int_int(Ga,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(Fa,K5)),groups4538972089207619220nt_int(Ga,K5)) ) ).

% sum_mono
tff(fact_7498_sum__mono,axiom,
    ! [K5: set_set_nat,Fa: fun_set_nat_rat,Ga: fun_set_nat_rat] :
      ( ! [I3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(Fa,I3)),aa_set_nat_rat(Ga,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7659867448343625626at_rat(Fa,K5)),groups7659867448343625626at_rat(Ga,K5)) ) ).

% sum_mono
tff(fact_7499_sum_Odistrib,axiom,
    ! [Ga: fun_nat_nat,Ha: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ov(Ga),Ha)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Ha),A4)) ).

% sum.distrib
tff(fact_7500_sum_Odistrib,axiom,
    ! [Ga: fun_int_int,Ha: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ow(Ga),Ha),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ga,A4)),groups4538972089207619220nt_int(Ha,A4)) ).

% sum.distrib
tff(fact_7501_sum__product,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat,Ga: fun_nat_nat,B4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),B4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_oy(Fa),Ga),B4)),A4) ).

% sum_product
tff(fact_7502_sum__product,axiom,
    ! [Fa: fun_int_int,A4: set_int,Ga: fun_int_int,B4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(Fa,A4)),groups4538972089207619220nt_int(Ga,B4)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_pa(Fa),Ga),B4),A4) ).

% sum_product
tff(fact_7503_sum__distrib__right,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat,R: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),R) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pb(Fa),R)),A4) ).

% sum_distrib_right
tff(fact_7504_sum__distrib__right,axiom,
    ! [Fa: fun_int_int,A4: set_int,R: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(Fa,A4)),R) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_pc(Fa),R),A4) ).

% sum_distrib_right
tff(fact_7505_sum__distrib__left,axiom,
    ! [R: nat,Fa: fun_nat_nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pd(R),Fa)),A4) ).

% sum_distrib_left
tff(fact_7506_sum__distrib__left,axiom,
    ! [R: int,Fa: fun_int_int,A4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R),groups4538972089207619220nt_int(Fa,A4)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pe(R),Fa),A4) ).

% sum_distrib_left
tff(fact_7507_sum__subtractf,axiom,
    ! [Fa: fun_int_int,Ga: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pf(Fa),Ga),A4) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(Fa,A4)),groups4538972089207619220nt_int(Ga,A4)) ).

% sum_subtractf
tff(fact_7508_sum__negf,axiom,
    ! [Fa: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aTP_Lamm_pg(Fa),A4) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(Fa,A4)) ).

% sum_negf
tff(fact_7509_sum__nonpos,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(Fa,(X3))),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups4406642042086082107nteger(Fa,A4)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_7510_sum__nonpos,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,X3)),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),A4)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_7511_sum__nonpos,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,X3)),zero_z3403309356797280102nteger) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(Fa,A4)),zero_z3403309356797280102nteger) ) ).

% sum_nonpos
tff(fact_7512_sum__nonpos,axiom,
    ! [A4: set_o,Fa: fun_o_rat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(X3))),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(Fa,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7513_sum__nonpos,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7514_sum__nonpos,axiom,
    ! [A4: set_int,Fa: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(Fa,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7515_sum__nonpos,axiom,
    ! [A4: set_o,Fa: fun_o_nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(X3))),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(Fa,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7516_sum__nonpos,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(Fa,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7517_sum__nonpos,axiom,
    ! [A4: set_o,Fa: fun_o_int] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(Fa,(X3))),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(Fa,A4)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_7518_sum__nonpos,axiom,
    ! [A4: set_nat,Fa: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,X3)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_7519_sum__nonneg,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(X3))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7520_sum__nonneg,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),A4)) ) ).

% sum_nonneg
tff(fact_7521_sum__nonneg,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7522_sum__nonneg,axiom,
    ! [A4: set_o,Fa: fun_o_rat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(X3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups7872700643590313910_o_rat(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7523_sum__nonneg,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)) ) ).

% sum_nonneg
tff(fact_7524_sum__nonneg,axiom,
    ! [A4: set_int,Fa: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7525_sum__nonneg,axiom,
    ! [A4: set_o,Fa: fun_o_nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(X3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups8507830703676809646_o_nat(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7526_sum__nonneg,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7527_sum__nonneg,axiom,
    ! [A4: set_o,Fa: fun_o_int] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Fa,(X3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups8505340233167759370_o_int(Fa,A4)) ) ).

% sum_nonneg
tff(fact_7528_sum__nonneg,axiom,
    ! [A4: set_nat,Fa: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)) ) ).

% sum_nonneg
tff(fact_7529_sum__subtractf__nat,axiom,
    ! [A4: set_o,Ga: fun_o_nat,Fa: fun_o_nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Ga,(X3))),aa_o_nat(Fa,(X3))) )
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_ph(Ga),Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(Fa,A4)),groups8507830703676809646_o_nat(Ga,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_7530_sum__subtractf__nat,axiom,
    ! [A4: set_se6260736226359567993nt_int,Ga: fun_se4101287982975402108nt_nat,Fa: fun_se4101287982975402108nt_nat] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4753215920113914285nt_nat(Ga,X3)),aa_set4753215920113914285nt_nat(Fa,X3)) )
     => ( groups185207323561075247nt_nat(aa_fun209864160807355708nt_nat(aTP_Lamm_pi(Ga),Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups185207323561075247nt_nat(Fa,A4)),groups185207323561075247nt_nat(Ga,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_7531_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,Ga: fun_set_nat_nat,Fa: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(Ga,X3)),aa_set_nat_nat(Fa,X3)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_pj(Ga),Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(Fa,A4)),groups8294997508430121362at_nat(Ga,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_7532_sum__subtractf__nat,axiom,
    ! [A4: set_int,Ga: fun_int_nat,Fa: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Ga,X3)),aa_int_nat(Fa,X3)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pk(Ga),Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(Fa,A4)),groups4541462559716669496nt_nat(Ga,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_7533_sum__subtractf__nat,axiom,
    ! [A4: set_nat,Ga: fun_nat_nat,Fa: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Ga,X3)),aa_nat_nat(Fa,X3)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pl(Ga),Fa)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_7534_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_Code_integer,Ga: fun_nat_nat,A4: set_nat] :
      ( ( aa_nat_Code_integer(Ha,zero_zero_nat) = zero_z3403309356797280102nteger )
     => ( ! [X3: nat,Y3: nat] : aa_nat_Code_integer(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Ha,X3)),aa_nat_Code_integer(Ha,Y3))
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(comp_n3898172953802868194er_nat(Ha,Ga)),A4) = aa_nat_Code_integer(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7535_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_rat,Ga: fun_nat_nat,A4: set_nat] :
      ( ( aa_nat_rat(Ha,zero_zero_nat) = zero_zero_rat )
     => ( ! [X3: nat,Y3: nat] : aa_nat_rat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ha,X3)),aa_nat_rat(Ha,Y3))
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(comp_nat_rat_nat(Ha,Ga)),A4) = aa_nat_rat(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7536_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_int,Ga: fun_nat_nat,A4: set_nat] :
      ( ( aa_nat_int(Ha,zero_zero_nat) = zero_zero_int )
     => ( ! [X3: nat,Y3: nat] : aa_nat_int(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y3)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ha,X3)),aa_nat_int(Ha,Y3))
       => ( aa_set_nat_int(groups3539618377306564664at_int(comp_nat_int_nat(Ha,Ga)),A4) = aa_nat_int(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7537_sum__comp__morphism,axiom,
    ! [Ha: fun_int_Code_integer,Ga: fun_int_int,A4: set_int] :
      ( ( aa_int_Code_integer(Ha,zero_zero_int) = zero_z3403309356797280102nteger )
     => ( ! [X3: int,Y3: int] : aa_int_Code_integer(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y3)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(Ha,X3)),aa_int_Code_integer(Ha,Y3))
       => ( groups7873554091576472773nteger(comp_i1585864551200866970er_int(Ha,Ga),A4) = aa_int_Code_integer(Ha,groups4538972089207619220nt_int(Ga,A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7538_sum__comp__morphism,axiom,
    ! [Ha: fun_int_rat,Ga: fun_int_int,A4: set_int] :
      ( ( aa_int_rat(Ha,zero_zero_int) = zero_zero_rat )
     => ( ! [X3: int,Y3: int] : aa_int_rat(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Ha,X3)),aa_int_rat(Ha,Y3))
       => ( groups3906332499630173760nt_rat(comp_int_rat_int(Ha,Ga),A4) = aa_int_rat(Ha,groups4538972089207619220nt_int(Ga,A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7539_sum__comp__morphism,axiom,
    ! [Ha: fun_int_nat,Ga: fun_int_int,A4: set_int] :
      ( ( aa_int_nat(Ha,zero_zero_int) = zero_zero_nat )
     => ( ! [X3: int,Y3: int] : aa_int_nat(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ha,X3)),aa_int_nat(Ha,Y3))
       => ( groups4541462559716669496nt_nat(comp_int_nat_int(Ha,Ga),A4) = aa_int_nat(Ha,groups4538972089207619220nt_int(Ga,A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7540_sum__comp__morphism,axiom,
    ! [Ha: fun_Code_integer_nat,Ga: fun_nat_Code_integer,A4: set_nat] :
      ( ( aa_Code_integer_nat(Ha,zero_z3403309356797280102nteger) = zero_zero_nat )
     => ( ! [X3: code_integer,Y3: code_integer] : aa_Code_integer_nat(Ha,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Ha,X3)),aa_Code_integer_nat(Ha,Y3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(comp_C4049556595663050210at_nat(Ha,Ga)),A4) = aa_Code_integer_nat(Ha,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7541_sum__comp__morphism,axiom,
    ! [Ha: fun_rat_nat,Ga: fun_nat_rat,A4: set_nat] :
      ( ( aa_rat_nat(Ha,zero_zero_rat) = zero_zero_nat )
     => ( ! [X3: rat,Y3: rat] : aa_rat_nat(Ha,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_rat_nat(Ha,X3)),aa_rat_nat(Ha,Y3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(comp_rat_nat_nat(Ha,Ga)),A4) = aa_rat_nat(Ha,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7542_sum__comp__morphism,axiom,
    ! [Ha: fun_int_nat,Ga: fun_nat_int,A4: set_nat] :
      ( ( aa_int_nat(Ha,zero_zero_int) = zero_zero_nat )
     => ( ! [X3: int,Y3: int] : aa_int_nat(Ha,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ha,X3)),aa_int_nat(Ha,Y3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(comp_int_nat_nat(Ha,Ga)),A4) = aa_int_nat(Ha,aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7543_sum__comp__morphism,axiom,
    ! [Ha: fun_nat_nat,Ga: fun_nat_nat,A4: set_nat] :
      ( ( aa_nat_nat(Ha,zero_zero_nat) = zero_zero_nat )
     => ( ! [X3: nat,Y3: nat] : aa_nat_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ha,X3)),aa_nat_nat(Ha,Y3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ha,Ga)),A4) = aa_nat_nat(Ha,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)) ) ) ) ).

% sum_comp_morphism
tff(fact_7544_sum__SucD,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat,N2: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4) = aa_nat_nat(suc,N2) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Fa,X3)) ) ) ).

% sum_SucD
tff(fact_7545_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Code_integer_nat(code_nat_of_integer,K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_pm,code_divmod_integer(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_7546_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_pn(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_7547_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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_po,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_7548_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_pp(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_7549_choose__even__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_pq(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_7550_choose__even__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( 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_pr,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_7551_choose__even__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_ps(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_7552_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_pt(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_7553_sum__gp,axiom,
    ! [X: rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),set_or1269000886237332187st_nat(M,N2)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),
        zero_zero_rat,
        $ite(X = 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,N2),one_one_nat)),M)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ) ).

% sum_gp
tff(fact_7554_Icc__eq__Icc,axiom,
    ! [L: set_int,Ha: set_int,L4: set_int,H_a: set_int] :
      ( ( set_or370866239135849197et_int(L,Ha) = set_or370866239135849197et_int(L4,H_a) )
    <=> ( ( ( L = L4 )
          & ( Ha = H_a ) )
        | ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),Ha)
          & ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L4),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_7555_Icc__eq__Icc,axiom,
    ! [L: rat,Ha: rat,L4: rat,H_a: rat] :
      ( ( set_or633870826150836451st_rat(L,Ha) = set_or633870826150836451st_rat(L4,H_a) )
    <=> ( ( ( L = L4 )
          & ( Ha = H_a ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),Ha)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L4),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_7556_Icc__eq__Icc,axiom,
    ! [L: num,Ha: num,L4: num,H_a: num] :
      ( ( set_or7049704709247886629st_num(L,Ha) = set_or7049704709247886629st_num(L4,H_a) )
    <=> ( ( ( L = L4 )
          & ( Ha = H_a ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),Ha)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L4),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_7557_Icc__eq__Icc,axiom,
    ! [L: nat,Ha: nat,L4: nat,H_a: nat] :
      ( ( set_or1269000886237332187st_nat(L,Ha) = set_or1269000886237332187st_nat(L4,H_a) )
    <=> ( ( ( L = L4 )
          & ( Ha = H_a ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Ha)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L4),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_7558_Icc__eq__Icc,axiom,
    ! [L: int,Ha: int,L4: int,H_a: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),Ha) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L4),H_a) )
    <=> ( ( ( L = L4 )
          & ( Ha = H_a ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Ha)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L4),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_7559_Icc__eq__Icc,axiom,
    ! [L: code_integer,Ha: code_integer,L4: code_integer,H_a: code_integer] :
      ( ( set_or189985376899183464nteger(L,Ha) = set_or189985376899183464nteger(L4,H_a) )
    <=> ( ( ( L = L4 )
          & ( Ha = H_a ) )
        | ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),Ha)
          & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L4),H_a) ) ) ) ).

% Icc_eq_Icc
tff(fact_7560_atLeastAtMost__iff,axiom,
    ! [I2: $o,L: $o,U: $o] :
      ( aa_set_o_o(member_o((I2)),set_or8904488021354931149Most_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(I2))
        & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(I2)),(U)) ) ) ).

% atLeastAtMost_iff
tff(fact_7561_atLeastAtMost__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,L: set_Pr958786334691620121nt_int,U: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),set_or2481441762145802318nt_int(L,U))
    <=> ( aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(L),I2)
        & aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7562_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7563_atLeastAtMost__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( member_set_int(I2,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),I2)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7564_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I2),set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7565_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I2),set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7566_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I2),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7567_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7568_atLeastAtMost__iff,axiom,
    ! [I2: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),set_or189985376899183464nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_7569_atMost__iff,axiom,
    ! [I2: $o,K2: $o] :
      ( aa_set_o_o(member_o((I2)),set_ord_atMost_o((K2)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(I2)),(K2)) ) ).

% atMost_iff
tff(fact_7570_atMost__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,K2: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),set_or2459421552957432928nt_int(K2))
    <=> aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(I2),K2) ) ).

% atMost_iff
tff(fact_7571_atMost__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or4236626031148496127et_nat(K2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I2),K2) ) ).

% atMost_iff
tff(fact_7572_atMost__iff,axiom,
    ! [I2: set_int,K2: set_int] :
      ( member_set_int(I2,set_or58775011639299419et_int(K2))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,I2),K2) ) ).

% atMost_iff
tff(fact_7573_atMost__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( aa_set_rat_o(member_rat(I2),set_ord_atMost_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),K2) ) ).

% atMost_iff
tff(fact_7574_atMost__iff,axiom,
    ! [I2: num,K2: num] :
      ( aa_set_num_o(member_num(I2),set_ord_atMost_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),K2) ) ).

% atMost_iff
tff(fact_7575_atMost__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_atMost_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2) ) ).

% atMost_iff
tff(fact_7576_atMost__iff,axiom,
    ! [I2: int,K2: int] :
      ( aa_set_int_o(member_int(I2),aa_int_set_int(set_ord_atMost_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2) ) ).

% atMost_iff
tff(fact_7577_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff
tff(fact_7578_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7579_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7580_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7581_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7582_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7583_atLeastatMost__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_7584_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_7585_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7586_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7587_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7588_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A,B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7589_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7590_atLeastatMost__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or189985376899183464nteger(A,B) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_7591_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_o,(B)),(A))
     => ( set_or8904488021354931149Most_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_7592_atLeastatMost__empty,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( set_or7959216805967363635t_assn(A,B) = bot_bot_set_assn ) ) ).

% atLeastatMost_empty
tff(fact_7593_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_7594_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_7595_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_7596_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_7597_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( set_or189985376899183464nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_7598_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C2,D2))
    <=> ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
        | ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7599_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C2,D2))
    <=> ( ~ 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,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7600_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C2,D2))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7601_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C2,D2))
    <=> ( ~ 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,C2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7602_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C2),D2))
    <=> ( ~ 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,C2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7603_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C2,D2))
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_7604_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y))
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y) ) ).

% atMost_subset_iff
tff(fact_7605_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_atMost_rat(X)),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% atMost_subset_iff
tff(fact_7606_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_atMost_num(X)),set_ord_atMost_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% atMost_subset_iff
tff(fact_7607_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% atMost_subset_iff
tff(fact_7608_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(set_ord_atMost_int,X)),aa_int_set_int(set_ord_atMost_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% atMost_subset_iff
tff(fact_7609_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,Ha: set_int,H_a: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,Ha),set_or58775011639299419et_int(H_a))
    <=> ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),Ha)
        | aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7610_Icc__subset__Iic__iff,axiom,
    ! [L: rat,Ha: rat,H_a: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,Ha)),set_ord_atMost_rat(H_a))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),Ha)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7611_Icc__subset__Iic__iff,axiom,
    ! [L: num,Ha: num,H_a: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(L,Ha)),set_ord_atMost_num(H_a))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),Ha)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7612_Icc__subset__Iic__iff,axiom,
    ! [L: nat,Ha: nat,H_a: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(L,Ha)),aa_nat_set_nat(set_ord_atMost_nat,H_a))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),Ha)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7613_Icc__subset__Iic__iff,axiom,
    ! [L: int,Ha: int,H_a: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),Ha)),aa_int_set_int(set_ord_atMost_int,H_a))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),Ha)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7614_Icc__subset__Iic__iff,axiom,
    ! [L: code_integer,Ha: code_integer,H_a: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(L,Ha)),set_or9101266186257409494nteger(H_a))
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),Ha)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ha),H_a) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7615_sum_OatMost__Suc,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat2240542744625857721at_nat(Ga,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_7616_sum_OatMost__Suc,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_rat(Ga,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_7617_sum_OatMost__Suc,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_int(Ga,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_7618_sum_OatMost__Suc,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_nat(Ga,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_7619_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K2) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_7620_sum_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_Code_integer(Ga,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_7621_sum_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,M: nat,N2: nat] :
      groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_z1048942125864253310at_nat,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2))),aa_nat2240542744625857721at_nat(Ga,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_7622_sum_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_rat(Ga,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_7623_sum_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_int(Ga,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_7624_sum_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_nat(Ga,aa_nat_nat(suc,N2)))) ).

% sum.cl_ivl_Suc
tff(fact_7625_not__Iic__le__Icc,axiom,
    ! [Ha: int,L4: int,H_a: int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(set_ord_atMost_int,Ha)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L4),H_a)) ).

% not_Iic_le_Icc
tff(fact_7626_atMost__def,axiom,
    ! [U: set_Pr958786334691620121nt_int] : set_or2459421552957432928nt_int(U) = collec5210948495886036740nt_int(aTP_Lamm_pu(U)) ).

% atMost_def
tff(fact_7627_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bp,U)) ).

% atMost_def
tff(fact_7628_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_bt,U)) ).

% atMost_def
tff(fact_7629_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_bv,U)) ).

% atMost_def
tff(fact_7630_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_pv,U)) ).

% atMost_def
tff(fact_7631_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bx,U)) ).

% atMost_def
tff(fact_7632_atMost__def,axiom,
    ! [U: int] : aa_int_set_int(set_ord_atMost_int,U) = collect_int(aa_int_fun_int_o(aTP_Lamm_bz,U)) ).

% atMost_def
tff(fact_7633_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,M),set_or633870826150836451st_rat(M,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7634_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M),set_or7049704709247886629st_num(M,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7635_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or1269000886237332187st_nat(L,M)),set_or1269000886237332187st_nat(M,U)) = set_or1269000886237332187st_nat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7636_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),M)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M),U)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7637_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or189985376899183464nteger(L,M),set_or189985376899183464nteger(M,U)) = set_or189985376899183464nteger(L,U) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7638_ex__nat__less,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
          & aa_nat_o(Pa,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,N2))
          & aa_nat_o(Pa,X4) ) ) ).

% ex_nat_less
tff(fact_7639_all__nat__less,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
         => aa_nat_o(Pa,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,N2))
         => aa_nat_o(Pa,X4) ) ) ).

% all_nat_less
tff(fact_7640_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,K2))),set_or1269000886237332187st_nat(M,N2)) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_7641_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7642_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Ga),K2)),set_or1269000886237332187st_nat(M,N2)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7643_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),set_or1269000886237332187st_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_7644_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),set_or1269000886237332187st_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_7645_atLeastatMost__psubset__iff,axiom,
    ! [A: assn,B: assn,C2: assn,D2: assn] :
      ( ord_less_set_assn(set_or7959216805967363635t_assn(A,B),set_or7959216805967363635t_assn(C2,D2))
    <=> ( ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
          | ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A)
            & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
            & ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C2),A)
              | aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),D2) ) ) )
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7646_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(A,B),set_or370866239135849197et_int(C2,D2))
    <=> ( ( ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
          | ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),D2)
            & ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,C2),A)
              | aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),D2) ) ) )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7647_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C2,D2))
    <=> ( ( ~ 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,C2),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D2) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7648_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C2,D2))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D2)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D2) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7649_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C2,D2))
    <=> ( ( ~ 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,C2),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D2) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7650_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,C2),D2))
    <=> ( ( ~ 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,C2),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D2) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7651_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A,B)),set_or189985376899183464nteger(C2,D2))
    <=> ( ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
          | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2)
            & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),A)
              | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D2) ) ) )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_7652_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( set_or1269000886237332187st_nat(M,N2) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_7653_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2)) = insert_nat(aa_nat_nat(suc,N2),set_or1269000886237332187st_nat(M,N2)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_7654_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2)) = set_or1269000886237332187st_nat(M,N2) ) ) ).

% atLeastAtMost_insertL
tff(fact_7655_sum_OatLeastAtMost__rev,axiom,
    ! [Ga: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(N2,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(Ga),N2),M)),set_or1269000886237332187st_nat(N2,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7656_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pz,M)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_7657_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,zero_zero_nat)),groups6857163185585827899at_nat(comp_n8698576032424989604at_nat(Ga,suc),set_or1269000886237332187st_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7658_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(comp_nat_rat_nat(Ga,suc)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7659_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(comp_nat_int_nat(Ga,suc)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7660_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,suc)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7661_sum_OatLeast0__atMost__Suc,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(zero_zero_nat,N2))),aa_nat2240542744625857721at_nat(Ga,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7662_sum_OatLeast0__atMost__Suc,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(zero_zero_nat,N2))),aa_nat_rat(Ga,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7663_sum_OatLeast0__atMost__Suc,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(zero_zero_nat,N2))),aa_nat_int(Ga,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7664_sum_OatLeast0__atMost__Suc,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(zero_zero_nat,N2))),aa_nat_nat(Ga,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7665_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,aa_nat_nat(suc,N2))),groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7666_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,aa_nat_nat(suc,N2))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7667_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,aa_nat_nat(suc,N2))),aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7668_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,aa_nat_nat(suc,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7669_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,M)),groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7670_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,M)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7671_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,M)),aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7672_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,M)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7673_sum_OatMost__Suc__shift,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,zero_zero_nat)),groups6857163185585827899at_nat(aTP_Lamm_qa(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_7674_sum_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qb(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_7675_sum_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_7676_sum_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_7677_sum__telescope,axiom,
    ! [Fa: fun_nat_rat,I2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qd(Fa)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,zero_zero_nat)),aa_nat_rat(Fa,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_7678_sum__telescope,axiom,
    ! [Fa: fun_nat_int,I2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qe(Fa)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,zero_zero_nat)),aa_nat_int(Fa,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_7679_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2))),aa_nat2240542744625857721at_nat(Ga,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,M)),groups6857163185585827899at_nat(aTP_Lamm_qa(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7680_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_rat(Ga,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,M)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qb(Ga)),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7681_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_int(Ga,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,M)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qc(Ga)),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7682_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_nat(Ga,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7683_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qf(Fa)),set_or1269000886237332187st_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,aa_nat_nat(suc,N2))),aa_nat_rat(Fa,M)) ) ) ).

% sum_Suc_diff
tff(fact_7684_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qg(Fa)),set_or1269000886237332187st_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,aa_nat_nat(suc,N2))),aa_nat_int(Fa,M)) ) ) ).

% sum_Suc_diff
tff(fact_7685_sum_OatLeast__atMost__pred__shift,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,aTP_Lamm_qh)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) ).

% sum.atLeast_atMost_pred_shift
tff(fact_7686_sum__choose__lower,axiom,
    ! [R: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,R)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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),N2))),N2) ).

% sum_choose_lower
tff(fact_7687_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,N2)),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,N2),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_7688_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,N2)),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,N2),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_7689_sum__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_Code_integer,A: nat,B: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),set_or1269000886237332187st_nat(A,B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_qk(Fa),A,B),zero_z3403309356797280102nteger) ).

% sum_atLeastAtMost_code
tff(fact_7690_sum__atLeastAtMost__code,axiom,
    ! [Fa: fun_na2338080690716861136at_nat,A: nat,B: nat] : groups6857163185585827899at_nat(Fa,set_or1269000886237332187st_nat(A,B)) = set_fo4497565046347964853at_nat(aTP_Lamm_ql(Fa),A,B,zero_z1048942125864253310at_nat) ).

% sum_atLeastAtMost_code
tff(fact_7691_sum__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_rat,A: nat,B: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Fa),set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_qm(Fa),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7692_sum__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups3539618377306564664at_int(Fa),set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_qn(Fa),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7693_sum__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Fa),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_qo(Fa),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7694_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_na2338080690716861136at_nat,P2: 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,N2),one_one_nat))
     => ( groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2))),groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7695_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7696_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2))),aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7697_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7698_sum_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,M))),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_7699_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qp(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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,N2))),one_one_rat)),N2) ).

% gbinomial_parallel_sum
tff(fact_7700_sum_Otriangle__reindex__eq,axiom,
    ! [Ga: fun_nat_fun_nat_nat,N2: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(Ga),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_qq(N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qs(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.triangle_reindex_eq
tff(fact_7701_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qt(M),N2)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),M) ) ) ).

% sum_choose_diagonal
tff(fact_7702_vandermonde,axiom,
    ! [M: nat,N2: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qu(M),N2),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),N2)),R) ).

% vandermonde
tff(fact_7703_sum__gp__basic,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_7704_sum__gp__basic,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_7705_sum__gp__basic,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = 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,X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_7706_sum__natinterval__diff,axiom,
    ! [Fa: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_qv(Fa)),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(Fa,M)),aa_nat_Code_integer(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))),zero_z3403309356797280102nteger) ).

% sum_natinterval_diff
tff(fact_7707_sum__natinterval__diff,axiom,
    ! [Fa: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qw(Fa)),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,M)),aa_nat_rat(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_7708_sum__natinterval__diff,axiom,
    ! [Fa: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qx(Fa)),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,M)),aa_nat_int(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_7709_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qy(Fa)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,N2)),aa_nat_rat(Fa,M)) ) ) ).

% sum_telescope''
tff(fact_7710_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qz(Fa)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,N2)),aa_nat_int(Fa,M)) ) ) ).

% sum_telescope''
tff(fact_7711_choose__row__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) ).

% choose_row_sum
tff(fact_7712_binomial,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ra(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial
tff(fact_7713_sum_Oin__pairs__0,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,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))),N2)))) = groups6857163185585827899at_nat(aTP_Lamm_rb(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7714_sum_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),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))),N2)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_rc(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7715_sum_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),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))),N2)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_rd(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7716_sum_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),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))),N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_re(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_7717_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,X)),aa_Code_integer_nat(code_nat_of_integer,Y))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_7718_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_rf(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(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),one_one_rat)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_7719_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N2) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rg(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7720_binomial__ring,axiom,
    ! [A: int,B: int,N2: 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)),N2) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rh(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7721_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),N2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_ri(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7722_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ra(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_7723_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),set_or1269000886237332187st_nat(M,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_7724_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),set_or1269000886237332187st_nat(M,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_7725_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),set_or1269000886237332187st_nat(M,N2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),M)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_7726_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),N2) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rj(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_7727_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: 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)),N2) = aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rk(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_7728_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),N2) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_rl(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_7729_sum_Oin__pairs,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,M: nat,N2: nat] : groups6857163185585827899at_nat(Ga,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))),N2)))) = groups6857163185585827899at_nat(aTP_Lamm_rb(Ga),set_or1269000886237332187st_nat(M,N2)) ).

% sum.in_pairs
tff(fact_7730_sum_Oin__pairs,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),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))),N2)))) = aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_rc(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% sum.in_pairs
tff(fact_7731_sum_Oin__pairs,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),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))),N2)))) = aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_rd(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% sum.in_pairs
tff(fact_7732_sum_Oin__pairs,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),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))),N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_re(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% sum.in_pairs
tff(fact_7733_choose__square__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rm,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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))),N2)),N2) ).

% choose_square_sum
tff(fact_7734_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_rn(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_ro(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7735_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_na2338080690716861136at_nat,Ha: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups6857163185585827899at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_rp(K2),Ga),Ha),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6857163185585827899at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_rq(K2),Ga),Ha),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7736_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rr(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rs(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7737_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rt(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ru(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7738_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rv(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rw(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7739_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_rx(M),A),X),Y)),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_ry(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_7740_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_rz(K2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7741_gauss__sum__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7742_sum__gp0,axiom,
    ! [X: rat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2)))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ).

% sum_gp0
tff(fact_7743_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_rx(M),A),X),Y)),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_sa(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_7744_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_sb(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_7745_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_sc,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_7746_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_sd(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_7747_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_se(M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_7748_double__arith__series,axiom,
    ! [A: rat,D2: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_sf(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = 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,N2)),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,aa_num_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,N2)),D2))) ).

% double_arith_series
tff(fact_7749_double__arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_sg(A),D2),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = times_2397367101498566445atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N2)),one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A)),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),D2))) ).

% double_arith_series
tff(fact_7750_double__arith__series,axiom,
    ! [A: int,D2: int,N2: 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_sh(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = 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,N2)),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,N2)),D2))) ).

% double_arith_series
tff(fact_7751_double__arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_si(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),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,N2)),D2))) ).

% double_arith_series
tff(fact_7752_double__arith__series,axiom,
    ! [A: nat,D2: nat,N2: 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_sj(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = 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,N2)),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,N2)),D2))) ).

% double_arith_series
tff(fact_7753_double__gauss__sum,axiom,
    ! [N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7754_double__gauss__sum,axiom,
    ! [N2: nat] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),groups6325495683096345652atural(semiri3763490453095760265atural,set_or1269000886237332187st_nat(zero_zero_nat,N2))) = times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N2)),one_one_Code_natural)) ).

% double_gauss_sum
tff(fact_7755_double__gauss__sum,axiom,
    ! [N2: 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),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int)) ).

% double_gauss_sum
tff(fact_7756_double__gauss__sum,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_7757_double__gauss__sum,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7758_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_7759_choose__linear__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sk,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),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,N2),one_one_nat))) ).

% choose_linear_sum
tff(fact_7760_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sl(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = 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,N2)),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,N2),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_7761_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(M,N2)) = 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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_7762_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_sm(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_7763_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_sn,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_7764_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_so(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_7765_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_set_nat_rat(groups2906978787729119204at_rat(semiri681578069525770553at_rat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7766_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),groups6325495683096345652atural(semiri3763490453095760265atural,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2))) = times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N2)),one_one_Code_natural)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7767_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: 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),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7768_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7769_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7770_arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] : groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_sg(A),D2),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N2)),one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),A)),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),D2))),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7771_arith__series,axiom,
    ! [A: int,D2: int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_sh(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = 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,N2)),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,N2)),D2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7772_arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_si(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),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,N2)),D2)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7773_arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sj(A),D2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = 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,N2)),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,N2)),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7774_gauss__sum,axiom,
    ! [N2: nat] : groups6325495683096345652atural(semiri3763490453095760265atural,set_or1269000886237332187st_nat(zero_zero_nat,N2)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N2)),one_one_Code_natural)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7775_gauss__sum,axiom,
    ! [N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = 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,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7776_gauss__sum,axiom,
    ! [N2: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7777_gauss__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7778_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = $ite(X = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ).

% sum_gp_offset
tff(fact_7779_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : groups6325495683096345652atural(semiri3763490453095760265atural,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = divide5121882707175180666atural(times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,N2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(semiri3763490453095760265atural,N2)),one_one_Code_natural)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7780_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(semiri1314217659103216013at_int),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = 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,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7781_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(semiri4939895301339042750nteger),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7782_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7783_gchoose__row__sum__weighted,axiom,
    ! [R: rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_pt(R)),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7784_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,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_7785_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z3),aa_nat_nat(suc,N2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N2))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_sp(Z3)),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))),N2)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7786_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
          aa_Pro4693618923521437820nteger(aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer),L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K2,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_sq(L)),code_divmod_abs(K2,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_7787_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),N2)
     => ( groups4538972089207619220nt_int(aTP_Lamm_fr,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M),N2)) = 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,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,N2),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_7788_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq),set_or4665077453230672383an_nat(M,N2)) = 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_7789_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),one_one_nat) ).

% sum_power2
tff(fact_7790_of__nat__id,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N2) = N2 ).

% of_nat_id
tff(fact_7791_apsnd__conv,axiom,
    ! [Fa: fun_int_int,X: int,Y: int] : aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),aa_int_int(Fa,Y)) ).

% apsnd_conv
tff(fact_7792_apsnd__conv,axiom,
    ! [Fa: fun_Pr7091840695141096449et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : aa_Pro5931322043027775230et_nat(produc1991759244683392747_nat_o(Fa),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y)) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),aa_Pro3679490825741336130et_nat(Fa,Y)) ).

% apsnd_conv
tff(fact_7793_apsnd__conv,axiom,
    ! [Fa: fun_Pr4885567672785483807et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : aa_Pro1979504083478021276et_nat(produc4622684310891945677_nat_o(Fa),aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),Y)) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro434340453207536480et_nat(Fa,Y)) ).

% apsnd_conv
tff(fact_7794_apsnd__conv,axiom,
    ! [Fa: fun_Pr2813956079162534559et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : produc174852792182663501_nat_o(Fa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y)) = aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,X),aa_Pro5541488491759012064et_nat(Fa,Y)) ).

% apsnd_conv
tff(fact_7795_apsnd__conv,axiom,
    ! [Fa: fun_Pr836988951548913597et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : produc2056317742435807279_nat_o(Fa,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),Y)) = aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,X),aa_Pro5931322043027775230et_nat(Fa,Y)) ).

% apsnd_conv
tff(fact_7796_apsnd__conv,axiom,
    ! [Fa: fun_Pr6333000437718031765it_nat,X: b,Y: produc6653097349344004940it_nat] : aa_Pro88785284265212210it_nat(produc701890989253350759_nat_b(Fa),aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),Y)) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,X),aa_Pro3544901040849163478it_nat(Fa,Y)) ).

% apsnd_conv
tff(fact_7797_apsnd__conv,axiom,
    ! [Fa: fun_Pr6333000437718031765it_nat,X: a,Y: produc6653097349344004940it_nat] : produc701890989253350758_nat_a(Fa,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),Y)) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,X),aa_Pro3544901040849163478it_nat(Fa,Y)) ).

% apsnd_conv
tff(fact_7798_apsnd__conv,axiom,
    ! [Fa: fun_Co8499238673658658103nteger,X: code_integer,Y: code_integer] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,Fa),produc1086072967326762835nteger(X,Y)) = produc1086072967326762835nteger(X,aa_Cod1783974652533396728nteger(Fa,Y)) ).

% apsnd_conv
tff(fact_7799_prod_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_sr),A4) = one_one_nat ).

% prod.neutral_const
tff(fact_7800_prod_Oneutral__const,axiom,
    ! [A4: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_ss),A4) = one_one_int ).

% prod.neutral_const
tff(fact_7801_prod_Oneutral__const,axiom,
    ! [A4: set_int] : groups1705073143266064639nt_int(aTP_Lamm_st,A4) = one_one_int ).

% prod.neutral_const
tff(fact_7802_of__nat__prod,axiom,
    ! [Fa: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(Fa,A4)) = groups1705073143266064639nt_int(aTP_Lamm_nv(Fa),A4) ).

% of_nat_prod
tff(fact_7803_of__nat__prod,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_nx(Fa)),A4) ).

% of_nat_prod
tff(fact_7804_of__nat__prod,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ny(Fa)),A4) ).

% of_nat_prod
tff(fact_7805_of__nat__prod,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_nw(Fa)),A4) ).

% of_nat_prod
tff(fact_7806_of__int__prod,axiom,
    ! [Fa: fun_nat_int,A4: set_nat] : aa_int_rat(ring_1_of_int_rat,aa_set_nat_int(groups705719431365010083at_int(Fa),A4)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_su(Fa)),A4) ).

% of_int_prod
tff(fact_7807_of__int__prod,axiom,
    ! [Fa: fun_nat_int,A4: set_nat] : aa_int_int(ring_1_of_int_int,aa_set_nat_int(groups705719431365010083at_int(Fa),A4)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_sv(Fa)),A4) ).

% of_int_prod
tff(fact_7808_of__int__prod,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(Fa,A4)) = groups1072433553688619179nt_rat(aTP_Lamm_nt(Fa),A4) ).

% of_int_prod
tff(fact_7809_of__int__prod,axiom,
    ! [Fa: fun_int_int,A4: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(Fa,A4)) = groups1705073143266064639nt_int(aTP_Lamm_nu(Fa),A4) ).

% of_int_prod
tff(fact_7810_atLeastLessThan__iff,axiom,
    ! [I2: $o,L: $o,U: $o] :
      ( aa_set_o_o(member_o((I2)),set_or7139685690850216873Than_o((L),(U)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(I2))
        & aa_o_o(aa_o_fun_o_o(ord_less_o,(I2)),(U)) ) ) ).

% atLeastLessThan_iff
tff(fact_7811_atLeastLessThan__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,L: set_Pr958786334691620121nt_int,U: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),set_or2826562449856406570nt_int(L,U))
    <=> ( aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(L),I2)
        & aa_set6170482117830564987_int_o(ord_le7563427860532173253nt_int(I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7812_atLeastLessThan__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or3540276404033026485et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7813_atLeastLessThan__iff,axiom,
    ! [I2: assn,L: assn,U: assn] :
      ( member_assn(I2,set_or5523502641901747543n_assn(L,U))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,L),I2)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7814_atLeastLessThan__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( member_set_int(I2,set_or8585797421378605585et_int(L,U))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,L),I2)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7815_atLeastLessThan__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I2),set_or4029947393144176647an_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7816_atLeastLessThan__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I2),set_or1222409239386451017an_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I2)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7817_atLeastLessThan__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I2),set_or4665077453230672383an_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7818_atLeastLessThan__iff,axiom,
    ! [I2: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),set_or8404916559141939852nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7819_atLeastLessThan__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I2),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),U) ) ) ).

% atLeastLessThan_iff
tff(fact_7820_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(A))
     => ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
tff(fact_7821_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),A)
     => ( set_or8585797421378605585et_int(A,B) = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_7822_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_7823_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_7824_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( set_or4665077453230672383an_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_7825_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_7826_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_7827_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or7139685690850216873Than_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7828_atLeastLessThan__empty__iff2,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_set_assn = set_or5523502641901747543n_assn(A,B) )
    <=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7829_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7830_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7831_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or4665077453230672383an_nat(A,B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7832_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7833_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_7834_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_7835_atLeastLessThan__empty__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( set_or5523502641901747543n_assn(A,B) = bot_bot_set_assn )
    <=> ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7836_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7837_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7838_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or4665077453230672383an_nat(A,B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7839_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7840_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_7841_ivl__subset,axiom,
    ! [I2: rat,J2: rat,M: rat,N2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(I2,J2)),set_or4029947393144176647an_rat(M,N2))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J2),I2)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),I2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J2),N2) ) ) ) ).

% ivl_subset
tff(fact_7842_ivl__subset,axiom,
    ! [I2: num,J2: num,M: num,N2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or1222409239386451017an_num(I2,J2)),set_or1222409239386451017an_num(M,N2))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J2),I2)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),I2)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J2),N2) ) ) ) ).

% ivl_subset
tff(fact_7843_ivl__subset,axiom,
    ! [I2: nat,J2: nat,M: nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or4665077453230672383an_nat(I2,J2)),set_or4665077453230672383an_nat(M,N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2)
        | ( 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,J2),N2) ) ) ) ).

% ivl_subset
tff(fact_7844_ivl__subset,axiom,
    ! [I2: code_integer,J2: code_integer,M: code_integer,N2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I2,J2)),set_or8404916559141939852nteger(M,N2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J2),I2)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),I2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J2),N2) ) ) ) ).

% ivl_subset
tff(fact_7845_ivl__subset,axiom,
    ! [I2: int,J2: int,M: int,N2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I2),J2)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,M),N2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),I2)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),N2) ) ) ) ).

% ivl_subset
tff(fact_7846_ivl__diff,axiom,
    ! [I2: rat,N2: rat,M: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),N2)
     => ( minus_minus_set_rat(set_or4029947393144176647an_rat(I2,M),set_or4029947393144176647an_rat(I2,N2)) = set_or4029947393144176647an_rat(N2,M) ) ) ).

% ivl_diff
tff(fact_7847_ivl__diff,axiom,
    ! [I2: num,N2: num,M: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),N2)
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I2,M),set_or1222409239386451017an_num(I2,N2)) = set_or1222409239386451017an_num(N2,M) ) ) ).

% ivl_diff
tff(fact_7848_ivl__diff,axiom,
    ! [I2: nat,N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_or4665077453230672383an_nat(I2,M)),set_or4665077453230672383an_nat(I2,N2)) = set_or4665077453230672383an_nat(N2,M) ) ) ).

% ivl_diff
tff(fact_7849_ivl__diff,axiom,
    ! [I2: code_integer,N2: code_integer,M: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),N2)
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I2,M),set_or8404916559141939852nteger(I2,N2)) = set_or8404916559141939852nteger(N2,M) ) ) ).

% ivl_diff
tff(fact_7850_ivl__diff,axiom,
    ! [I2: int,N2: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),N2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I2),M)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,I2),N2)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,N2),M) ) ) ).

% ivl_diff
tff(fact_7851_sum_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_Code_integer(Ga,N2))) ).

% sum.op_ivl_Suc
tff(fact_7852_sum_Oop__ivl__Suc,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,M: nat,N2: nat] :
      groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z1048942125864253310at_nat,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,N2))),aa_nat2240542744625857721at_nat(Ga,N2))) ).

% sum.op_ivl_Suc
tff(fact_7853_sum_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_rat(Ga,N2))) ).

% sum.op_ivl_Suc
tff(fact_7854_sum_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_int(Ga,N2))) ).

% sum.op_ivl_Suc
tff(fact_7855_sum_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_nat(Ga,N2))) ).

% sum.op_ivl_Suc
tff(fact_7856_prod_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_Code_integer(Ga,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_7857_prod_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_assn,M: nat,N2: nat] :
      groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2))),aa_nat_assn(Ga,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_7858_prod_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_rat(Ga,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_7859_prod_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_nat(Ga,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_7860_prod_Ocl__ivl__Suc,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N2)),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_int(Ga,aa_nat_nat(suc,N2)))) ).

% prod.cl_ivl_Suc
tff(fact_7861_prod_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_Code_integer(Ga,N2))) ).

% prod.op_ivl_Suc
tff(fact_7862_prod_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_assn,M: nat,N2: nat] :
      groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,N2))),aa_nat_assn(Ga,N2))) ).

% prod.op_ivl_Suc
tff(fact_7863_prod_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_rat(Ga,N2))) ).

% prod.op_ivl_Suc
tff(fact_7864_prod_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_nat(Ga,N2))) ).

% prod.op_ivl_Suc
tff(fact_7865_prod_Oop__ivl__Suc,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_int(Ga,N2))) ).

% prod.op_ivl_Suc
tff(fact_7866_prod_Oshift__bounds__nat__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Ga),K2)),set_or4665077453230672383an_nat(M,N2)) ).

% prod.shift_bounds_nat_ivl
tff(fact_7867_prod_Oshift__bounds__nat__ivl,axiom,
    ! [Ga: fun_nat_int,M: nat,K2: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sw(Ga),K2)),set_or4665077453230672383an_nat(M,N2)) ).

% prod.shift_bounds_nat_ivl
tff(fact_7868_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,K2))),set_or4665077453230672383an_nat(M,N2)) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_7869_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [Ga: fun_nat_int,M: nat,K2: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,K2))),set_or4665077453230672383an_nat(M,N2)) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_7870_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,N2))),groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(N2,P2))) = groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7871_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(N2,P2))) = aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7872_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(N2,P2))) = aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7873_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2))),aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(N2,P2))) = aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_7874_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),set_or4665077453230672383an_nat(M,N2)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_7875_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),set_or4665077453230672383an_nat(M,N2)) ).

% prod.shift_bounds_Suc_ivl
tff(fact_7876_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: 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,C2),D2)
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C2,D2) )
        <=> ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7877_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D2)
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C2,D2) )
        <=> ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7878_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: 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,C2),D2)
       => ( ( set_or4665077453230672383an_nat(A,B) = set_or4665077453230672383an_nat(C2,D2) )
        <=> ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7879_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
       => ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C2,D2) )
        <=> ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7880_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D2: 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,C2),D2)
       => ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2) )
        <=> ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_7881_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C2,D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7882_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C2,D2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D2)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7883_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( set_or4665077453230672383an_nat(A,B) = set_or4665077453230672383an_nat(C2,D2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7884_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C2,D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7885_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_7886_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C2,D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7887_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C2,D2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7888_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( set_or4665077453230672383an_nat(A,B) = set_or4665077453230672383an_nat(C2,D2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7889_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C2,D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7890_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_7891_prod_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat,Ga: fun_nat_nat,Ha: fun_nat_nat] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D2)
               => ( aa_nat_nat(Ga,X3) = aa_nat_nat(Ha,X3) ) ) )
         => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(A,B)) = aa_set_nat_nat(groups708209901874060359at_nat(Ha),set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_7892_prod_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D2)
               => ( aa_nat_int(Ga,X3) = aa_nat_int(Ha,X3) ) ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(A,B)) = aa_set_nat_int(groups705719431365010083at_int(Ha),set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_7893_prod_Oivl__cong,axiom,
    ! [A: int,C2: int,B: int,D2: int,Ga: fun_int_int,Ha: fun_int_int] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),X3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D2)
               => ( aa_int_int(Ga,X3) = aa_int_int(Ha,X3) ) ) )
         => ( groups1705073143266064639nt_int(Ga,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B)) = groups1705073143266064639nt_int(Ha,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2)) ) ) ) ) ).

% prod.ivl_cong
tff(fact_7894_prod_Oswap,axiom,
    ! [Ga: fun_nat_fun_nat_nat,B4: set_nat,A4: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_sx(Ga),B4)),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_sy(Ga),A4)),B4) ).

% prod.swap
tff(fact_7895_prod_Oswap,axiom,
    ! [Ga: fun_nat_fun_nat_int,B4: set_nat,A4: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_sz(Ga),B4)),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_set2359242532610849469at_int(aTP_Lamm_tb(Ga),A4)),B4) ).

% prod.swap
tff(fact_7896_prod_Oswap,axiom,
    ! [Ga: fun_nat_fun_int_int,B4: set_int,A4: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_tc(Ga),B4)),A4) = groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_te(Ga),A4),B4) ).

% prod.swap
tff(fact_7897_prod_Oswap,axiom,
    ! [Ga: fun_int_fun_nat_int,B4: set_nat,A4: set_int] : groups1705073143266064639nt_int(aa_set3358596244511904025nt_int(aTP_Lamm_tf(Ga),B4),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_set6310687161026558745at_int(aTP_Lamm_th(Ga),A4)),B4) ).

% prod.swap
tff(fact_7898_prod_Oswap,axiom,
    ! [Ga: fun_int_fun_int_int,B4: set_int,A4: set_int] : groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_ti(Ga),B4),A4) = groups1705073143266064639nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_tj(Ga),A4),B4) ).

% prod.swap
tff(fact_7899_prod_Odistrib,axiom,
    ! [Ga: fun_nat_nat,Ha: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tk(Ga),Ha)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),A4)),aa_set_nat_nat(groups708209901874060359at_nat(Ha),A4)) ).

% prod.distrib
tff(fact_7900_prod_Odistrib,axiom,
    ! [Ga: fun_nat_int,Ha: fun_nat_int,A4: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_tl(Ga),Ha)),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),A4)),aa_set_nat_int(groups705719431365010083at_int(Ha),A4)) ).

% prod.distrib
tff(fact_7901_prod_Odistrib,axiom,
    ! [Ga: fun_int_int,Ha: fun_int_int,A4: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tm(Ga),Ha),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(Ga,A4)),groups1705073143266064639nt_int(Ha,A4)) ).

% prod.distrib
tff(fact_7902_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,M)),groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7903_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,M)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7904_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,M)),aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7905_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,M)),aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_7906_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(A,B))),aa_nat_assn(Ga,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7907_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(A,B))),aa_nat_rat(Ga,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7908_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(A,B))),aa_nat_nat(Ga,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7909_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(A,B))),aa_nat_int(Ga,B)) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_7910_prod__power__distrib,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tn(Fa),N2)),A4) ).

% prod_power_distrib
tff(fact_7911_prod__power__distrib,axiom,
    ! [Fa: fun_nat_int,A4: set_nat,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_set_nat_int(groups705719431365010083at_int(Fa),A4)),N2) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_to(Fa),N2)),A4) ).

% prod_power_distrib
tff(fact_7912_prod__power__distrib,axiom,
    ! [Fa: fun_int_int,A4: set_int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,groups1705073143266064639nt_int(Fa,A4)),N2) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_tp(Fa),N2),A4) ).

% prod_power_distrib
tff(fact_7913_prod_OatLeastLessThan__shift__0,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,M))),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ).

% prod.atLeastLessThan_shift_0
tff(fact_7914_prod_OatLeastLessThan__shift__0,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,M))),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ).

% prod.atLeastLessThan_shift_0
tff(fact_7915_mod__prod__eq,axiom,
    ! [Fa: fun_nat_nat,A: nat,A4: 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_nz(Fa),A)),A4)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)),A) ).

% mod_prod_eq
tff(fact_7916_mod__prod__eq,axiom,
    ! [Fa: fun_nat_int,A: int,A4: 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_tq(Fa),A)),A4)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_set_nat_int(groups705719431365010083at_int(Fa),A4)),A) ).

% mod_prod_eq
tff(fact_7917_mod__prod__eq,axiom,
    ! [Fa: fun_int_int,A: int,A4: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_oa(Fa),A),A4)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(Fa,A4)),A) ).

% mod_prod_eq
tff(fact_7918_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,N2)),groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,N2))) ) ) ).

% prod.last_plus
tff(fact_7919_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,N2)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,N2))) ) ) ).

% prod.last_plus
tff(fact_7920_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,N2)),aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2))) ) ) ).

% prod.last_plus
tff(fact_7921_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,N2)),aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2))) ) ) ).

% prod.last_plus
tff(fact_7922_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(Ga,aTP_Lamm_qh)),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2)) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7923_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(Ga,aTP_Lamm_qh)),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2)) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7924_prod_OatLeastLessThan__rev,axiom,
    ! [Ga: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(N2,M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_tr(Ga),N2),M)),set_or4665077453230672383an_nat(N2,M)) ).

% prod.atLeastLessThan_rev
tff(fact_7925_prod_OatLeastLessThan__rev,axiom,
    ! [Ga: fun_nat_int,N2: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(N2,M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ts(Ga),N2),M)),set_or4665077453230672383an_nat(N2,M)) ).

% prod.atLeastLessThan_rev
tff(fact_7926_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_tt(A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu(A),N2)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% prod.nested_swap
tff(fact_7927_prod_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_tv(A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_tw(A),N2)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% prod.nested_swap
tff(fact_7928_apsnd__compose,axiom,
    ! [Fa: fun_Co1561183069634336733nteger,Ga: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger] : produc5043883716779652157nteger(Fa,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,Ga),X)) = produc5043883716779652157nteger(comp_C7848664941760151970nteger(Fa,Ga),X) ).

% apsnd_compose
tff(fact_7929_apsnd__compose,axiom,
    ! [Fa: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,Fa),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,Ga),X)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,aa_fun4435975922079263404nteger(comp_C7449957260575251196nteger(Fa),Ga)),X) ).

% apsnd_compose
tff(fact_7930_prod__nonneg,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) ) ).

% prod_nonneg
tff(fact_7931_prod__nonneg,axiom,
    ! [A4: set_nat,Fa: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(Fa),A4)) ) ).

% prod_nonneg
tff(fact_7932_prod__nonneg,axiom,
    ! [A4: set_int,Fa: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(Fa,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(Fa,A4)) ) ).

% prod_nonneg
tff(fact_7933_prod__mono,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer,Ga: fun_o_Code_integer] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A4)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I3)))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(Fa,(I3))),aa_o_Code_integer(Ga,(I3))) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(Fa,A4)),groups7694694392188491536nteger(Ga,A4)) ) ).

% prod_mono
tff(fact_7934_prod__mono,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer,Ga: fun_nat_Code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A4)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,I3)),aa_nat_Code_integer(Ga,I3)) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),A4)),aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),A4)) ) ).

% prod_mono
tff(fact_7935_prod__mono,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer,Ga: fun_int_Code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A4)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,I3)),aa_int_Code_integer(Ga,I3)) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(Fa,A4)),groups3827104343326376752nteger(Ga,A4)) ) ).

% prod_mono
tff(fact_7936_prod__mono,axiom,
    ! [A4: set_o,Fa: fun_o_rat,Ga: fun_o_rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(I3)))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(I3))),aa_o_rat(Ga,(I3))) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(Fa,A4)),groups2869687844427037835_o_rat(Ga,A4)) ) ).

% prod_mono
tff(fact_7937_prod__mono,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat,Ga: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I3)),aa_nat_rat(Ga,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(Fa),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4)) ) ).

% prod_mono
tff(fact_7938_prod__mono,axiom,
    ! [A4: set_int,Fa: fun_int_rat,Ga: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I3)),aa_int_rat(Ga,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(Fa,A4)),groups1072433553688619179nt_rat(Ga,A4)) ) ).

% prod_mono
tff(fact_7939_prod__mono,axiom,
    ! [A4: set_o,Fa: fun_o_nat,Ga: fun_o_nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(I3)))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(I3))),aa_o_nat(Ga,(I3))) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3504817904513533571_o_nat(Fa,A4)),groups3504817904513533571_o_nat(Ga,A4)) ) ).

% prod_mono
tff(fact_7940_prod__mono,axiom,
    ! [A4: set_int,Fa: fun_int_nat,Ga: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,I3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,I3)),aa_int_nat(Ga,I3)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(Fa,A4)),groups1707563613775114915nt_nat(Ga,A4)) ) ).

% prod_mono
tff(fact_7941_prod__mono,axiom,
    ! [A4: set_o,Fa: fun_o_int,Ga: fun_o_int] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Fa,(I3)))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(Fa,(I3))),aa_o_int(Ga,(I3))) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(Fa,A4)),groups3502327434004483295_o_int(Ga,A4)) ) ).

% prod_mono
tff(fact_7942_prod__mono,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat,Ga: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,I3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,I3)),aa_nat_nat(Ga,I3)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)),aa_set_nat_nat(groups708209901874060359at_nat(Ga),A4)) ) ).

% prod_mono
tff(fact_7943_prod_Ohead__if,axiom,
    ! [Ga: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_Code_integer(Ga,N2))) ).

% prod.head_if
tff(fact_7944_prod_Ohead__if,axiom,
    ! [Ga: fun_nat_assn,M: nat,N2: nat] :
      groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or4665077453230672383an_nat(M,N2))),aa_nat_assn(Ga,N2))) ).

% prod.head_if
tff(fact_7945_prod_Ohead__if,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_rat(Ga,N2))) ).

% prod.head_if
tff(fact_7946_prod_Ohead__if,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_nat(Ga,N2))) ).

% prod.head_if
tff(fact_7947_prod_Ohead__if,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_int(Ga,N2))) ).

% prod.head_if
tff(fact_7948_prod__pos,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Fa,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) ) ).

% prod_pos
tff(fact_7949_prod__pos,axiom,
    ! [A4: set_nat,Fa: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(Fa,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups705719431365010083at_int(Fa),A4)) ) ).

% prod_pos
tff(fact_7950_prod__pos,axiom,
    ! [A4: set_int,Fa: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(Fa,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(Fa,A4)) ) ).

% prod_pos
tff(fact_7951_prod__ge__1,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(Fa,(X3))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups7694694392188491536nteger(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7952_prod__ge__1,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(Fa,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),A4)) ) ).

% prod_ge_1
tff(fact_7953_prod__ge__1,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(Fa,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups3827104343326376752nteger(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7954_prod__ge__1,axiom,
    ! [A4: set_o,Fa: fun_o_rat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(Fa,(X3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups2869687844427037835_o_rat(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7955_prod__ge__1,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(Fa,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(Fa),A4)) ) ).

% prod_ge_1
tff(fact_7956_prod__ge__1,axiom,
    ! [A4: set_int,Fa: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(Fa,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7957_prod__ge__1,axiom,
    ! [A4: set_o,Fa: fun_o_nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_o_nat(Fa,(X3))) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups3504817904513533571_o_nat(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7958_prod__ge__1,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(Fa,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7959_prod__ge__1,axiom,
    ! [A4: set_o,Fa: fun_o_int] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(Fa,(X3))) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups3502327434004483295_o_int(Fa,A4)) ) ).

% prod_ge_1
tff(fact_7960_prod__ge__1,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(Fa,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) ) ).

% prod_ge_1
tff(fact_7961_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B)),set_or4029947393144176647an_rat(C2,D2))
     => ( 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,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7962_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or1222409239386451017an_num(A,B)),set_or1222409239386451017an_num(C2,D2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7963_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or4665077453230672383an_nat(A,B)),set_or4665077453230672383an_nat(C2,D2))
     => ( 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,C2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7964_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A,B)),set_or8404916559141939852nteger(C2,D2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7965_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2))
     => ( 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,C2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_7966_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,K2))),set_or1269000886237332187st_nat(M,N2)) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_7967_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [Ga: fun_nat_int,M: nat,K2: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,K2))),set_or1269000886237332187st_nat(M,N2)) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_7968_ivl__disj__un__two_I3_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( sup_sup_set_rat(set_or4029947393144176647an_rat(L,M),set_or4029947393144176647an_rat(M,U)) = set_or4029947393144176647an_rat(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7969_ivl__disj__un__two_I3_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or1222409239386451017an_num(L,M),set_or1222409239386451017an_num(M,U)) = set_or1222409239386451017an_num(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7970_ivl__disj__un__two_I3_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or4665077453230672383an_nat(L,M)),set_or4665077453230672383an_nat(M,U)) = set_or4665077453230672383an_nat(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7971_ivl__disj__un__two_I3_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,M),set_or8404916559141939852nteger(M,U)) = set_or8404916559141939852nteger(L,U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7972_ivl__disj__un__two_I3_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),M)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,M),U)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7973_all__nat__less__eq,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
         => aa_nat_o(Pa,M3) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,N2))
         => aa_nat_o(Pa,X4) ) ) ).

% all_nat_less_eq
tff(fact_7974_ex__nat__less__eq,axiom,
    ! [N2: nat,Pa: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
          & aa_nat_o(Pa,M3) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,N2))
          & aa_nat_o(Pa,X4) ) ) ).

% ex_nat_less_eq
tff(fact_7975_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [Ga: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or4665077453230672383an_nat(N2,M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(Ga),N2),M)),set_or1269000886237332187st_nat(aa_nat_nat(suc,N2),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7976_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [Ga: fun_nat_int,N2: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or4665077453230672383an_nat(N2,M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_tx(Ga),N2),M)),set_or1269000886237332187st_nat(aa_nat_nat(suc,N2),M)) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7977_pochhammer__prod,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ty(A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% pochhammer_prod
tff(fact_7978_pochhammer__prod,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_tz(A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% pochhammer_prod
tff(fact_7979_pochhammer__prod,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% pochhammer_prod
tff(fact_7980_pochhammer__prod,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ub,A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% pochhammer_prod
tff(fact_7981_fact__prod__rev,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2)),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% fact_prod_rev
tff(fact_7982_fact__prod__rev,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2)),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% fact_prod_rev
tff(fact_7983_fact__prod__rev,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2)),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% fact_prod_rev
tff(fact_7984_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,K2))),set_or4665077453230672383an_nat(M,N2)) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_7985_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7986_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7987_power__sum,axiom,
    ! [C2: nat,Fa: fun_nat_nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C2),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_uc(C2),Fa)),A4) ).

% power_sum
tff(fact_7988_power__sum,axiom,
    ! [C2: int,Fa: fun_nat_nat,A4: set_nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C2),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_ud(C2),Fa)),A4) ).

% power_sum
tff(fact_7989_power__sum,axiom,
    ! [C2: int,Fa: fun_int_nat,A4: set_int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,C2),groups4541462559716669496nt_nat(Fa,A4)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_ue(C2),Fa),A4) ).

% power_sum
tff(fact_7990_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Ga),K2)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7991_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [Ga: fun_nat_int,M: nat,K2: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sw(Ga),K2)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7992_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),set_or4665077453230672383an_nat(M,N2)) ).

% sum.shift_bounds_Suc_ivl
tff(fact_7993_sum_Oshift__bounds__nat__ivl,axiom,
    ! [Ga: fun_nat_nat,M: nat,K2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Ga),K2)),set_or4665077453230672383an_nat(M,N2)) ).

% sum.shift_bounds_nat_ivl
tff(fact_7994_prod__le__1,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(X3)))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(Fa,(X3))),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(Fa,A4)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7995_prod__le__1,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),A4)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7996_prod__le__1,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(Fa,A4)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7997_prod__le__1,axiom,
    ! [A4: set_o,Fa: fun_o_rat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(X3)))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(X3))),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(Fa,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7998_prod__le__1,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(Fa),A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7999_prod__le__1,axiom,
    ! [A4: set_int,Fa: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(Fa,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_8000_prod__le__1,axiom,
    ! [A4: set_o,Fa: fun_o_nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(X3)))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(X3))),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups3504817904513533571_o_nat(Fa,A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_8001_prod__le__1,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(Fa,A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_8002_prod__le__1,axiom,
    ! [A4: set_o,Fa: fun_o_int] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Fa,(X3)))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(Fa,(X3))),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(Fa,A4)),one_one_int) ) ).

% prod_le_1
tff(fact_8003_prod__le__1,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_8004_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2),N2)))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))) ) ) ).

% fact_split
tff(fact_8005_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2),N2)))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))) ) ) ).

% fact_split
tff(fact_8006_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2),N2)))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))) ) ) ).

% fact_split
tff(fact_8007_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2),N2)))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K2))) ) ) ).

% fact_split
tff(fact_8008_sum_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat,Ga: fun_nat_nat,Ha: fun_nat_nat] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),D2)
               => ( aa_nat_nat(Ga,X3) = aa_nat_nat(Ha,X3) ) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(A,B)) = aa_set_nat_nat(groups3542108847815614940at_nat(Ha),set_or4665077453230672383an_nat(C2,D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_8009_sum_Oivl__cong,axiom,
    ! [A: int,C2: int,B: int,D2: int,Ga: fun_int_int,Ha: fun_int_int] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),X3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),D2)
               => ( aa_int_int(Ga,X3) = aa_int_int(Ha,X3) ) ) )
         => ( groups4538972089207619220nt_int(Ga,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,A),B)) = groups4538972089207619220nt_int(Ha,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_8010_gbinomial__altdef__of__nat,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_uf(A),K2)),set_or4665077453230672383an_nat(zero_zero_nat,K2)) ).

% gbinomial_altdef_of_nat
tff(fact_8011_binomial__altdef__of__nat,axiom,
    ! [K2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N2)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ug(K2),N2)),set_or4665077453230672383an_nat(zero_zero_nat,K2)) ) ) ).

% binomial_altdef_of_nat
tff(fact_8012_ivl__disj__un__two_I7_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),U)
       => ( sup_sup_set_rat(set_or4029947393144176647an_rat(L,M),set_or633870826150836451st_rat(M,U)) = set_or633870826150836451st_rat(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_8013_ivl__disj__un__two_I7_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),U)
       => ( sup_sup_set_num(set_or1222409239386451017an_num(L,M),set_or7049704709247886629st_num(M,U)) = set_or7049704709247886629st_num(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_8014_ivl__disj__un__two_I7_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or4665077453230672383an_nat(L,M)),set_or1269000886237332187st_nat(M,U)) = set_or1269000886237332187st_nat(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_8015_ivl__disj__un__two_I7_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,M),set_or189985376899183464nteger(M,U)) = set_or189985376899183464nteger(L,U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_8016_ivl__disj__un__two_I7_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),M)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M),U)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_8017_gbinomial__mult__fact,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_uh(A)),set_or4665077453230672383an_nat(zero_zero_nat,K2)) ).

% gbinomial_mult_fact
tff(fact_8018_gbinomial__mult__fact_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_uh(A)),set_or4665077453230672383an_nat(zero_zero_nat,K2)) ).

% gbinomial_mult_fact'
tff(fact_8019_gbinomial__prod__rev,axiom,
    ! [A: code_natural,K2: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),K2) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_ui(A),set_or4665077453230672383an_nat(zero_zero_nat,K2)),aa_nat_Code_natural(semiri2447717529341329178atural,K2)) ).

% gbinomial_prod_rev
tff(fact_8020_gbinomial__prod__rev,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_uh(A)),set_or4665077453230672383an_nat(zero_zero_nat,K2))),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_prod_rev
tff(fact_8021_gbinomial__prod__rev,axiom,
    ! [A: code_integer,K2: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_uj(A)),set_or4665077453230672383an_nat(zero_zero_nat,K2))),aa_nat_Code_integer(semiri3624122377584611663nteger,K2)) ).

% gbinomial_prod_rev
tff(fact_8022_gbinomial__prod__rev,axiom,
    ! [A: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uk,A)),set_or4665077453230672383an_nat(zero_zero_nat,K2))),aa_nat_nat(semiri1408675320244567234ct_nat,K2)) ).

% gbinomial_prod_rev
tff(fact_8023_gbinomial__prod__rev,axiom,
    ! [A: int,K2: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),K2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ul,A)),set_or4665077453230672383an_nat(zero_zero_nat,K2))),aa_nat_int(semiri1406184849735516958ct_int,K2)) ).

% gbinomial_prod_rev
tff(fact_8024_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,N2))),groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(N2,P2))) = groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8025_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(N2,P2))) = aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8026_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,N2))),aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(N2,P2))) = aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8027_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(N2,P2))) = aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,P2)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_8028_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P2: nat,Fa: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),set_or4665077453230672383an_nat(M,P2))),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),set_or4665077453230672383an_nat(M,N2))) = aa_set_nat_rat(groups2906978787729119204at_rat(Fa),set_or4665077453230672383an_nat(N2,P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8029_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P2: nat,Fa: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),P2)
       => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),set_or4665077453230672383an_nat(M,P2))),aa_set_nat_int(groups3539618377306564664at_int(Fa),set_or4665077453230672383an_nat(M,N2))) = aa_set_nat_int(groups3539618377306564664at_int(Fa),set_or4665077453230672383an_nat(N2,P2)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_8030_bset_I1_J,axiom,
    ! [D4: int,B4: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Qa,X3)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X5)
                & aa_int_o(Qa,X5) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4))
                & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_8031_bset_I2_J,axiom,
    ! [D4: int,B4: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Qa,X3)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X5)
                | aa_int_o(Qa,X5) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4))
                | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_8032_aset_I1_J,axiom,
    ! [D4: int,A4: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Qa,X3)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X5)
                & aa_int_o(Qa,X5) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4))
                & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_8033_aset_I2_J,axiom,
    ! [D4: int,A4: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(Pa,X3)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(Qa,X3)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X5)
                | aa_int_o(Qa,X5) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4))
                | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_8034_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( set_or4665077453230672383an_nat(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or4665077453230672383an_nat(I2,J2)),set_or4665077453230672383an_nat(J2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_8035_prod_OatLeastAtMost__rev,axiom,
    ! [Ga: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(N2,M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(Ga),N2),M)),set_or1269000886237332187st_nat(N2,M)) ).

% prod.atLeastAtMost_rev
tff(fact_8036_prod_OatLeastAtMost__rev,axiom,
    ! [Ga: fun_nat_int,N2: nat,M: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(N2,M)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_tx(Ga),N2),M)),set_or1269000886237332187st_nat(N2,M)) ).

% prod.atLeastAtMost_rev
tff(fact_8037_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,M)),groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8038_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,M)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8039_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,M)),aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8040_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,M)),aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8041_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,aa_nat_nat(suc,N2))),groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8042_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,aa_nat_nat(suc,N2))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8043_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,aa_nat_nat(suc,N2))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8044_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N2))
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,aa_nat_nat(suc,N2))),aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8045_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: assn,B: assn,C2: assn,D2: assn] :
      ( ord_less_eq_set_assn(set_or7959216805967363635t_assn(A,B),set_or5523502641901747543n_assn(C2,D2))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
       => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C2),A)
          & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8046_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(A,B),set_or8585797421378605585et_int(C2,D2))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C2),A)
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8047_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or4029947393144176647an_rat(C2,D2))
    <=> ( 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,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8048_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or1222409239386451017an_num(C2,D2))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_num,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8049_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(A,B)),set_or4665077453230672383an_nat(C2,D2))
    <=> ( 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,C2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8050_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A,B)),set_or8404916559141939852nteger(C2,D2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8051_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,A),B)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,C2),D2))
    <=> ( 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,C2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_8052_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B)),set_or633870826150836451st_rat(C2,D2))
    <=> ( 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,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D2) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_8053_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),U)
       => ( sup_sup_set_rat(set_or633870826150836451st_rat(L,M),set_or4029947393144176647an_rat(M,U)) = set_or4029947393144176647an_rat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_8054_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: num,M: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),M)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,M),U)
       => ( sup_sup_set_num(set_or7049704709247886629st_num(L,M),set_or1222409239386451017an_num(M,U)) = set_or1222409239386451017an_num(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_8055_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),U)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or1269000886237332187st_nat(L,M)),set_or4665077453230672383an_nat(M,U)) = set_or4665077453230672383an_nat(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_8056_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: code_integer,M: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,M),U)
       => ( sup_su848401254843788991nteger(set_or189985376899183464nteger(L,M),set_or8404916559141939852nteger(M,U)) = set_or8404916559141939852nteger(L,U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_8057_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: int,M: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),U)
       => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),M)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,M),U)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_8058_sum_OatLeast0__lessThan__Suc,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(zero_zero_nat,N2))),aa_nat2240542744625857721at_nat(Ga,N2)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8059_sum_OatLeast0__lessThan__Suc,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(zero_zero_nat,N2))),aa_nat_rat(Ga,N2)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8060_sum_OatLeast0__lessThan__Suc,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(zero_zero_nat,N2))),aa_nat_int(Ga,N2)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8061_sum_OatLeast0__lessThan__Suc,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(zero_zero_nat,N2))),aa_nat_nat(Ga,N2)) ).

% sum.atLeast0_lessThan_Suc
tff(fact_8062_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,zero_zero_nat)),groups6857163185585827899at_nat(comp_n8698576032424989604at_nat(Ga,suc),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_8063_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(comp_nat_rat_nat(Ga,suc)),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_8064_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(comp_nat_int_nat(Ga,suc)),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_8065_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,suc)),set_or4665077453230672383an_nat(zero_zero_nat,N2))) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_8066_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,M)),groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8067_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,M)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8068_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,M)),aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8069_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,M)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),N2))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_8070_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(A,aa_nat_nat(suc,B))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(A,B))),aa_nat2240542744625857721at_nat(Ga,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8071_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),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(Ga),set_or4665077453230672383an_nat(A,B))),aa_nat_rat(Ga,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8072_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),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(Ga),set_or4665077453230672383an_nat(A,B))),aa_nat_int(Ga,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8073_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),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(Ga),set_or4665077453230672383an_nat(A,B))),aa_nat_nat(Ga,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_8074_sum_OatLeastLessThan__shift__0,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,M))),set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ).

% sum.atLeastLessThan_shift_0
tff(fact_8075_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_na2338080690716861136at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,N2)),groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,N2))) ) ) ).

% sum.last_plus
tff(fact_8076_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,N2)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,N2))) ) ) ).

% sum.last_plus
tff(fact_8077_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,N2)),aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,N2))) ) ) ).

% sum.last_plus
tff(fact_8078_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,N2)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2))) ) ) ).

% sum.last_plus
tff(fact_8079_aset_I10_J,axiom,
    ! [D2: int,D4: int,A4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T6))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)),T6)) ) ) ) ).

% aset(10)
tff(fact_8080_aset_I9_J,axiom,
    ! [D2: int,D4: int,A4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T6))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)),T6)) ) ) ) ).

% aset(9)
tff(fact_8081_bset_I10_J,axiom,
    ! [D2: int,D4: int,B4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T6))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)),T6)) ) ) ) ).

% bset(10)
tff(fact_8082_bset_I9_J,axiom,
    ! [D2: int,D4: int,B4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),T6))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)),T6)) ) ) ) ).

% bset(9)
tff(fact_8083_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2))),aa_nat_assn(Ga,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,M)),groups6906906614972039071t_assn(aTP_Lamm_um(Ga),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8084_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_rat(Ga,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,M)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qb(Ga)),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8085_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_nat(Ga,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8086_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2))),aa_nat_int(Ga,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,M)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),set_or1269000886237332187st_nat(M,N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8087_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2))
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,M),N2)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8088_prod_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(Ga,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_um(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8089_prod_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qb(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8090_prod_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8091_prod_OatMost__Suc__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8092_prod_OatLeast__atMost__pred__shift,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(Ga,aTP_Lamm_qh)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) ).

% prod.atLeast_atMost_pred_shift
tff(fact_8093_prod_OatLeast__atMost__pred__shift,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(Ga,aTP_Lamm_qh)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2)) ).

% prod.atLeast_atMost_pred_shift
tff(fact_8094_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      set_or4665077453230672383an_nat(M,aa_nat_nat(suc,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2),insert_nat(N2,set_or4665077453230672383an_nat(M,N2)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_8095_fact__prod,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(one_one_nat,N2))) ).

% fact_prod
tff(fact_8096_fact__prod,axiom,
    ! [N2: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,N2) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(one_one_nat,N2))) ).

% fact_prod
tff(fact_8097_fact__prod,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(one_one_nat,N2))) ).

% fact_prod
tff(fact_8098_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qf(Fa)),set_or4665077453230672383an_nat(M,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,N2)),aa_nat_rat(Fa,M)) ) ) ).

% sum_Suc_diff'
tff(fact_8099_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,Fa: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qg(Fa)),set_or4665077453230672383an_nat(M,N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,N2)),aa_nat_int(Fa,M)) ) ) ).

% sum_Suc_diff'
tff(fact_8100_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(comp_nat_nat_nat(Ga,aTP_Lamm_qh)),set_or4665077453230672383an_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2)) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_8101_sum_OatLeastLessThan__rev,axiom,
    ! [Ga: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(N2,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_tr(Ga),N2),M)),set_or4665077453230672383an_nat(N2,M)) ).

% sum.atLeastLessThan_rev
tff(fact_8102_sum_Onested__swap,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_un(A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(A),N2)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% sum.nested_swap
tff(fact_8103_prod__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_Code_integer,A: nat,B: nat] : aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),set_or1269000886237332187st_nat(A,B)) = aa_Cod1783974652533396728nteger(set_fo1084959871951514735nteger(aTP_Lamm_up(Fa),A,B),one_one_Code_integer) ).

% prod_atLeastAtMost_code
tff(fact_8104_prod__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_assn,A: nat,B: nat] : groups6906906614972039071t_assn(Fa,set_or1269000886237332187st_nat(A,B)) = aa_assn_assn(set_fo1959793692361082170t_assn(aTP_Lamm_uq(Fa),A,B),one_one_assn) ).

% prod_atLeastAtMost_code
tff(fact_8105_prod__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_rat,A: nat,B: nat] : aa_set_nat_rat(groups73079841787564623at_rat(Fa),set_or1269000886237332187st_nat(A,B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ur(Fa),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8106_prod__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Fa),set_or1269000886237332187st_nat(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_us(Fa),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8107_prod__atLeastAtMost__code,axiom,
    ! [Fa: fun_nat_int,A: nat,B: nat] : aa_set_nat_int(groups705719431365010083at_int(Fa),set_or1269000886237332187st_nat(A,B)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ut(Fa),A),B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8108_prod_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(comp_nat_nat_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,M))),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_8109_prod_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(Ga,aa_nat_fun_nat_nat(plus_plus_nat,M))),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_8110_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_assn,P2: 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,N2),one_one_nat))
     => ( groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(M,N2))),groups6906906614972039071t_assn(Ga,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8111_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_rat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8112_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_nat,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(M,N2))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8113_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,Ga: fun_nat_int,P2: 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,N2),one_one_nat))
     => ( aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(M,N2))),aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8114_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: $o,U: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(L)),(U))
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_or7139685690850216873Than_o((L),(U))),insert_o((U),bot_bot_set_o)) = set_or8904488021354931149Most_o((L),(U)) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_8115_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( sup_sup_set_rat(set_or4029947393144176647an_rat(L,U),insert_rat(U,bot_bot_set_rat)) = set_or633870826150836451st_rat(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_8116_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),U)
     => ( sup_sup_set_num(set_or1222409239386451017an_num(L,U),insert_num(U,bot_bot_set_num)) = set_or7049704709247886629st_num(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_8117_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_or4665077453230672383an_nat(L,U)),insert_nat(U,bot_bot_set_nat)) = set_or1269000886237332187st_nat(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_8118_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
     => ( sup_su848401254843788991nteger(set_or8404916559141939852nteger(L,U),insert_Code_integer(U,bot_bo3990330152332043303nteger)) = set_or189985376899183464nteger(L,U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_8119_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U)),insert_int(U,bot_bot_set_int)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_8120_sum_Ohead__if,axiom,
    ! [Ga: fun_nat_Code_integer,M: nat,N2: nat] :
      aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_Code_integer(Ga,N2))) ).

% sum.head_if
tff(fact_8121_sum_Ohead__if,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,M: nat,N2: nat] :
      groups6857163185585827899at_nat(Ga,set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_z1048942125864253310at_nat,aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,set_or4665077453230672383an_nat(M,N2))),aa_nat2240542744625857721at_nat(Ga,N2))) ).

% sum.head_if
tff(fact_8122_sum_Ohead__if,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_rat(Ga,N2))) ).

% sum.head_if
tff(fact_8123_sum_Ohead__if,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] :
      aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_int(Ga,N2))) ).

% sum.head_if
tff(fact_8124_sum_Ohead__if,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(M,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(M,N2))),aa_nat_nat(Ga,N2))) ).

% sum.head_if
tff(fact_8125_periodic__finite__ex,axiom,
    ! [D2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K: int] :
            ( aa_int_o(Pa,X3)
          <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2))) )
       => ( ? [X_12: int] : aa_int_o(Pa,X_12)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D2))
              & aa_int_o(Pa,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_8126_bset_I3_J,axiom,
    ! [D4: int,T6: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T6),one_one_int)),B4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X5 = T6 )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4) = T6 ) ) ) ) ) ).

% bset(3)
tff(fact_8127_bset_I4_J,axiom,
    ! [D4: int,T6: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T6),B4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X5 != T6 )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4) != T6 ) ) ) ) ) ).

% bset(4)
tff(fact_8128_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T6)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)),T6) ) ) ) ).

% bset(5)
tff(fact_8129_bset_I7_J,axiom,
    ! [D4: int,T6: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T6),B4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T6),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T6),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)) ) ) ) ) ).

% bset(7)
tff(fact_8130_aset_I3_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T6),one_one_int)),A4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X5 = T6 )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4) = T6 ) ) ) ) ) ).

% aset(3)
tff(fact_8131_aset_I4_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T6),A4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X5 != T6 )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4) != T6 ) ) ) ) ) ).

% aset(4)
tff(fact_8132_aset_I5_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T6),A4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X5),T6)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)),T6) ) ) ) ) ).

% aset(5)
tff(fact_8133_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T6),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T6),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)) ) ) ) ).

% aset(7)
tff(fact_8134_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K2: num] :
      set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K2)),insert_nat(aa_num_nat(pred_numeral,K2),set_or4665077453230672383an_nat(M,aa_num_nat(pred_numeral,K2))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_8135_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(aa_nat_nat(suc,N2),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8136_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [Ga: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or4665077453230672383an_nat(N2,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(Ga),N2),M)),set_or1269000886237332187st_nat(aa_nat_nat(suc,N2),M)) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_8137_simp__from__to,axiom,
    ! [I2: int,J2: int] :
      aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,I2),J2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I2),bot_bot_set_int,insert_int(I2,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2))) ).

% simp_from_to
tff(fact_8138_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T6),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T6),aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)) ) ) ) ).

% aset(8)
tff(fact_8139_aset_I6_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T6),one_one_int)),A4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T6)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X5),D4)),T6) ) ) ) ) ).

% aset(6)
tff(fact_8140_bset_I8_J,axiom,
    ! [D4: int,T6: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T6),one_one_int)),B4)
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B4)
                   => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T6),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T6),aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)) ) ) ) ) ).

% bset(8)
tff(fact_8141_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,T6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B4)
                 => ( X5 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),T6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X5),D4)),T6) ) ) ) ).

% bset(6)
tff(fact_8142_cpmi,axiom,
    ! [D4: int,Pa: fun_int_o,P3: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(Pa,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B4)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
             => ( aa_int_o(Pa,X3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
         => ( ! [X3: int,K: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
           => ( ? [X_12: int] : aa_int_o(Pa,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B4)
                        & aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_8143_cppi,axiom,
    ! [D4: int,Pa: fun_int_o,P3: fun_int_o,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(Pa,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A4)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
             => ( aa_int_o(Pa,X3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
         => ( ! [X3: int,K: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D4))) )
           => ( ? [X_12: int] : aa_int_o(Pa,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,one_one_int),D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A4)
                        & aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_8144_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_ty(A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod
tff(fact_8145_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_tz(A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod
tff(fact_8146_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod
tff(fact_8147_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ub,A)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod
tff(fact_8148_pochhammer__prod__rev,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_uu(A),N2)),set_or1269000886237332187st_nat(one_one_nat,N2)) ).

% pochhammer_prod_rev
tff(fact_8149_pochhammer__prod__rev,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),N2) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_uv(A),N2)),set_or1269000886237332187st_nat(one_one_nat,N2)) ).

% pochhammer_prod_rev
tff(fact_8150_pochhammer__prod__rev,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw(A),N2)),set_or1269000886237332187st_nat(one_one_nat,N2)) ).

% pochhammer_prod_rev
tff(fact_8151_pochhammer__prod__rev,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),N2) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ux(A),N2)),set_or1269000886237332187st_nat(one_one_nat,N2)) ).

% pochhammer_prod_rev
tff(fact_8152_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fq),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_8153_prod_Oin__pairs,axiom,
    ! [Ga: fun_nat_assn,M: nat,N2: nat] : groups6906906614972039071t_assn(Ga,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))),N2)))) = groups6906906614972039071t_assn(aTP_Lamm_uy(Ga),set_or1269000886237332187st_nat(M,N2)) ).

% prod.in_pairs
tff(fact_8154_prod_Oin__pairs,axiom,
    ! [Ga: fun_nat_rat,M: nat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(Ga),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))),N2)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_uz(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.in_pairs
tff(fact_8155_prod_Oin__pairs,axiom,
    ! [Ga: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),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))),N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_va(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.in_pairs
tff(fact_8156_prod_Oin__pairs,axiom,
    ! [Ga: fun_nat_int,M: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),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))),N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_vb(Ga)),set_or1269000886237332187st_nat(M,N2)) ).

% prod.in_pairs
tff(fact_8157_prod_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(Ga,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))),N2)))) = groups6906906614972039071t_assn(aTP_Lamm_uy(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8158_prod_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(Ga),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))),N2)))) = aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_uz(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8159_prod_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),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))),N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_va(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8160_prod_Oin__pairs__0,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),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))),N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_vb(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8161_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_uu(A),N2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8162_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,N2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_uv(A),N2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8163_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw(A),N2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8164_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ux(A),N2)),set_or1269000886237332187st_nat(zero_zero_nat,N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8165_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = 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),K2),code_divmod_abs(K2,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_vc(L)),code_divmod_abs(K2,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),code_divmod_abs(K2,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_vd(L)),code_divmod_abs(K2,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_8166_horner__sum__eq__sum,axiom,
    ! [Fa: fun_nat_rat,A: rat,Xs: list_nat] : aa_list_nat_rat(groups6853238114764508677at_rat(Fa,A),Xs) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_lis7014846069874721091at_rat(aa_rat4396832037704353944at_rat(aTP_Lamm_ve(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8167_horner__sum__eq__sum,axiom,
    ! [Fa: fun_o_rat,A: rat,Xs: list_o] : groups8483887719401441109_o_rat(Fa,A,Xs) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_lis2932070769283852167at_rat(aa_rat5396404679109611328at_rat(aTP_Lamm_vf(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_o(Xs))) ).

% horner_sum_eq_sum
tff(fact_8168_horner__sum__eq__sum,axiom,
    ! [Fa: fun_int_rat,A: rat,Xs: list_int] : groups7852591826665563233nt_rat(Fa,A,Xs) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_lis1742918661435654559at_rat(aa_rat8348276666120063220at_rat(aTP_Lamm_vg(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8169_horner__sum__eq__sum,axiom,
    ! [Fa: fun_nat_int,A: int,Xs: list_nat] : groups7485877704341954137at_int(Fa,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_vh(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8170_horner__sum__eq__sum,axiom,
    ! [Fa: fun_int_int,A: int,Xs: list_int] : groups8485231416243008693nt_int(Fa,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_vi(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8171_horner__sum__eq__sum,axiom,
    ! [Fa: fun_nat_nat,A: nat,Xs: list_nat] : aa_list_nat_nat(groups7488368174851004413at_nat(Fa,A),Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis1299702591413342011at_nat(aa_nat9125522253416007816at_nat(aTP_Lamm_vj(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs))) ).

% horner_sum_eq_sum
tff(fact_8172_horner__sum__eq__sum,axiom,
    ! [Fa: fun_o_nat,A: nat,Xs: list_o] : groups9119017779487936845_o_nat(Fa,A,Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis6440299327677248895at_nat(aa_nat4797045371981825344at_nat(aTP_Lamm_vk(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_o(Xs))) ).

% horner_sum_eq_sum
tff(fact_8173_horner__sum__eq__sum,axiom,
    ! [Fa: fun_int_nat,A: nat,Xs: list_int] : groups8487721886752058969nt_nat(Fa,A,Xs) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_lis5251147219829051287at_nat(aa_nat3853594844976941284at_nat(aTP_Lamm_vl(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_int(Xs))) ).

% horner_sum_eq_sum
tff(fact_8174_horner__sum__eq__sum,axiom,
    ! [Fa: fun_o_int,A: int,Xs: list_o] : groups9116527308978886569_o_int(Fa,A,Xs) = aa_set_nat_int(groups3539618377306564664at_int(aa_lis2262448308168052187at_int(aa_int7290165588788099136at_int(aTP_Lamm_vm(Fa),A),Xs)),set_or4665077453230672383an_nat(zero_zero_nat,size_size_list_o(Xs))) ).

% horner_sum_eq_sum
tff(fact_8175_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_vn(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aa_fun3918145391673965625nteger(aTP_Lamm_ro(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8176_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_assn,Ha: fun_nat_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_vo(K2),Ga),Ha),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_vp(K2),Ga),Ha),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8177_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_vq(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rs(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8178_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vr(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rw(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8179_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vs(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_int(groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ru(K2),Ga),Ha)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8180_gbinomial__Suc,axiom,
    ! [A: code_natural,K2: nat] : aa_nat_Code_natural(gbinom7368847122466276068atural(A),aa_nat_nat(suc,K2)) = divide5121882707175180666atural(groups2279045934846249631atural(aTP_Lamm_ui(A),set_or1269000886237332187st_nat(zero_zero_nat,K2)),aa_nat_Code_natural(semiri2447717529341329178atural,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8181_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_uh(A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8182_gbinomial__Suc,axiom,
    ! [A: code_integer,K2: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A),aa_nat_nat(suc,K2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(aTP_Lamm_uj(A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8183_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uk,A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8184_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),aa_nat_nat(suc,K2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_ul,A)),set_or1269000886237332187st_nat(zero_zero_nat,K2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_8185_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      code_bit_cut_integer(K2) = $ite(K2 = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_vt(K2),code_divmod_abs(K2,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_8186_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_8187_sum_Otriangle__reindex,axiom,
    ! [Ga: fun_nat_fun_nat_nat,N2: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(Ga),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vu(N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qs(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.triangle_reindex
tff(fact_8188_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,M: nat] :
      nat_prod_decode_aux(K2,M) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),K2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),M)),nat_prod_decode_aux(aa_nat_nat(suc,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_8189_lessThan__iff,axiom,
    ! [I2: $o,K2: $o] :
      ( aa_set_o_o(member_o((I2)),set_ord_lessThan_o((K2)))
    <=> aa_o_o(aa_o_fun_o_o(ord_less_o,(I2)),(K2)) ) ).

% lessThan_iff
tff(fact_8190_lessThan__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,K2: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(I2),set_or4940836740269066044nt_int(K2))
    <=> aa_set6170482117830564987_int_o(ord_le7563427860532173253nt_int(I2),K2) ) ).

% lessThan_iff
tff(fact_8191_lessThan__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I2),set_or890127255671739683et_nat(K2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I2),K2) ) ).

% lessThan_iff
tff(fact_8192_lessThan__iff,axiom,
    ! [I2: assn,K2: assn] :
      ( member_assn(I2,set_or7637083652282234053n_assn(K2))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,I2),K2) ) ).

% lessThan_iff
tff(fact_8193_lessThan__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( aa_set_rat_o(member_rat(I2),set_ord_lessThan_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),K2) ) ).

% lessThan_iff
tff(fact_8194_lessThan__iff,axiom,
    ! [I2: num,K2: num] :
      ( aa_set_num_o(member_num(I2),set_ord_lessThan_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I2),K2) ) ).

% lessThan_iff
tff(fact_8195_lessThan__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_lessThan_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K2) ) ).

% lessThan_iff
tff(fact_8196_lessThan__iff,axiom,
    ! [I2: int,K2: int] :
      ( aa_set_int_o(member_int(I2),aa_int_set_int(set_ord_lessThan_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2) ) ).

% lessThan_iff
tff(fact_8197_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8198_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8199_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8200_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(set_ord_lessThan_int,X)),aa_int_set_int(set_ord_lessThan_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ).

% lessThan_subset_iff
tff(fact_8201_sum_OlessThan__Suc,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat2240542744625857721at_nat(Ga,N2)) ).

% sum.lessThan_Suc
tff(fact_8202_sum_OlessThan__Suc,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_rat(Ga,N2)) ).

% sum.lessThan_Suc
tff(fact_8203_sum_OlessThan__Suc,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_int(Ga,N2)) ).

% sum.lessThan_Suc
tff(fact_8204_sum_OlessThan__Suc,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_nat(Ga,N2)) ).

% sum.lessThan_Suc
tff(fact_8205_int__prod,axiom,
    ! [Fa: fun_int_nat,A4: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(Fa,A4)) = groups1705073143266064639nt_int(aTP_Lamm_nv(Fa),A4) ).

% int_prod
tff(fact_8206_int__prod,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_nw(Fa)),A4) ).

% int_prod
tff(fact_8207_lessThan__def,axiom,
    ! [U: set_Pr958786334691620121nt_int] : set_or4940836740269066044nt_int(U) = collec5210948495886036740nt_int(aTP_Lamm_vv(U)) ).

% lessThan_def
tff(fact_8208_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bq,U)) ).

% lessThan_def
tff(fact_8209_lessThan__def,axiom,
    ! [U: assn] : set_or7637083652282234053n_assn(U) = collect_assn(aa_assn_fun_assn_o(aTP_Lamm_bs,U)) ).

% lessThan_def
tff(fact_8210_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_bw,U)) ).

% lessThan_def
tff(fact_8211_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_dr,U)) ).

% lessThan_def
tff(fact_8212_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,U)) ).

% lessThan_def
tff(fact_8213_lessThan__def,axiom,
    ! [U: int] : aa_int_set_int(set_ord_lessThan_int,U) = collect_int(aa_int_fun_int_o(aTP_Lamm_ca,U)) ).

% lessThan_def
tff(fact_8214_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(N2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8215_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( aa_set_num_o(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(N2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8216_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8217_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_int_set_int(set_ord_lessThan_int,M)),aa_int_set_int(set_ord_lessThan_int,N2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,M),N2) ) ).

% lessThan_strict_subset_iff
tff(fact_8218_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_8219_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_8220_sum_Onat__diff__reindex,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw(Ga),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nat_diff_reindex
tff(fact_8221_prod_Onat__diff__reindex,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw(Ga),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nat_diff_reindex
tff(fact_8222_prod_Onat__diff__reindex,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_vx(Ga),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_int(groups705719431365010083at_int(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nat_diff_reindex
tff(fact_8223_prod__int__eq,axiom,
    ! [I2: nat,J2: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),set_or1269000886237332187st_nat(I2,J2)) = groups1705073143266064639nt_int(aTP_Lamm_fr,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J2))) ).

% prod_int_eq
tff(fact_8224_ivl__disj__un__one_I2_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( sup_sup_set_rat(set_ord_lessThan_rat(L),set_or4029947393144176647an_rat(L,U)) = set_ord_lessThan_rat(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8225_ivl__disj__un__one_I2_J,axiom,
    ! [L: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),U)
     => ( sup_sup_set_num(set_ord_lessThan_num(L),set_or1222409239386451017an_num(L,U)) = set_ord_lessThan_num(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8226_ivl__disj__un__one_I2_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),set_or4665077453230672383an_nat(L,U)) = aa_nat_set_nat(set_ord_lessThan_nat,U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8227_ivl__disj__un__one_I2_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
     => ( sup_su848401254843788991nteger(set_or5754767410780653050nteger(L),set_or8404916559141939852nteger(L,U)) = set_or5754767410780653050nteger(U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8228_ivl__disj__un__one_I2_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(set_ord_lessThan_int,L)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U)) = aa_int_set_int(set_ord_lessThan_int,U) ) ) ).

% ivl_disj_un_one(2)
tff(fact_8229_prod_Otriangle__reindex,axiom,
    ! [Ga: fun_nat_fun_nat_nat,N2: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(Ga),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vu(N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_vy(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.triangle_reindex
tff(fact_8230_prod_Otriangle__reindex,axiom,
    ! [Ga: fun_nat_fun_nat_int,N2: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(Ga),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vu(N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_wa(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.triangle_reindex
tff(fact_8231_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_atMost_rat(A)),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8232_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_atMost_num(A)),set_ord_lessThan_num(B))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8233_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8234_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(set_ord_atMost_int,A)),aa_int_set_int(set_ord_lessThan_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8235_sum__diff__distrib,axiom,
    ! [Qa: fun_int_nat,Pa: fun_int_nat,N2: int] :
      ( ! [X3: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Qa,X3)),aa_int_nat(Pa,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(Pa,aa_int_set_int(set_ord_lessThan_int,N2))),groups4541462559716669496nt_nat(Qa,aa_int_set_int(set_ord_lessThan_int,N2))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pk(Qa),Pa),aa_int_set_int(set_ord_lessThan_int,N2)) ) ) ).

% sum_diff_distrib
tff(fact_8236_sum__diff__distrib,axiom,
    ! [Qa: fun_nat_nat,Pa: fun_nat_nat,N2: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Qa,X3)),aa_nat_nat(Pa,X3))
     => ( 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,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(Qa),aa_nat_set_nat(set_ord_lessThan_nat,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pl(Qa),Pa)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ) ) ).

% sum_diff_distrib
tff(fact_8237_sum_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,zero_zero_nat)),groups6857163185585827899at_nat(aTP_Lamm_qa(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8238_sum_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qb(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8239_sum_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8240_sum_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_8241_sum__lessThan__telescope_H,axiom,
    ! [Fa: fun_nat_rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qd(Fa)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,zero_zero_nat)),aa_nat_rat(Fa,M)) ).

% sum_lessThan_telescope'
tff(fact_8242_sum__lessThan__telescope_H,axiom,
    ! [Fa: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qe(Fa)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,zero_zero_nat)),aa_nat_int(Fa,M)) ).

% sum_lessThan_telescope'
tff(fact_8243_sum__lessThan__telescope,axiom,
    ! [Fa: fun_nat_rat,M: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qf(Fa)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Fa,M)),aa_nat_rat(Fa,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8244_sum__lessThan__telescope,axiom,
    ! [Fa: fun_nat_int,M: nat] : aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qg(Fa)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,M)),aa_nat_int(Fa,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8245_prod_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(Ga,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_um(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8246_prod_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qb(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8247_prod_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8248_prod_OlessThan__Suc__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_8249_sum_OatLeast1__atMost__eq,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.atLeast1_atMost_eq
tff(fact_8250_prod_OatLeast1__atMost__eq,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.atLeast1_atMost_eq
tff(fact_8251_prod_OatLeast1__atMost__eq,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N2)) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.atLeast1_atMost_eq
tff(fact_8252_sum__bounds__lt__plus1,axiom,
    ! [Fa: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Fa)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(Fa),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8253_prod__int__plus__eq,axiom,
    ! [I2: nat,J2: nat] : aa_set_nat_int(groups705719431365010083at_int(semiri1314217659103216013at_int),set_or1269000886237332187st_nat(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2))) = groups1705073143266064639nt_int(aTP_Lamm_fr,aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J2)))) ).

% prod_int_plus_eq
tff(fact_8254_sum_Onat__group,axiom,
    ! [Ga: fun_nat_nat,K2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wb(Ga),K2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2))) ).

% sum.nat_group
tff(fact_8255_prod_Onat__group,axiom,
    ! [Ga: fun_nat_nat,K2: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wc(Ga),K2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2))) ).

% prod.nat_group
tff(fact_8256_prod_Onat__group,axiom,
    ! [Ga: fun_nat_int,K2: nat,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_wd(Ga),K2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_int(groups705719431365010083at_int(Ga),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K2))) ).

% prod.nat_group
tff(fact_8257_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_we(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nested_swap'
tff(fact_8258_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_wf(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nested_swap'
tff(fact_8259_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_wg(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_tw(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nested_swap'
tff(fact_8260_Iio__Int__singleton,axiom,
    ! [K2: $o,X: $o] :
      inf_inf_set_o(set_ord_lessThan_o((K2)),insert_o((X),bot_bot_set_o)) = $ite(aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(K2)),insert_o((X),bot_bot_set_o),bot_bot_set_o) ).

% Iio_Int_singleton
tff(fact_8261_Iio__Int__singleton,axiom,
    ! [K2: assn,X: assn] :
      inf_inf_set_assn(set_or7637083652282234053n_assn(K2),insert_assn(X,bot_bot_set_assn)) = $ite(aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),K2),insert_assn(X,bot_bot_set_assn),bot_bot_set_assn) ).

% Iio_Int_singleton
tff(fact_8262_Iio__Int__singleton,axiom,
    ! [K2: rat,X: rat] :
      inf_inf_set_rat(set_ord_lessThan_rat(K2),insert_rat(X,bot_bot_set_rat)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),K2),insert_rat(X,bot_bot_set_rat),bot_bot_set_rat) ).

% Iio_Int_singleton
tff(fact_8263_Iio__Int__singleton,axiom,
    ! [K2: num,X: num] :
      inf_inf_set_num(set_ord_lessThan_num(K2),insert_num(X,bot_bot_set_num)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,X),K2),insert_num(X,bot_bot_set_num),bot_bot_set_num) ).

% Iio_Int_singleton
tff(fact_8264_Iio__Int__singleton,axiom,
    ! [K2: nat,X: nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,K2)),insert_nat(X,bot_bot_set_nat)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),K2),insert_nat(X,bot_bot_set_nat),bot_bot_set_nat) ).

% Iio_Int_singleton
tff(fact_8265_Iio__Int__singleton,axiom,
    ! [K2: int,X: int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_int_set_int(set_ord_lessThan_int,K2)),insert_int(X,bot_bot_set_int)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),K2),insert_int(X,bot_bot_set_int),bot_bot_set_int) ).

% Iio_Int_singleton
tff(fact_8266_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: $o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,set_ord_lessThan_o((U))),insert_o((U),bot_bot_set_o)) = set_ord_atMost_o((U)) ).

% ivl_disj_un_singleton(2)
tff(fact_8267_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)),insert_nat(U,bot_bot_set_nat)) = aa_nat_set_nat(set_ord_atMost_nat,U) ).

% ivl_disj_un_singleton(2)
tff(fact_8268_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(set_ord_lessThan_int,U)),insert_int(U,bot_bot_set_int)) = aa_int_set_int(set_ord_atMost_int,U) ).

% ivl_disj_un_singleton(2)
tff(fact_8269_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),U)
     => ( sup_sup_set_rat(set_ord_lessThan_rat(L),set_or633870826150836451st_rat(L,U)) = set_ord_atMost_rat(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8270_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),U)
     => ( sup_sup_set_num(set_ord_lessThan_num(L),set_or7049704709247886629st_num(L,U)) = set_ord_atMost_num(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8271_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),U)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,L)),set_or1269000886237332187st_nat(L,U)) = aa_nat_set_nat(set_ord_atMost_nat,U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8272_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),U)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(set_ord_lessThan_int,L)),aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)) = aa_int_set_int(set_ord_atMost_int,U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8273_ivl__disj__un__one_I4_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),U)
     => ( sup_su848401254843788991nteger(set_or5754767410780653050nteger(L),set_or189985376899183464nteger(L,U)) = set_or9101266186257409494nteger(U) ) ) ).

% ivl_disj_un_one(4)
tff(fact_8274_one__diff__power__eq,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_8275_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_8276_one__diff__power__eq,axiom,
    ! [X: int,N2: 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,X),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_8277_power__diff__1__eq,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),one_one_Code_integer)),aa_set871585252387937313nteger(groups7501900531339628137nteger(power_8256067586552552935nteger(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_8278_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N2)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)),aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_8279_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_8280_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N2)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8281_sum_OatMost__shift,axiom,
    ! [Ga: fun_na2338080690716861136at_nat,N2: nat] : groups6857163185585827899at_nat(Ga,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Ga,zero_zero_nat)),groups6857163185585827899at_nat(aTP_Lamm_qa(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8282_sum_OatMost__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_qb(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8283_sum_OatMost__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8284_sum_OatMost__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8285_prod_OatMost__shift,axiom,
    ! [Ga: fun_nat_assn,N2: nat] : groups6906906614972039071t_assn(Ga,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Ga,zero_zero_nat)),groups6906906614972039071t_assn(aTP_Lamm_um(Ga),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8286_prod_OatMost__shift,axiom,
    ! [Ga: fun_nat_rat,N2: nat] : aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Ga,zero_zero_nat)),aa_set_nat_rat(groups73079841787564623at_rat(aTP_Lamm_qb(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8287_prod_OatMost__shift,axiom,
    ! [Ga: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Ga,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_pw(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8288_prod_OatMost__shift,axiom,
    ! [Ga: fun_nat_int,N2: nat] : aa_set_nat_int(groups705719431365010083at_int(Ga),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Ga,zero_zero_nat)),aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_qc(Ga)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8289_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(power_power_rat(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N2),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N2))),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X))) ).

% sum_gp_strict
tff(fact_8290_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),N2)),aa_nat_rat(power_power_rat(Y),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_wh(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_8291_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_wi(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_8292_diff__power__eq__sum,axiom,
    ! [X: rat,N2: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_wj(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_8293_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_nat(suc,N2))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_wk(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_8294_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),X)),aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_wl(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_8295_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),aa_nat_rat(power_power_rat(X),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),X)),aa_set_nat_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_wm(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_8296_one__diff__power__eq_H,axiom,
    ! [X: int,N2: 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,X),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_wn(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_8297_prod_Otriangle__reindex__eq,axiom,
    ! [Ga: fun_nat_fun_nat_nat,N2: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(Ga),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_qq(N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_vy(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.triangle_reindex_eq
tff(fact_8298_prod_Otriangle__reindex__eq,axiom,
    ! [Ga: fun_nat_fun_nat_int,N2: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(Ga),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_qq(N2)))) = aa_set_nat_int(groups705719431365010083at_int(aTP_Lamm_wa(Ga)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.triangle_reindex_eq
tff(fact_8299_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_8300_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_8301_signed__take__bit__code,axiom,
    ! [N2: nat,A: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N2),A) = $let(
        l: code_integer,
        l:= aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(aa_nat_nat(suc,N2)),A),
        $ite(aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,l),N2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_nat_nat(suc,N2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))),l) ) ).

% signed_take_bit_code
tff(fact_8302_signed__take__bit__code,axiom,
    ! [N2: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,N2),A) = $let(
        l: int,
        l:= aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,N2)),A),
        $ite(aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,l),N2),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,N2)),aa_int_int(uminus_uminus_int,one_one_int))),l) ) ).

% signed_take_bit_code
tff(fact_8303_sum__zero__power_H,axiom,
    ! [C2: fun_nat_rat,D2: fun_nat_rat,A4: set_nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wo(C2),D2)),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(C2,zero_zero_nat)),aa_nat_rat(D2,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_8304_or__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),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))),K2)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_8305_execute__of__list,axiom,
    ! [Xs: list_nat,Ha: heap_e7401611519738050253t_unit] : heap_T2203789307644377761ay_nat(array_of_list_nat(Xs),Ha) = some_P8206036975937309155it_nat(produc8657510640128716596it_nat(aTP_Lamm_wp(Xs),array_alloc_nat(Xs,Ha))) ).

% execute_of_list
tff(fact_8306_execute__of__list,axiom,
    ! [Xs: list_o,Ha: heap_e7401611519738050253t_unit] : heap_T8140152426084121245rray_o(array_of_list_o(Xs),Ha) = some_P3509045262911171395it_nat(produc5512802776183222702it_nat(aTP_Lamm_wq(Xs),array_alloc_o(Xs,Ha))) ).

% execute_of_list
tff(fact_8307_execute__of__list,axiom,
    ! [Xs: list_int,Ha: heap_e7401611519738050253t_unit] : heap_T7249310324989956861ay_int(array_of_list_int(Xs),Ha) = some_P8650484732927216191it_nat(produc3016605123915955252it_nat(aTP_Lamm_wr(Xs),array_alloc_int(Xs,Ha))) ).

% execute_of_list
tff(fact_8308_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% push_bit_nonnegative_int_iff
tff(fact_8309_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_8310_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),A) ).

% push_bit_push_bit
tff(fact_8311_push__bit__push__bit,axiom,
    ! [M: nat,N2: 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,N2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),A) ).

% push_bit_push_bit
tff(fact_8312_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_8313_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Ico_iff
tff(fact_8314_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_8315_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_8316_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_ws((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_Code_integer(B,(A)),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_8317_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_wt(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_8318_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_wu(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_8319_sum_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups879477027807139574nteger(aa_fun4435975922079263404nteger(aTP_Lamm_wv(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta
tff(fact_8320_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_ww((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_rat(B,(A)),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8321_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wx(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8322_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_wy(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8323_sum_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_wz(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8324_sum_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_xa((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_nat(B,(A)),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_8325_sum_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_xb(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_8326_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups4406642042086082107nteger(aa_fun8520278569782659900nteger(aTP_Lamm_xc((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_Code_integer(B,(A)),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_8327_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_xd(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_8328_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_xe(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_8329_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups879477027807139574nteger(aa_fun4435975922079263404nteger(aTP_Lamm_xf(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),zero_z3403309356797280102nteger) ) ) ).

% sum.delta'
tff(fact_8330_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_xg((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_rat(B,(A)),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8331_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_xh(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8332_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_xi(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8333_sum_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_xj(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8334_sum_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_xk((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_nat(B,(A)),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_8335_sum_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_xl(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_nat(B,A),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_8336_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7694694392188491536nteger(aa_fun8520278569782659900nteger(aTP_Lamm_xm((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_Code_integer(B,(A)),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_8337_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aTP_Lamm_xn(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_8338_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aTP_Lamm_xo(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_8339_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3674199335183972705nteger(aa_fun4435975922079263404nteger(aTP_Lamm_xp(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),one_one_Code_integer) ) ) ).

% prod.delta'
tff(fact_8340_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups5301882518646026715o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_xq((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_assn(B,(A)),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8341_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xr(A),B),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8342_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xs(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8343_prod_Odelta_H,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xt(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_assn(B,A),one_one_assn) ) ) ).

% prod.delta'
tff(fact_8344_prod_Odelta_H,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_xu((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_rat(B,(A)),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8345_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_xv(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_8346_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7694694392188491536nteger(aa_fun8520278569782659900nteger(aTP_Lamm_xw((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_Code_integer(B,(A)),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_8347_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun589076786913273746nteger(aTP_Lamm_xx(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_8348_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3827104343326376752nteger(aa_fun5304498221708291146nteger(aTP_Lamm_xy(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_Code_integer(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_8349_prod_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3674199335183972705nteger(aa_fun4435975922079263404nteger(aTP_Lamm_xz(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Cod1783974652533396728nteger(B,A),one_one_Code_integer) ) ) ).

% prod.delta
tff(fact_8350_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups5301882518646026715o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_ya((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_assn(B,(A)),one_one_assn) ) ) ).

% prod.delta
tff(fact_8351_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aTP_Lamm_yb(A),B),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8352_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aTP_Lamm_yc(A),B),S) = $ite(aa_set_int_o(member_int(A),S),aa_int_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8353_prod_Odelta,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_yd(A),B),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_Code_integer_assn(B,A),one_one_assn) ) ) ).

% prod.delta
tff(fact_8354_prod_Odelta,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_ye((A)),B),S) = $ite(aa_set_o_o(member_o((A)),S),aa_o_rat(B,(A)),one_one_rat) ) ) ).

% prod.delta
tff(fact_8355_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_yf(A),B)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_nat_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_8356_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X)),A4)
       => ( groups7872700643590313910_o_rat(Ga,insert_o((X),A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Ga,(X))),groups7872700643590313910_o_rat(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8357_sum_Oinsert,axiom,
    ! [A4: set_nat,X: nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(X),A4)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),insert_nat(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,X)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)) ) ) ) ).

% sum.insert
tff(fact_8358_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(X),A4)
       => ( groups3906332499630173760nt_rat(Ga,insert_int(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Ga,X)),groups3906332499630173760nt_rat(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8359_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
       => ( groups6602215022474089585er_rat(Ga,insert_Code_integer(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Ga,X)),groups6602215022474089585er_rat(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8360_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X)),A4)
       => ( groups8507830703676809646_o_nat(Ga,insert_o((X),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Ga,(X))),groups8507830703676809646_o_nat(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8361_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(X),A4)
       => ( groups4541462559716669496nt_nat(Ga,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ga,X)),groups4541462559716669496nt_nat(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8362_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
       => ( groups7237345082560585321er_nat(Ga,insert_Code_integer(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Ga,X)),groups7237345082560585321er_nat(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8363_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,Ga: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((X)),A4)
       => ( groups8505340233167759370_o_int(Ga,insert_o((X),A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(Ga,(X))),groups8505340233167759370_o_int(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8364_sum_Oinsert,axiom,
    ! [A4: set_nat,X: nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(X),A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),insert_nat(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,X)),aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)) ) ) ) ).

% sum.insert
tff(fact_8365_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
       => ( groups7234854612051535045er_int(Ga,insert_Code_integer(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(Ga,X)),groups7234854612051535045er_int(Ga,A4)) ) ) ) ).

% sum.insert
tff(fact_8366_prod__pos__nat__iff,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(Fa,A4))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Fa,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8367_prod__pos__nat__iff,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups3190895334310489300er_nat(Fa,A4))
      <=> ! [X4: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Fa,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8368_prod__pos__nat__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Fa: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4077766827762148844at_nat(Fa,A4))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(Fa,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8369_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Fa,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8370_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,Pa: fun_int_o,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_yg(Pa),Fa),A4) = groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8371_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_yh(Pa),Fa),A4) = groups6602215022474089585er_rat(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8372_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_yi(Pa),Fa)),A4) = aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8373_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,Pa: fun_int_o,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_yj(Pa),Fa),A4) = groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8374_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aTP_Lamm_yk(Pa),Fa),A4) = groups7237345082560585321er_nat(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8375_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(aa_fun3291479149765983562er_int(aTP_Lamm_yl(Pa),Fa),A4) = groups7234854612051535045er_int(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8376_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ym(Pa),Fa)),A4) = aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8377_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,Pa: fun_int_o,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_yn(Pa),Fa),A4) = groups7873554091576472773nteger(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8378_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups879477027807139574nteger(aa_fun4435975922079263404nteger(aTP_Lamm_yo(Pa),Fa),A4) = groups879477027807139574nteger(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8379_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_yp(Pa),Fa)),A4) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))) ) ) ).

% sum_of_bool_mult_eq
tff(fact_8380_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,Fa: fun_int_rat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_yq(Fa),Pa),A4) = groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8381_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_rat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(aa_fun25512833367352522er_rat(aTP_Lamm_yr(Fa),Pa),A4) = groups6602215022474089585er_rat(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8382_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ys(Fa),Pa)),A4) = aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8383_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,Fa: fun_int_nat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_yt(Fa),Pa),A4) = groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8384_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(aa_fun3533741391760749250er_nat(aTP_Lamm_yu(Fa),Pa),A4) = groups7237345082560585321er_nat(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8385_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_int,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(aa_fun8579262409106328350er_int(aTP_Lamm_yv(Fa),Pa),A4) = groups7234854612051535045er_int(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8386_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,Fa: fun_nat_int,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun27009930614080094at_int(aTP_Lamm_yw(Fa),Pa)),A4) = aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8387_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_yx(Fa),Pa),A4) = groups7873554091576472773nteger(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8388_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Fa: fun_Co8499238673658658103nteger,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups879477027807139574nteger(aa_fun791458909507501263nteger(aTP_Lamm_yy(Fa),Pa),A4) = groups879477027807139574nteger(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8389_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_yz(Fa),Pa)),A4) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa))) ) ) ).

% sum_mult_of_bool_eq
tff(fact_8390_sum__zero__power,axiom,
    ! [C2: fun_nat_rat,A4: set_nat] :
      aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_za(C2)),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_rat(C2,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_8391_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% or_numerals(4)
tff(fact_8392_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = 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,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% or_numerals(4)
tff(fact_8393_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(4)
tff(fact_8394_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(4)
tff(fact_8395_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% or_numerals(6)
tff(fact_8396_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Y))) = 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,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% or_numerals(6)
tff(fact_8397_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(6)
tff(fact_8398_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(6)
tff(fact_8399_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,X)),aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),times_2397367101498566445atural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one)),bit_se9127793120404214118atural(aa_num_Code_natural(numera5444537566228673987atural,X),aa_num_Code_natural(numera5444537566228673987atural,Y)))) ).

% or_numerals(7)
tff(fact_8400_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,X))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Y))) = 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,X)),aa_num_Code_integer(numera6620942414471956472nteger,Y)))) ).

% or_numerals(7)
tff(fact_8401_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).

% or_numerals(7)
tff(fact_8402_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% or_numerals(7)
tff(fact_8403_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),N2) )
     => aa_set_nat_o(finite_finite_nat,N7) ) ).

% bounded_nat_set_is_finite
tff(fact_8404_finite__nat__set__iff__bounded,axiom,
    ! [N7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N7)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_8405_push__bit__add,axiom,
    ! [N2: nat,A: int,B: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),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,N2),A)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),B)) ).

% push_bit_add
tff(fact_8406_push__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N2),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,N2),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N2),B)) ).

% push_bit_add
tff(fact_8407_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,S)
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_zb(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8408_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_zc(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8409_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_zd(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8410_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set4044230130517860621eger_o(aTP_Lamm_ze(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8411_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_zf(S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8412_finite__nat__set__iff__bounded__le,axiom,
    ! [N7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N7)
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M3) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_8413_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun_nat_o,I2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_zg(Pa),I2))) ).

% finite_M_bounded_by_nat
tff(fact_8414_finite__less__ub,axiom,
    ! [Fa: fun_nat_nat,U: nat] :
      ( ! [N5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),aa_nat_nat(Fa,N5))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_zh(Fa),U))) ) ).

% finite_less_ub
tff(fact_8415_sum_Oswap__restrict,axiom,
    ! [A4: set_o,B4: set_nat,Ga: fun_o_fun_nat_nat,Ra: fun_o_fun_nat_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_zj(B4),Ga),Ra),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_zm(A4),Ga),Ra)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8416_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,Ga: fun_int_fun_nat_nat,Ra: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_zo(B4),Ga),Ra),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_zr(A4),Ga),Ra)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8417_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_nat,Ga: fun_Co3312552843764780407at_nat,Ra: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_zt(B4),Ga),Ra),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_zw(A4),Ga),Ra)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8418_sum_Oswap__restrict,axiom,
    ! [A4: set_o,B4: set_int,Ga: fun_o_fun_int_int,Ra: fun_o_fun_int_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups8505340233167759370_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_zy(B4),Ga),Ra),A4) = groups4538972089207619220nt_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_aab(A4),Ga),Ra),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8419_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_int,Ga: fun_nat_fun_int_int,Ra: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_aad(B4),Ga),Ra)),A4) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_aaf(A4),Ga),Ra),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8420_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_int,Ga: fun_Co3009391273526075439nt_int,Ra: fun_Co8353275306710069249_int_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups7234854612051535045er_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_aah(B4),Ga),Ra),A4) = groups4538972089207619220nt_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_aak(A4),Ga),Ra),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8421_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_o,Ga: fun_nat_fun_o_nat,Ra: fun_nat_fun_o_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_aam(B4),Ga),Ra)),A4) = groups8507830703676809646_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_aap(A4),Ga),Ra),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8422_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_int,Ga: fun_nat_fun_int_nat,Ra: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_aaq(B4),Ga),Ra)),A4) = groups4541462559716669496nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_aas(A4),Ga),Ra),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8423_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_Code_integer,Ga: fun_na5492297256294505719er_nat,Ra: fun_na6318815019539281885eger_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_aau(B4),Ga),Ra)),A4) = groups7237345082560585321er_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_aax(A4),Ga),Ra),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8424_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_fun_nat_nat,Ra: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_aaz(B4),Ga),Ra)),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun8857679445107423627at_nat(aa_fun7975631875395124402at_nat(aTP_Lamm_abb(A4),Ga),Ra)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_8425_prod_Oswap__restrict,axiom,
    ! [A4: set_o,B4: set_nat,Ga: fun_o_fun_nat_nat,Ra: fun_o_fun_nat_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3504817904513533571_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_abc(B4),Ga),Ra),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_abd(A4),Ga),Ra)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8426_prod_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,Ga: fun_int_fun_nat_nat,Ra: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1707563613775114915nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_abe(B4),Ga),Ra),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_abf(A4),Ga),Ra)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8427_prod_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_nat,Ga: fun_Co3312552843764780407at_nat,Ra: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3190895334310489300er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_abg(B4),Ga),Ra),A4) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_abh(A4),Ga),Ra)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8428_prod_Oswap__restrict,axiom,
    ! [A4: set_o,B4: set_nat,Ga: fun_o_fun_nat_int,Ra: fun_o_fun_nat_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3502327434004483295_o_int(aa_fun2051904251736260813_o_int(aa_fun7508458648369839740_o_int(aTP_Lamm_abi(B4),Ga),Ra),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_fun6872859990233652739at_int(aa_fun8947132502270365866at_int(aTP_Lamm_abk(A4),Ga),Ra)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8429_prod_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_nat,Ga: fun_Co822469698833053651at_int,Ra: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups3188404863801439024er_int(aa_fun2505341505705002151er_int(aa_fun493137163814837797er_int(aTP_Lamm_abl(B4),Ga),Ra),A4) = aa_set_nat_int(groups705719431365010083at_int(aa_fun3278309868433077274at_int(aa_fun1276769135904165912at_int(aTP_Lamm_abn(A4),Ga),Ra)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8430_prod_Oswap__restrict,axiom,
    ! [A4: set_o,B4: set_int,Ga: fun_o_fun_int_int,Ra: fun_o_fun_int_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3502327434004483295_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_abo(B4),Ga),Ra),A4) = groups1705073143266064639nt_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_abp(A4),Ga),Ra),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8431_prod_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_int,Ga: fun_Co3009391273526075439nt_int,Ra: fun_Co8353275306710069249_int_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3188404863801439024er_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_abq(B4),Ga),Ra),A4) = groups1705073143266064639nt_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_abr(A4),Ga),Ra),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8432_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_o,Ga: fun_nat_fun_o_nat,Ra: fun_nat_fun_o_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_abs(B4),Ga),Ra)),A4) = groups3504817904513533571_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_abt(A4),Ga),Ra),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8433_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_int,Ga: fun_nat_fun_int_nat,Ra: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_abu(B4),Ga),Ra)),A4) = groups1707563613775114915nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_abv(A4),Ga),Ra),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8434_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_Code_integer,Ga: fun_na5492297256294505719er_nat,Ra: fun_na6318815019539281885eger_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_abw(B4),Ga),Ra)),A4) = groups3190895334310489300er_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_abx(A4),Ga),Ra),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8435_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)) ) ) ).

% OR_lower
tff(fact_8436_or__greater__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)) ) ).

% or_greater_eq
tff(fact_8437_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N5: nat] :
          ( ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N5)
          | ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,B),N5) )
     => ( 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_8438_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N5: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N5)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N5) )
     => ( 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_8439_infinite__growing,axiom,
    ! [X6: set_Code_integer] :
      ( ( X6 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X6)
           => ? [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),X6)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Xa2) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X6) ) ) ).

% infinite_growing
tff(fact_8440_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X6)
           => ? [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),X6)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(Xa2)) ) )
       => ~ aa_set_o_o(finite_finite_o,X6) ) ) ).

% infinite_growing
tff(fact_8441_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),X6)
           => ? [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),X6)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa2) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X6) ) ) ).

% infinite_growing
tff(fact_8442_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( aa_set_num_o(member_num(X3),X6)
           => ? [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),X6)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Xa2) ) )
       => ~ aa_set_num_o(finite_finite_num,X6) ) ) ).

% infinite_growing
tff(fact_8443_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X6)
           => ? [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),X6)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa2) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X6) ) ) ).

% infinite_growing
tff(fact_8444_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X6)
           => ? [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),X6)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa2) ) )
       => ~ aa_set_int_o(finite_finite_int,X6) ) ) ).

% infinite_growing
tff(fact_8445_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
            & ~ ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),S)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8446_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),S)
            & ~ ? [Xa2: $o] :
                  ( aa_set_o_o(member_o((Xa2)),S)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa2)),(X3)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8447_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( finite_finite_assn(S)
     => ( ( S != bot_bot_set_assn )
       => ? [X3: assn] :
            ( member_assn(X3,S)
            & ~ ? [Xa2: assn] :
                  ( member_assn(Xa2,S)
                  & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8448_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S)
     => ( ( S != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),S)
            & ~ ? [Xa2: rat] :
                  ( aa_set_rat_o(member_rat(Xa2),S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8449_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( aa_set_num_o(finite_finite_num,S)
     => ( ( S != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),S)
            & ~ ? [Xa2: num] :
                  ( aa_set_num_o(member_num(Xa2),S)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8450_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),S)
            & ~ ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8451_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),S)
            & ~ ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8452_plus__and__or,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) ).

% plus_and_or
tff(fact_8453_sum__mono__inv,axiom,
    ! [Fa: fun_o_rat,I5: set_o,Ga: fun_o_rat,I2: $o] :
      ( ( groups7872700643590313910_o_rat(Fa,I5) = groups7872700643590313910_o_rat(Ga,I5) )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(I3))),aa_o_rat(Ga,(I3))) )
       => ( aa_set_o_o(member_o((I2)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_rat(Fa,(I2)) = aa_o_rat(Ga,(I2)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8454_sum__mono__inv,axiom,
    ! [Fa: fun_nat_rat,I5: set_nat,Ga: fun_nat_rat,I2: nat] :
      ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),I5) = aa_set_nat_rat(groups2906978787729119204at_rat(Ga),I5) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I3)),aa_nat_rat(Ga,I3)) )
       => ( aa_set_nat_o(member_nat(I2),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(Fa,I2) = aa_nat_rat(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8455_sum__mono__inv,axiom,
    ! [Fa: fun_int_rat,I5: set_int,Ga: fun_int_rat,I2: int] :
      ( ( groups3906332499630173760nt_rat(Fa,I5) = groups3906332499630173760nt_rat(Ga,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I3)),aa_int_rat(Ga,I3)) )
       => ( aa_set_int_o(member_int(I2),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(Fa,I2) = aa_int_rat(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8456_sum__mono__inv,axiom,
    ! [Fa: fun_Code_integer_rat,I5: set_Code_integer,Ga: fun_Code_integer_rat,I2: code_integer] :
      ( ( groups6602215022474089585er_rat(Fa,I5) = groups6602215022474089585er_rat(Ga,I5) )
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,I3)),aa_Code_integer_rat(Ga,I3)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_rat(Fa,I2) = aa_Code_integer_rat(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8457_sum__mono__inv,axiom,
    ! [Fa: fun_o_nat,I5: set_o,Ga: fun_o_nat,I2: $o] :
      ( ( groups8507830703676809646_o_nat(Fa,I5) = groups8507830703676809646_o_nat(Ga,I5) )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(I3))),aa_o_nat(Ga,(I3))) )
       => ( aa_set_o_o(member_o((I2)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_nat(Fa,(I2)) = aa_o_nat(Ga,(I2)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8458_sum__mono__inv,axiom,
    ! [Fa: fun_int_nat,I5: set_int,Ga: fun_int_nat,I2: int] :
      ( ( groups4541462559716669496nt_nat(Fa,I5) = groups4541462559716669496nt_nat(Ga,I5) )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,I3)),aa_int_nat(Ga,I3)) )
       => ( aa_set_int_o(member_int(I2),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(Fa,I2) = aa_int_nat(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8459_sum__mono__inv,axiom,
    ! [Fa: fun_Code_integer_nat,I5: set_Code_integer,Ga: fun_Code_integer_nat,I2: code_integer] :
      ( ( groups7237345082560585321er_nat(Fa,I5) = groups7237345082560585321er_nat(Ga,I5) )
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(Fa,I3)),aa_Code_integer_nat(Ga,I3)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_nat(Fa,I2) = aa_Code_integer_nat(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8460_sum__mono__inv,axiom,
    ! [Fa: fun_o_int,I5: set_o,Ga: fun_o_int,I2: $o] :
      ( ( groups8505340233167759370_o_int(Fa,I5) = groups8505340233167759370_o_int(Ga,I5) )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(Fa,(I3))),aa_o_int(Ga,(I3))) )
       => ( aa_set_o_o(member_o((I2)),I5)
         => ( aa_set_o_o(finite_finite_o,I5)
           => ( aa_o_int(Fa,(I2)) = aa_o_int(Ga,(I2)) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8461_sum__mono__inv,axiom,
    ! [Fa: fun_nat_int,I5: set_nat,Ga: fun_nat_int,I2: nat] :
      ( ( aa_set_nat_int(groups3539618377306564664at_int(Fa),I5) = aa_set_nat_int(groups3539618377306564664at_int(Ga),I5) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,I3)),aa_nat_int(Ga,I3)) )
       => ( aa_set_nat_o(member_nat(I2),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_int(Fa,I2) = aa_nat_int(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8462_sum__mono__inv,axiom,
    ! [Fa: fun_Code_integer_int,I5: set_Code_integer,Ga: fun_Code_integer_int,I2: code_integer] :
      ( ( groups7234854612051535045er_int(Fa,I5) = groups7234854612051535045er_int(Ga,I5) )
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(Fa,I3)),aa_Code_integer_int(Ga,I3)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_int(Fa,I2) = aa_Code_integer_int(Ga,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8463_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_8464_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A,B)) ) ).

% infinite_Ico
tff(fact_8465_push__bit__take__bit,axiom,
    ! [M: nat,N2: 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,N2),A)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),A)) ).

% push_bit_take_bit
tff(fact_8466_push__bit__take__bit,axiom,
    ! [M: nat,N2: 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,N2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),A)) ).

% push_bit_take_bit
tff(fact_8467_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_Code_integer,Y: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_aby(I5),X)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_aby(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_abz(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8468_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,Y: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_aca(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_aca(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_acb(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8469_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,Y: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_acc(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_acc(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_acd(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8470_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Co8499238673658658103nteger,Y: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ace(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ace(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_acf(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8471_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_rat,Y: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_acg(I5),X)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_acg(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_ach(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8472_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_aci(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_aci(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_acj(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8473_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ack(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ack(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_acl(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8474_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Code_integer_rat,Y: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_acm(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_acm(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_acn(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8475_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_nat,Y: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aco(I5),X)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aco(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aa_fun5833554292436378194un_o_o(aTP_Lamm_acp(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8476_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_nat,Y: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_acq(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_acq(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_acr(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8477_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_Code_integer,Y: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_acs(I5),X)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_acs(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_act(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8478_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,Y: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_acu(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_acu(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aa_fun2745573882733218024_nat_o(aTP_Lamm_acv(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8479_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,Y: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_acw(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_acw(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aa_fun8818272713621706452_int_o(aTP_Lamm_acx(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8480_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Co8499238673658658103nteger,Y: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_acy(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_acy(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_acz(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8481_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_assn,Y: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8382565201914911066un_o_o(aTP_Lamm_ada(I5),X)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8382565201914911066un_o_o(aTP_Lamm_ada(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun8382565201914911066un_o_o(aa_fun5794833117214906962un_o_o(aTP_Lamm_adb(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8482_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_assn,Y: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_adc(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_adc(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_add(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8483_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_assn,Y: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_ade(I5),X)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_ade(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_adf(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8484_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: fun_Co4490034186337852034r_assn,Y: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6538703224075513620eger_o(aTP_Lamm_adg(I5),X)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6538703224075513620eger_o(aTP_Lamm_adg(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6538703224075513620eger_o(aa_fun4422309413433313771eger_o(aTP_Lamm_adh(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8485_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: fun_o_rat,Y: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_adi(I5),X)))
     => ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_adi(I5),Y)))
       => aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_adj(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8486_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_adk(I5),X)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_adk(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_adl(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8487_sum_Ointer__filter,axiom,
    ! [A4: set_o,Ga: fun_o_Code_integer,Pa: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups4406642042086082107nteger(Ga,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) = groups4406642042086082107nteger(aa_fun6542514109734600263nteger(aTP_Lamm_adm(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8488_sum_Ointer__filter,axiom,
    ! [A4: set_nat,Ga: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aa_fun3969837373441324047nteger(aTP_Lamm_adn(Ga),Pa)),A4) ) ) ).

% sum.inter_filter
tff(fact_8489_sum_Ointer__filter,axiom,
    ! [A4: set_int,Ga: fun_int_Code_integer,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7873554091576472773nteger(Ga,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))) = groups7873554091576472773nteger(aa_fun6019004165459918095nteger(aTP_Lamm_ado(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8490_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups879477027807139574nteger(Ga,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_adp(A4),Pa))) = groups879477027807139574nteger(aa_fun791458909507501263nteger(aTP_Lamm_adq(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8491_sum_Ointer__filter,axiom,
    ! [A4: set_o,Ga: fun_o_rat,Pa: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(Ga,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) = groups7872700643590313910_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_adr(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8492_sum_Ointer__filter,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))) = aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ads(Ga),Pa)),A4) ) ) ).

% sum.inter_filter
tff(fact_8493_sum_Ointer__filter,axiom,
    ! [A4: set_int,Ga: fun_int_rat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(Ga,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_adt(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8494_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(Ga,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_adp(A4),Pa))) = groups6602215022474089585er_rat(aa_fun25512833367352522er_rat(aTP_Lamm_adu(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8495_sum_Ointer__filter,axiom,
    ! [A4: set_o,Ga: fun_o_nat,Pa: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8507830703676809646_o_nat(Ga,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) = groups8507830703676809646_o_nat(aa_fun_o_o_fun_o_nat(aTP_Lamm_adv(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8496_sum_Ointer__filter,axiom,
    ! [A4: set_int,Ga: fun_int_nat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(Ga,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))) = groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_adw(Ga),Pa),A4) ) ) ).

% sum.inter_filter
tff(fact_8497_filter__preserves__multiset,axiom,
    ! [M6: fun_list_nat_nat,Pa: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_adx(M6)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ady(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8498_filter__preserves__multiset,axiom,
    ! [M6: fun_se4101287982975402108nt_nat,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_adz(M6)))
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_aea(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8499_filter__preserves__multiset,axiom,
    ! [M6: fun_set_nat_nat,Pa: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_aeb(M6)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_aec(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8500_filter__preserves__multiset,axiom,
    ! [M6: fun_nat_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aed(M6)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aee(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8501_filter__preserves__multiset,axiom,
    ! [M6: fun_int_nat,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_aef(M6)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aeg(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8502_filter__preserves__multiset,axiom,
    ! [M6: fun_Code_integer_nat,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_aeh(M6)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_aei(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8503_filter__preserves__multiset,axiom,
    ! [M6: fun_Pr2602117396471556196at_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_aej(M6)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_aek(M6),Pa))) ) ).

% filter_preserves_multiset
tff(fact_8504_prod_Ointer__filter,axiom,
    ! [A4: set_o,Ga: fun_o_Code_integer,Pa: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7694694392188491536nteger(Ga,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) = groups7694694392188491536nteger(aa_fun6542514109734600263nteger(aTP_Lamm_ael(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8505_prod_Ointer__filter,axiom,
    ! [A4: set_nat,Ga: fun_nat_Code_integer,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(aa_fun3969837373441324047nteger(aTP_Lamm_aem(Ga),Pa)),A4) ) ) ).

% prod.inter_filter
tff(fact_8506_prod_Ointer__filter,axiom,
    ! [A4: set_int,Ga: fun_int_Code_integer,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3827104343326376752nteger(Ga,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))) = groups3827104343326376752nteger(aa_fun6019004165459918095nteger(aTP_Lamm_aen(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8507_prod_Ointer__filter,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups3674199335183972705nteger(Ga,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_adp(A4),Pa))) = groups3674199335183972705nteger(aa_fun791458909507501263nteger(aTP_Lamm_aeo(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8508_prod_Ointer__filter,axiom,
    ! [A4: set_o,Ga: fun_o_assn,Pa: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups5301882518646026715o_assn(Ga,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) = groups5301882518646026715o_assn(aa_fun8560762093604944786o_assn(aTP_Lamm_aep(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8509_prod_Ointer__filter,axiom,
    ! [A4: set_nat,Ga: fun_nat_assn,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6906906614972039071t_assn(Ga,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))) = groups6906906614972039071t_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_aeq(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8510_prod_Ointer__filter,axiom,
    ! [A4: set_int,Ga: fun_int_assn,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7882442080178216443t_assn(Ga,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_an(A4),Pa))) = groups7882442080178216443t_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_aer(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8511_prod_Ointer__filter,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1304777262505850412r_assn(Ga,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_adp(A4),Pa))) = groups1304777262505850412r_assn(aa_fun6392223008197429786r_assn(aTP_Lamm_aes(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8512_prod_Ointer__filter,axiom,
    ! [A4: set_o,Ga: fun_o_rat,Pa: fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups2869687844427037835_o_rat(Ga,collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(A4),Pa))) = groups2869687844427037835_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_aet(Ga),Pa),A4) ) ) ).

% prod.inter_filter
tff(fact_8513_prod_Ointer__filter,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(A4),Pa))) = aa_set_nat_rat(groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_aeu(Ga),Pa)),A4) ) ) ).

% prod.inter_filter
tff(fact_8514_finite__int__segment,axiom,
    ! [A: rat,B: rat] : aa_set_rat_o(finite_finite_rat,collect_rat(aa_rat_fun_rat_o(aTP_Lamm_aev(A),B))) ).

% finite_int_segment
tff(fact_8515_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(X3))) )
       => ( ( groups4406642042086082107nteger(Fa,A4) = zero_z3403309356797280102nteger )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => ( aa_o_Code_integer(Fa,(X4)) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8516_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,X3)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),A4) = zero_z3403309356797280102nteger )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => ( aa_nat_Code_integer(Fa,X4) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8517_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,X3)) )
       => ( ( groups7873554091576472773nteger(Fa,A4) = zero_z3403309356797280102nteger )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_Code_integer(Fa,X4) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8518_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,X3)) )
       => ( ( groups879477027807139574nteger(Fa,A4) = zero_z3403309356797280102nteger )
        <=> ! [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A4)
             => ( aa_Cod1783974652533396728nteger(Fa,X4) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8519_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(X3))) )
       => ( ( groups7872700643590313910_o_rat(Fa,A4) = zero_zero_rat )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => ( aa_o_rat(Fa,(X4)) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8520_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X3)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => ( aa_nat_rat(Fa,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8521_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X3)) )
       => ( ( groups3906332499630173760nt_rat(Fa,A4) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_rat(Fa,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8522_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,X3)) )
       => ( ( groups6602215022474089585er_rat(Fa,A4) = zero_zero_rat )
        <=> ! [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A4)
             => ( aa_Code_integer_rat(Fa,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8523_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(X3))) )
       => ( ( groups8507830703676809646_o_nat(Fa,A4) = zero_zero_nat )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => ( aa_o_nat(Fa,(X4)) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8524_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,X3)) )
       => ( ( groups4541462559716669496nt_nat(Fa,A4) = zero_zero_nat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => ( aa_int_nat(Fa,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8525_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_nat,Ga: fun_nat_Code_integer,I2: fun_nat_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,T6)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Ga,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),T6)
                    & ( aa_nat_nat(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,X3)),aa_nat_Code_integer(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8526_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_int,Ga: fun_int_Code_integer,I2: fun_int_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Ga,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),T6)
                    & ( aa_int_nat(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,X3)),aa_int_Code_integer(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),S2)),groups7873554091576472773nteger(Ga,T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8527_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_Code_integer,Ga: fun_Co8499238673658658103nteger,I2: fun_Code_integer_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Ga,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),T6)
                    & ( aa_Code_integer_nat(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,X3)),aa_Cod1783974652533396728nteger(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),S2)),groups879477027807139574nteger(Ga,T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8528_sum__le__included,axiom,
    ! [S2: set_int,T6: set_nat,Ga: fun_nat_Code_integer,I2: fun_nat_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_nat_o(finite_finite_nat,T6)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Ga,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),T6)
                    & ( aa_nat_int(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,X3)),aa_nat_Code_integer(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(Fa,S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8529_sum__le__included,axiom,
    ! [S2: set_int,T6: set_int,Ga: fun_int_Code_integer,I2: fun_int_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Ga,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),T6)
                    & ( aa_int_int(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,X3)),aa_int_Code_integer(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(Fa,S2)),groups7873554091576472773nteger(Ga,T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8530_sum__le__included,axiom,
    ! [S2: set_int,T6: set_Code_integer,Ga: fun_Co8499238673658658103nteger,I2: fun_Code_integer_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Ga,X3)) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S2)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),T6)
                    & ( aa_Code_integer_int(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,X3)),aa_Cod1783974652533396728nteger(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(Fa,S2)),groups879477027807139574nteger(Ga,T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8531_sum__le__included,axiom,
    ! [S2: set_Code_integer,T6: set_nat,Ga: fun_nat_Code_integer,I2: fun_nat_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_nat_o(finite_finite_nat,T6)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Ga,X3)) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),T6)
                    & ( aa_nat_Code_integer(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(Fa,X3)),aa_nat_Code_integer(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(Fa,S2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8532_sum__le__included,axiom,
    ! [S2: set_Code_integer,T6: set_int,Ga: fun_int_Code_integer,I2: fun_int_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Ga,X3)) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),T6)
                    & ( aa_int_Code_integer(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(Fa,X3)),aa_int_Code_integer(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(Fa,S2)),groups7873554091576472773nteger(Ga,T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8533_sum__le__included,axiom,
    ! [S2: set_Code_integer,T6: set_Code_integer,Ga: fun_Co8499238673658658103nteger,I2: fun_Co8499238673658658103nteger,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Ga,X3)) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),T6)
                    & ( aa_Cod1783974652533396728nteger(I2,Xa2) = X3 )
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(Fa,X3)),aa_Cod1783974652533396728nteger(Ga,Xa2)) ) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(Fa,S2)),groups879477027807139574nteger(Ga,T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8534_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_nat,Ga: fun_nat_rat,I2: fun_nat_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,T6)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Ga,X3)) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S2)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),T6)
                    & ( aa_nat_nat(I2,Xa2) = X3 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Ga,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),S2)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),T6)) ) ) ) ) ).

% sum_le_included
tff(fact_8535_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,Pa: fun_se7289222521248229150eger_o,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S6)
             => ( ! [Y5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y5),S6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,Y5)),aa_Code_integer_rat(Fa,X3)) )
               => ( aa_set5643387332196575111eger_o(Pa,S6)
                 => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(X3,S6)) ) ) )
         => aa_set5643387332196575111eger_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8536_finite__ranking__induct,axiom,
    ! [S: set_o,Pa: fun_set_o_o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X3: $o,S6: set_o] :
              ( aa_set_o_o(finite_finite_o,S6)
             => ( ! [Y5: $o] :
                    ( aa_set_o_o(member_o((Y5)),S6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(Y5))),aa_o_rat(Fa,(X3))) )
               => ( aa_set_o_o(Pa,S6)
                 => aa_set_o_o(Pa,insert_o((X3),S6)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8537_finite__ranking__induct,axiom,
    ! [S: set_nat,Pa: fun_set_nat_o,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [X3: nat,S6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S6)
             => ( ! [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),S6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,Y5)),aa_nat_rat(Fa,X3)) )
               => ( aa_set_nat_o(Pa,S6)
                 => aa_set_nat_o(Pa,insert_nat(X3,S6)) ) ) )
         => aa_set_nat_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8538_finite__ranking__induct,axiom,
    ! [S: set_int,Pa: fun_set_int_o,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [X3: int,S6: set_int] :
              ( aa_set_int_o(finite_finite_int,S6)
             => ( ! [Y5: int] :
                    ( aa_set_int_o(member_int(Y5),S6)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,Y5)),aa_int_rat(Fa,X3)) )
               => ( aa_set_int_o(Pa,S6)
                 => aa_set_int_o(Pa,insert_int(X3,S6)) ) ) )
         => aa_set_int_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8539_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,Pa: fun_se7289222521248229150eger_o,Fa: fun_Code_integer_num] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S6)
             => ( ! [Y5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y5),S6)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_Code_integer_num(Fa,Y5)),aa_Code_integer_num(Fa,X3)) )
               => ( aa_set5643387332196575111eger_o(Pa,S6)
                 => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(X3,S6)) ) ) )
         => aa_set5643387332196575111eger_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8540_finite__ranking__induct,axiom,
    ! [S: set_o,Pa: fun_set_o_o,Fa: fun_o_num] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X3: $o,S6: set_o] :
              ( aa_set_o_o(finite_finite_o,S6)
             => ( ! [Y5: $o] :
                    ( aa_set_o_o(member_o((Y5)),S6)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_o_num(Fa,(Y5))),aa_o_num(Fa,(X3))) )
               => ( aa_set_o_o(Pa,S6)
                 => aa_set_o_o(Pa,insert_o((X3),S6)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8541_finite__ranking__induct,axiom,
    ! [S: set_nat,Pa: fun_set_nat_o,Fa: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [X3: nat,S6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S6)
             => ( ! [Y5: nat] :
                    ( aa_set_nat_o(member_nat(Y5),S6)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,Y5)),aa_nat_num(Fa,X3)) )
               => ( aa_set_nat_o(Pa,S6)
                 => aa_set_nat_o(Pa,insert_nat(X3,S6)) ) ) )
         => aa_set_nat_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8542_finite__ranking__induct,axiom,
    ! [S: set_int,Pa: fun_set_int_o,Fa: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [X3: int,S6: set_int] :
              ( aa_set_int_o(finite_finite_int,S6)
             => ( ! [Y5: int] :
                    ( aa_set_int_o(member_int(Y5),S6)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(Fa,Y5)),aa_int_num(Fa,X3)) )
               => ( aa_set_int_o(Pa,S6)
                 => aa_set_int_o(Pa,insert_int(X3,S6)) ) ) )
         => aa_set_int_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8543_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,Pa: fun_se7289222521248229150eger_o,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S6)
             => ( ! [Y5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y5),S6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(Fa,Y5)),aa_Code_integer_nat(Fa,X3)) )
               => ( aa_set5643387332196575111eger_o(Pa,S6)
                 => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(X3,S6)) ) ) )
         => aa_set5643387332196575111eger_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8544_finite__ranking__induct,axiom,
    ! [S: set_o,Pa: fun_set_o_o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [X3: $o,S6: set_o] :
              ( aa_set_o_o(finite_finite_o,S6)
             => ( ! [Y5: $o] :
                    ( aa_set_o_o(member_o((Y5)),S6)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(Y5))),aa_o_nat(Fa,(X3))) )
               => ( aa_set_o_o(Pa,S6)
                 => aa_set_o_o(Pa,insert_o((X3),S6)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8545_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Ga,X3)) )
       => ( ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X5)),aa_nat_rat(Ga,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8546_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,Fa: fun_int_rat,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,X3)),aa_int_rat(Ga,X3)) )
       => ( ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,X5)),aa_int_rat(Ga,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(Fa,A4)),groups3906332499630173760nt_rat(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8547_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,X3)),aa_Code_integer_rat(Ga,X3)) )
       => ( ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(Fa,X5)),aa_Code_integer_rat(Ga,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(Fa,A4)),groups6602215022474089585er_rat(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8548_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,Fa: fun_int_nat,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X3)),aa_int_nat(Ga,X3)) )
       => ( ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Fa,X5)),aa_int_nat(Ga,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(Fa,A4)),groups4541462559716669496nt_nat(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8549_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(Fa,X3)),aa_Code_integer_nat(Ga,X3)) )
       => ( ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(Fa,X5)),aa_Code_integer_nat(Ga,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(Fa,A4)),groups7237345082560585321er_nat(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8550_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,Fa: fun_nat_int,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,X3)),aa_nat_int(Ga,X3)) )
       => ( ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,X5)),aa_nat_int(Ga,X5)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8551_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_int,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(Fa,X3)),aa_Code_integer_int(Ga,X3)) )
       => ( ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(Fa,X5)),aa_Code_integer_int(Ga,X5)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups7234854612051535045er_int(Fa,A4)),groups7234854612051535045er_int(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8552_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,X3)),aa_nat_nat(Ga,X3)) )
       => ( ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,X5)),aa_nat_nat(Ga,X5)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8553_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,Fa: fun_int_int,Ga: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(Fa,X3)),aa_int_int(Ga,X3)) )
       => ( ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(Fa,X5)),aa_int_int(Ga,X5)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(Fa,A4)),groups4538972089207619220nt_int(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8554_sum__strict__mono__ex1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Fa: fun_Pr8317260874932935276at_rat,Ga: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Pro3608043709957902237at_rat(Fa,X3)),aa_Pro3608043709957902237at_rat(Ga,X3)) )
       => ( ? [X5: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Pro3608043709957902237at_rat(Fa,X5)),aa_Pro3608043709957902237at_rat(Ga,X5)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups342789780944988191at_rat(Fa,A4)),groups342789780944988191at_rat(Ga,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8555_finite__linorder__max__induct,axiom,
    ! [A4: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),B3) )
               => ( aa_set5643387332196575111eger_o(Pa,A7)
                 => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(B3,A7)) ) ) )
         => aa_set5643387332196575111eger_o(Pa,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8556_finite__linorder__max__induct,axiom,
    ! [A4: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [B3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X5: $o] :
                    ( aa_set_o_o(member_o((X5)),A7)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X5)),(B3)) )
               => ( aa_set_o_o(Pa,A7)
                 => aa_set_o_o(Pa,insert_o((B3),A7)) ) ) )
         => aa_set_o_o(Pa,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8557_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,Pa: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(Pa,bot_bot_set_rat)
       => ( ! [B3: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X5: rat] :
                    ( aa_set_rat_o(member_rat(X5),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),B3) )
               => ( aa_set_rat_o(Pa,A7)
                 => aa_set_rat_o(Pa,insert_rat(B3,A7)) ) ) )
         => aa_set_rat_o(Pa,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8558_finite__linorder__max__induct,axiom,
    ! [A4: set_num,Pa: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(Pa,bot_bot_set_num)
       => ( ! [B3: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X5: num] :
                    ( aa_set_num_o(member_num(X5),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X5),B3) )
               => ( aa_set_num_o(Pa,A7)
                 => aa_set_num_o(Pa,insert_num(B3,A7)) ) ) )
         => aa_set_num_o(Pa,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8559_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [B3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X5: nat] :
                    ( aa_set_nat_o(member_nat(X5),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),B3) )
               => ( aa_set_nat_o(Pa,A7)
                 => aa_set_nat_o(Pa,insert_nat(B3,A7)) ) ) )
         => aa_set_nat_o(Pa,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8560_finite__linorder__max__induct,axiom,
    ! [A4: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [B3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X5: int] :
                    ( aa_set_int_o(member_int(X5),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X5),B3) )
               => ( aa_set_int_o(Pa,A7)
                 => aa_set_int_o(Pa,insert_int(B3,A7)) ) ) )
         => aa_set_int_o(Pa,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8561_finite__linorder__min__induct,axiom,
    ! [A4: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B3),X5) )
               => ( aa_set5643387332196575111eger_o(Pa,A7)
                 => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(B3,A7)) ) ) )
         => aa_set5643387332196575111eger_o(Pa,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8562_finite__linorder__min__induct,axiom,
    ! [A4: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [B3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X5: $o] :
                    ( aa_set_o_o(member_o((X5)),A7)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B3)),(X5)) )
               => ( aa_set_o_o(Pa,A7)
                 => aa_set_o_o(Pa,insert_o((B3),A7)) ) ) )
         => aa_set_o_o(Pa,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8563_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,Pa: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(Pa,bot_bot_set_rat)
       => ( ! [B3: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X5: rat] :
                    ( aa_set_rat_o(member_rat(X5),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),X5) )
               => ( aa_set_rat_o(Pa,A7)
                 => aa_set_rat_o(Pa,insert_rat(B3,A7)) ) ) )
         => aa_set_rat_o(Pa,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8564_finite__linorder__min__induct,axiom,
    ! [A4: set_num,Pa: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(Pa,bot_bot_set_num)
       => ( ! [B3: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X5: num] :
                    ( aa_set_num_o(member_num(X5),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B3),X5) )
               => ( aa_set_num_o(Pa,A7)
                 => aa_set_num_o(Pa,insert_num(B3,A7)) ) ) )
         => aa_set_num_o(Pa,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8565_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [B3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X5: nat] :
                    ( aa_set_nat_o(member_nat(X5),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X5) )
               => ( aa_set_nat_o(Pa,A7)
                 => aa_set_nat_o(Pa,insert_nat(B3,A7)) ) ) )
         => aa_set_nat_o(Pa,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8566_finite__linorder__min__induct,axiom,
    ! [A4: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [B3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X5: int] :
                    ( aa_set_int_o(member_int(X5),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X5) )
               => ( aa_set_int_o(Pa,A7)
                 => aa_set_int_o(Pa,insert_int(B3,A7)) ) ) )
         => aa_set_int_o(Pa,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8567_sum_Orelated,axiom,
    ! [Ra: fun_Co2081214437782444048eger_o,S: set_nat,Ha: fun_nat_Code_integer,Ga: fun_nat_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X22),Y22)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_nat_Code_integer(Ha,X3)),aa_nat_Code_integer(Ga,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),S)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),S)) ) ) ) ) ).

% sum.related
tff(fact_8568_sum_Orelated,axiom,
    ! [Ra: fun_Co2081214437782444048eger_o,S: set_int,Ha: fun_int_Code_integer,Ga: fun_int_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X22),Y22)) )
       => ( aa_set_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_int_Code_integer(Ha,X3)),aa_int_Code_integer(Ga,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,groups7873554091576472773nteger(Ha,S)),groups7873554091576472773nteger(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8569_sum_Orelated,axiom,
    ! [Ra: fun_Co2081214437782444048eger_o,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)
     => ( ! [X12: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,X12),X22)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,Y12),Y22) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X22),Y22)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,aa_Cod1783974652533396728nteger(Ha,X3)),aa_Cod1783974652533396728nteger(Ga,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(Ra,groups879477027807139574nteger(Ha,S)),groups879477027807139574nteger(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8570_sum_Orelated,axiom,
    ! [Ra: fun_rat_fun_rat_o,S: set_nat,Ha: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(Ra,zero_zero_rat),zero_zero_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(Ra,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(Ra,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,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,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_nat_rat(Ha,X3)),aa_nat_rat(Ga,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_set_nat_rat(groups2906978787729119204at_rat(Ha),S)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),S)) ) ) ) ) ).

% sum.related
tff(fact_8571_sum_Orelated,axiom,
    ! [Ra: fun_rat_fun_rat_o,S: set_int,Ha: fun_int_rat,Ga: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(Ra,zero_zero_rat),zero_zero_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(Ra,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(Ra,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,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_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_int_rat(Ha,X3)),aa_int_rat(Ga,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,groups3906332499630173760nt_rat(Ha,S)),groups3906332499630173760nt_rat(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8572_sum_Orelated,axiom,
    ! [Ra: fun_rat_fun_rat_o,S: set_Code_integer,Ha: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(Ra,zero_zero_rat),zero_zero_rat)
     => ( ! [X12: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(Ra,X12),X22)
              & aa_rat_o(aa_rat_fun_rat_o(Ra,Y12),Y22) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,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_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(Ra,aa_Code_integer_rat(Ha,X3)),aa_Code_integer_rat(Ga,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(Ra,groups6602215022474089585er_rat(Ha,S)),groups6602215022474089585er_rat(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8573_sum_Orelated,axiom,
    ! [Ra: fun_nat_fun_nat_o,S: set_int,Ha: fun_int_nat,Ga: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Ra,zero_zero_nat),zero_zero_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(Ra,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(Ra,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(Ra,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_int_o(finite_finite_int,S)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,aa_int_nat(Ha,X3)),aa_int_nat(Ga,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(Ra,groups4541462559716669496nt_nat(Ha,S)),groups4541462559716669496nt_nat(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8574_sum_Orelated,axiom,
    ! [Ra: fun_nat_fun_nat_o,S: set_Code_integer,Ha: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Ra,zero_zero_nat),zero_zero_nat)
     => ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(Ra,X12),X22)
              & aa_nat_o(aa_nat_fun_nat_o(Ra,Y12),Y22) )
           => aa_nat_o(aa_nat_fun_nat_o(Ra,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_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_nat_o(aa_nat_fun_nat_o(Ra,aa_Code_integer_nat(Ha,X3)),aa_Code_integer_nat(Ga,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(Ra,groups7237345082560585321er_nat(Ha,S)),groups7237345082560585321er_nat(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8575_sum_Orelated,axiom,
    ! [Ra: fun_int_fun_int_o,S: set_nat,Ha: fun_nat_int,Ga: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(Ra,zero_zero_int),zero_zero_int)
     => ( ! [X12: int,Y12: int,X22: int,Y22: int] :
            ( ( aa_int_o(aa_int_fun_int_o(Ra,X12),X22)
              & aa_int_o(aa_int_fun_int_o(Ra,Y12),Y22) )
           => aa_int_o(aa_int_fun_int_o(Ra,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,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_int_o(aa_int_fun_int_o(Ra,aa_nat_int(Ha,X3)),aa_nat_int(Ga,X3)) )
           => aa_int_o(aa_int_fun_int_o(Ra,aa_set_nat_int(groups3539618377306564664at_int(Ha),S)),aa_set_nat_int(groups3539618377306564664at_int(Ga),S)) ) ) ) ) ).

% sum.related
tff(fact_8576_sum_Orelated,axiom,
    ! [Ra: fun_int_fun_int_o,S: set_Code_integer,Ha: fun_Code_integer_int,Ga: fun_Code_integer_int] :
      ( aa_int_o(aa_int_fun_int_o(Ra,zero_zero_int),zero_zero_int)
     => ( ! [X12: int,Y12: int,X22: int,Y22: int] :
            ( ( aa_int_o(aa_int_fun_int_o(Ra,X12),X22)
              & aa_int_o(aa_int_fun_int_o(Ra,Y12),Y22) )
           => aa_int_o(aa_int_fun_int_o(Ra,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_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_int_o(aa_int_fun_int_o(Ra,aa_Code_integer_int(Ha,X3)),aa_Code_integer_int(Ga,X3)) )
           => aa_int_o(aa_int_fun_int_o(Ra,groups7234854612051535045er_int(Ha,S)),groups7234854612051535045er_int(Ga,S)) ) ) ) ) ).

% sum.related
tff(fact_8577_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(Fa,X3)),aa_Code_integer_rat(Ga,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(Fa,A4)),groups6602215022474089585er_rat(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8578_sum__strict__mono,axiom,
    ! [A4: set_o,Fa: fun_o_rat,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(Fa,(X3))),aa_o_rat(Ga,(X3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(Fa,A4)),groups7872700643590313910_o_rat(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8579_sum__strict__mono,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(Ga,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8580_sum__strict__mono,axiom,
    ! [A4: set_int,Fa: fun_int_rat,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,X3)),aa_int_rat(Ga,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(Fa,A4)),groups3906332499630173760nt_rat(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8581_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(Fa,X3)),aa_Code_integer_nat(Ga,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(Fa,A4)),groups7237345082560585321er_nat(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8582_sum__strict__mono,axiom,
    ! [A4: set_o,Fa: fun_o_nat,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(Fa,(X3))),aa_o_nat(Ga,(X3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(Fa,A4)),groups8507830703676809646_o_nat(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8583_sum__strict__mono,axiom,
    ! [A4: set_int,Fa: fun_int_nat,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Fa,X3)),aa_int_nat(Ga,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(Fa,A4)),groups4541462559716669496nt_nat(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8584_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_int,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(Fa,X3)),aa_Code_integer_int(Ga,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups7234854612051535045er_int(Fa,A4)),groups7234854612051535045er_int(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8585_sum__strict__mono,axiom,
    ! [A4: set_o,Fa: fun_o_int,Ga: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(Fa,(X3))),aa_o_int(Ga,(X3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(Fa,A4)),groups8505340233167759370_o_int(Ga,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8586_sum__strict__mono,axiom,
    ! [A4: set_nat,Fa: fun_nat_int,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,X3)),aa_nat_int(Ga,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)) ) ) ) ).

% sum_strict_mono
tff(fact_8587_sum_Oinsert__if,axiom,
    ! [A4: set_o,Ga: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(Ga,insert_o((X),A4)) = $ite(aa_set_o_o(member_o((X)),A4),groups7872700643590313910_o_rat(Ga,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Ga,(X))),groups7872700643590313910_o_rat(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8588_sum_Oinsert__if,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),insert_nat(X,A4)) = $ite(aa_set_nat_o(member_nat(X),A4),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,X)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4))) ) ) ).

% sum.insert_if
tff(fact_8589_sum_Oinsert__if,axiom,
    ! [A4: set_int,Ga: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(Ga,insert_int(X,A4)) = $ite(aa_set_int_o(member_int(X),A4),groups3906332499630173760nt_rat(Ga,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Ga,X)),groups3906332499630173760nt_rat(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8590_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(Ga,insert_Code_integer(X,A4)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A4),groups6602215022474089585er_rat(Ga,A4),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Ga,X)),groups6602215022474089585er_rat(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8591_sum_Oinsert__if,axiom,
    ! [A4: set_o,Ga: fun_o_nat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8507830703676809646_o_nat(Ga,insert_o((X),A4)) = $ite(aa_set_o_o(member_o((X)),A4),groups8507830703676809646_o_nat(Ga,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Ga,(X))),groups8507830703676809646_o_nat(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8592_sum_Oinsert__if,axiom,
    ! [A4: set_int,Ga: fun_int_nat,X: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(Ga,insert_int(X,A4)) = $ite(aa_set_int_o(member_int(X),A4),groups4541462559716669496nt_nat(Ga,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ga,X)),groups4541462559716669496nt_nat(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8593_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_nat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(Ga,insert_Code_integer(X,A4)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A4),groups7237345082560585321er_nat(Ga,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Ga,X)),groups7237345082560585321er_nat(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8594_sum_Oinsert__if,axiom,
    ! [A4: set_o,Ga: fun_o_int,X: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8505340233167759370_o_int(Ga,insert_o((X),A4)) = $ite(aa_set_o_o(member_o((X)),A4),groups8505340233167759370_o_int(Ga,A4),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(Ga,(X))),groups8505340233167759370_o_int(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8595_sum_Oinsert__if,axiom,
    ! [A4: set_nat,Ga: fun_nat_int,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),insert_nat(X,A4)) = $ite(aa_set_nat_o(member_nat(X),A4),aa_set_nat_int(groups3539618377306564664at_int(Ga),A4),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,X)),aa_set_nat_int(groups3539618377306564664at_int(Ga),A4))) ) ) ).

% sum.insert_if
tff(fact_8596_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_int,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(Ga,insert_Code_integer(X,A4)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A4),groups7234854612051535045er_int(Ga,A4),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(Ga,X)),groups7234854612051535045er_int(Ga,A4))) ) ) ).

% sum.insert_if
tff(fact_8597_bit__push__bit__iff__int,axiom,
    ! [M: nat,K2: int,N2: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K2)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_8598_prod__dvd__prod__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(Fa,A4)),groups3190895334310489300er_nat(Fa,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8599_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Fa: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4077766827762148844at_nat(Fa,A4)),groups4077766827762148844at_nat(Fa,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8600_prod__dvd__prod__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(Fa,A4)),groups3188404863801439024er_int(Fa,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8601_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Fa: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4075276357253098568at_int(Fa,A4)),groups4075276357253098568at_int(Fa,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8602_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(Fa,A4)),groups1707563613775114915nt_nat(Fa,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8603_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)),aa_set_nat_nat(groups708209901874060359at_nat(Fa),B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8604_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(Fa),A4)),aa_set_nat_int(groups705719431365010083at_int(Fa),B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8605_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(Fa,A4)),groups1705073143266064639nt_int(Fa,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8606_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_nat,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_o_nat(Fa,(A3))),aa_o_nat(Ga,(A3))) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3504817904513533571_o_nat(Fa,A4)),groups3504817904513533571_o_nat(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8607_prod__dvd__prod__subset2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_Code_integer_nat(Fa,A3)),aa_Code_integer_nat(Ga,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups3190895334310489300er_nat(Fa,A4)),groups3190895334310489300er_nat(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8608_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_int,Ga: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_o_int(Fa,(A3))),aa_o_int(Ga,(A3))) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3502327434004483295_o_int(Fa,A4)),groups3502327434004483295_o_int(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8609_prod__dvd__prod__subset2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_int,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_Code_integer_int(Fa,A3)),aa_Code_integer_int(Ga,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups3188404863801439024er_int(Fa,A4)),groups3188404863801439024er_int(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8610_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_nat,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(Fa,A3)),aa_int_nat(Ga,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(Fa,A4)),groups1707563613775114915nt_nat(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8611_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(Fa,A3)),aa_nat_nat(Ga,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(Fa),A4)),aa_set_nat_nat(groups708209901874060359at_nat(Ga),B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8612_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_int,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(Fa,A3)),aa_nat_int(Ga,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_set_nat_int(groups705719431365010083at_int(Fa),A4)),aa_set_nat_int(groups705719431365010083at_int(Ga),B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8613_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_int,Ga: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(Fa,A3)),aa_int_int(Ga,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(Fa,A4)),groups1705073143266064639nt_int(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8614_prod__dvd__prod__subset2,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Fa: fun_set_nat_nat,Ga: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( ! [A3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(Fa,A3)),aa_set_nat_nat(Ga,A3)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4248547760180025341at_nat(Fa,A4)),groups4248547760180025341at_nat(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8615_prod__dvd__prod__subset2,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Fa: fun_set_nat_int,Ga: fun_set_nat_int] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( ! [A3: set_nat] :
              ( aa_set_set_nat_o(member_set_nat(A3),A4)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_set_nat_int(Fa,A3)),aa_set_nat_int(Ga,A3)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4246057289670975065at_int(Fa,A4)),groups4246057289670975065at_int(Ga,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8616_concat__bit__eq,axiom,
    ! [N2: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N2),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),K2)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,N2),L)) ).

% concat_bit_eq
tff(fact_8617_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q4)),N2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8618_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,Fa: fun_o_Code_integer,B4: code_integer,I2: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I3))) )
       => ( ( groups4406642042086082107nteger(Fa,S2) = B4 )
         => ( aa_set_o_o(member_o((I2)),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(Fa,(I2))),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8619_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,Fa: fun_nat_Code_integer,B4: code_integer,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I3)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),S2) = B4 )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8620_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,Fa: fun_int_Code_integer,B4: code_integer,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I3)) )
       => ( ( groups7873554091576472773nteger(Fa,S2) = B4 )
         => ( aa_set_int_o(member_int(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8621_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,Fa: fun_Co8499238673658658103nteger,B4: code_integer,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,I3)) )
       => ( ( groups879477027807139574nteger(Fa,S2) = B4 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8622_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,Fa: fun_o_rat,B4: rat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(I3))) )
       => ( ( groups7872700643590313910_o_rat(Fa,S2) = B4 )
         => ( aa_set_o_o(member_o((I2)),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(I2))),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8623_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,Fa: fun_nat_rat,B4: rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I3)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),S2) = B4 )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8624_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,Fa: fun_int_rat,B4: rat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I3)) )
       => ( ( groups3906332499630173760nt_rat(Fa,S2) = B4 )
         => ( aa_set_int_o(member_int(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8625_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,Fa: fun_Code_integer_rat,B4: rat,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I3)) )
       => ( ( groups6602215022474089585er_rat(Fa,S2) = B4 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8626_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,Fa: fun_o_nat,B4: nat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(I3))) )
       => ( ( groups8507830703676809646_o_nat(Fa,S2) = B4 )
         => ( aa_set_o_o(member_o((I2)),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(I2))),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8627_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,Fa: fun_int_nat,B4: nat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,I3)) )
       => ( ( groups4541462559716669496nt_nat(Fa,S2) = B4 )
         => ( aa_set_int_o(member_int(I2),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,I2)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8628_sum__nonneg__0,axiom,
    ! [S2: set_o,Fa: fun_o_Code_integer,I2: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I3))) )
       => ( ( groups4406642042086082107nteger(Fa,S2) = zero_z3403309356797280102nteger )
         => ( aa_set_o_o(member_o((I2)),S2)
           => ( aa_o_Code_integer(Fa,(I2)) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8629_sum__nonneg__0,axiom,
    ! [S2: set_nat,Fa: fun_nat_Code_integer,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I3)) )
       => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),S2) = zero_z3403309356797280102nteger )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => ( aa_nat_Code_integer(Fa,I2) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8630_sum__nonneg__0,axiom,
    ! [S2: set_int,Fa: fun_int_Code_integer,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I3)) )
       => ( ( groups7873554091576472773nteger(Fa,S2) = zero_z3403309356797280102nteger )
         => ( aa_set_int_o(member_int(I2),S2)
           => ( aa_int_Code_integer(Fa,I2) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8631_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,Fa: fun_Co8499238673658658103nteger,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,I3)) )
       => ( ( groups879477027807139574nteger(Fa,S2) = zero_z3403309356797280102nteger )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => ( aa_Cod1783974652533396728nteger(Fa,I2) = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8632_sum__nonneg__0,axiom,
    ! [S2: set_o,Fa: fun_o_rat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(I3))) )
       => ( ( groups7872700643590313910_o_rat(Fa,S2) = zero_zero_rat )
         => ( aa_set_o_o(member_o((I2)),S2)
           => ( aa_o_rat(Fa,(I2)) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8633_sum__nonneg__0,axiom,
    ! [S2: set_nat,Fa: fun_nat_rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I3)) )
       => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),S2) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => ( aa_nat_rat(Fa,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8634_sum__nonneg__0,axiom,
    ! [S2: set_int,Fa: fun_int_rat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I3)) )
       => ( ( groups3906332499630173760nt_rat(Fa,S2) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I2),S2)
           => ( aa_int_rat(Fa,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8635_sum__nonneg__0,axiom,
    ! [S2: set_Code_integer,Fa: fun_Code_integer_rat,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I3)) )
       => ( ( groups6602215022474089585er_rat(Fa,S2) = zero_zero_rat )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S2)
           => ( aa_Code_integer_rat(Fa,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8636_sum__nonneg__0,axiom,
    ! [S2: set_o,Fa: fun_o_nat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(I3))) )
       => ( ( groups8507830703676809646_o_nat(Fa,S2) = zero_zero_nat )
         => ( aa_set_o_o(member_o((I2)),S2)
           => ( aa_o_nat(Fa,(I2)) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8637_sum__nonneg__0,axiom,
    ! [S2: set_int,Fa: fun_int_nat,I2: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,I3)) )
       => ( ( groups4541462559716669496nt_nat(Fa,S2) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I2),S2)
           => ( aa_int_nat(Fa,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8638_add__mset__in__multiset,axiom,
    ! [M6: fun_list_nat_nat,A: list_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_adx(M6)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_lis557685189327716851_nat_o(aTP_Lamm_aew(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8639_add__mset__in__multiset,axiom,
    ! [M6: fun_se4101287982975402108nt_nat,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_adz(M6)))
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_set4993530631398190431_int_o(aTP_Lamm_aex(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8640_add__mset__in__multiset,axiom,
    ! [M6: fun_set_nat_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_aeb(M6)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_aey(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8641_add__mset__in__multiset,axiom,
    ! [M6: fun_nat_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aed(M6)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aez(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8642_add__mset__in__multiset,axiom,
    ! [M6: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_aef(M6)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_afa(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8643_add__mset__in__multiset,axiom,
    ! [M6: fun_Code_integer_nat,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_aeh(M6)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_afb(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8644_add__mset__in__multiset,axiom,
    ! [M6: fun_Pr2602117396471556196at_nat,A: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_aej(M6)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_Pro989595756709598047_nat_o(aTP_Lamm_afc(M6),A))) ) ).

% add_mset_in_multiset
tff(fact_8645_sum_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_Code_integer,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups4406642042086082107nteger(Ga,inf_inf_set_o(A4,B4)) = groups4406642042086082107nteger(aa_set9049949509649958604nteger(aTP_Lamm_afd(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8646_sum_Ointer__restrict,axiom,
    ! [A4: set_int,Ga: fun_int_Code_integer,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7873554091576472773nteger(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) = groups7873554091576472773nteger(aa_set5517701783735052070nteger(aTP_Lamm_afe(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8647_sum_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups879477027807139574nteger(Ga,inf_in1364745209274528805nteger(A4,B4)) = groups879477027807139574nteger(aa_set4294055438810586376nteger(aTP_Lamm_aff(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8648_sum_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_rat,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(Ga,inf_inf_set_o(A4,B4)) = groups7872700643590313910_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_afg(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8649_sum_Ointer__restrict,axiom,
    ! [A4: set_int,Ga: fun_int_rat,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) = groups3906332499630173760nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_afh(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8650_sum_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(Ga,inf_in1364745209274528805nteger(A4,B4)) = groups6602215022474089585er_rat(aa_set7344733238996822275er_rat(aTP_Lamm_afi(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8651_sum_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_nat,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8507830703676809646_o_nat(Ga,inf_inf_set_o(A4,B4)) = groups8507830703676809646_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_afj(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8652_sum_Ointer__restrict,axiom,
    ! [A4: set_int,Ga: fun_int_nat,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) = groups4541462559716669496nt_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_afk(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8653_sum_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_nat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(Ga,inf_in1364745209274528805nteger(A4,B4)) = groups7237345082560585321er_nat(aa_set1629589760535443195er_nat(aTP_Lamm_afl(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8654_sum_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_int,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8505340233167759370_o_int(Ga,inf_inf_set_o(A4,B4)) = groups8505340233167759370_o_int(aa_set_o_fun_o_int(aTP_Lamm_afm(Ga),B4),A4) ) ) ).

% sum.inter_restrict
tff(fact_8655_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),collect_nat(aTP_Lamm_afn(Ga)))) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8656_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7873554091576472773nteger(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_afo(Ga)))) = groups7873554091576472773nteger(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8657_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups879477027807139574nteger(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_afp(Ga)))) = groups879477027807139574nteger(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8658_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),collect_nat(aTP_Lamm_afq(Ga)))) = aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8659_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_afr(Ga)))) = groups3906332499630173760nt_rat(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8660_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_afs(Ga)))) = groups6602215022474089585er_rat(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8661_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_aft(Ga)))) = groups4541462559716669496nt_nat(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8662_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_afu(Ga)))) = groups7237345082560585321er_nat(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8663_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),collect_nat(aTP_Lamm_afv(Ga)))) = aa_set_nat_int(groups3539618377306564664at_int(Ga),A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8664_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_afw(Ga)))) = groups7234854612051535045er_int(Ga,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8665_diff__preserves__multiset,axiom,
    ! [M6: fun_list_nat_nat,N7: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_adx(M6)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun7915899554530022_nat_o(aTP_Lamm_afx(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8666_diff__preserves__multiset,axiom,
    ! [M6: fun_se4101287982975402108nt_nat,N7: fun_se4101287982975402108nt_nat] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_adz(M6)))
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun3785083049450726210_int_o(aTP_Lamm_afy(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8667_diff__preserves__multiset,axiom,
    ! [M6: fun_set_nat_nat,N7: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_aeb(M6)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun1332161875654541722_nat_o(aTP_Lamm_afz(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8668_diff__preserves__multiset,axiom,
    ! [M6: fun_nat_nat,N7: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_aed(M6)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_aga(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8669_diff__preserves__multiset,axiom,
    ! [M6: fun_int_nat,N7: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_aef(M6)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_agb(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8670_diff__preserves__multiset,axiom,
    ! [M6: fun_Code_integer_nat,N7: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_aeh(M6)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_agc(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8671_diff__preserves__multiset,axiom,
    ! [M6: fun_Pr2602117396471556196at_nat,N7: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aTP_Lamm_aej(M6)))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun741873263805325634_nat_o(aTP_Lamm_agd(M6),N7))) ) ).

% diff_preserves_multiset
tff(fact_8672_prod_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_Code_integer,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7694694392188491536nteger(Ga,inf_inf_set_o(A4,B4)) = groups7694694392188491536nteger(aa_set9049949509649958604nteger(aTP_Lamm_age(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8673_prod_Ointer__restrict,axiom,
    ! [A4: set_int,Ga: fun_int_Code_integer,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3827104343326376752nteger(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) = groups3827104343326376752nteger(aa_set5517701783735052070nteger(aTP_Lamm_agf(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8674_prod_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups3674199335183972705nteger(Ga,inf_in1364745209274528805nteger(A4,B4)) = groups3674199335183972705nteger(aa_set4294055438810586376nteger(aTP_Lamm_agg(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8675_prod_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_assn,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups5301882518646026715o_assn(Ga,inf_inf_set_o(A4,B4)) = groups5301882518646026715o_assn(aa_set_o_fun_o_assn(aTP_Lamm_agh(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8676_prod_Ointer__restrict,axiom,
    ! [A4: set_int,Ga: fun_int_assn,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) = groups7882442080178216443t_assn(aa_set5310648174735401585t_assn(aTP_Lamm_agi(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8677_prod_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1304777262505850412r_assn(Ga,inf_in1364745209274528805nteger(A4,B4)) = groups1304777262505850412r_assn(aa_set2437433236263590995r_assn(aTP_Lamm_agj(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8678_prod_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_rat,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups2869687844427037835_o_rat(Ga,inf_inf_set_o(A4,B4)) = groups2869687844427037835_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_agk(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8679_prod_Ointer__restrict,axiom,
    ! [A4: set_int,Ga: fun_int_rat,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4)) = groups1072433553688619179nt_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_agl(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8680_prod_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups2555765274223993564er_rat(Ga,inf_in1364745209274528805nteger(A4,B4)) = groups2555765274223993564er_rat(aa_set7344733238996822275er_rat(aTP_Lamm_agm(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8681_prod_Ointer__restrict,axiom,
    ! [A4: set_o,Ga: fun_o_nat,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups3504817904513533571_o_nat(Ga,inf_inf_set_o(A4,B4)) = groups3504817904513533571_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_agn(Ga),B4),A4) ) ) ).

% prod.inter_restrict
tff(fact_8682_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),collect_nat(aTP_Lamm_ago(Ga)))) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8683_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3827104343326376752nteger(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_agp(Ga)))) = groups3827104343326376752nteger(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8684_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups3674199335183972705nteger(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_agq(Ga)))) = groups3674199335183972705nteger(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8685_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),collect_nat(aTP_Lamm_agr(Ga)))) = groups6906906614972039071t_assn(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8686_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_ags(Ga)))) = groups7882442080178216443t_assn(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8687_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1304777262505850412r_assn(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_agt(Ga)))) = groups1304777262505850412r_assn(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8688_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),collect_nat(aTP_Lamm_agu(Ga)))) = aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8689_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_agv(Ga)))) = groups1072433553688619179nt_rat(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8690_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups2555765274223993564er_rat(Ga,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_agw(Ga)))) = groups2555765274223993564er_rat(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8691_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),collect_int(aTP_Lamm_agx(Ga)))) = groups1707563613775114915nt_nat(Ga,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8692_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_jl,M))) ) ).

% finite_divisors_nat
tff(fact_8693_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N7),set_or1269000886237332187st_nat(zero_zero_nat,N2))
     => aa_set_nat_o(finite_finite_nat,N7) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_8694_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N7),set_or4665077453230672383an_nat(zero_zero_nat,N2))
     => aa_set_nat_o(finite_finite_nat,N7) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_8695_finite__abs__int__segment,axiom,
    ! [A: rat] : aa_set_rat_o(finite_finite_rat,collect_rat(aa_rat_fun_rat_o(aTP_Lamm_agy,A))) ).

% finite_abs_int_segment
tff(fact_8696_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I2)),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8697_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8698_sum__pos2,axiom,
    ! [I5: set_int,I2: int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8699_sum__pos2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups879477027807139574nteger(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8700_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I2)),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(Fa,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8701_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(Fa,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8702_sum__pos2,axiom,
    ! [I5: set_int,I2: int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(Fa,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8703_sum__pos2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8704_sum__pos2,axiom,
    ! [I5: set_o,I2: $o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I2)),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(Fa,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(I3))) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8705_sum__pos2,axiom,
    ! [I5: set_int,I2: int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Fa,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8706_sum__pos,axiom,
    ! [I5: set_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups879477027807139574nteger(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8707_sum__pos,axiom,
    ! [I5: set_o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups4406642042086082107nteger(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8708_sum__pos,axiom,
    ! [I5: set_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),I5)) ) ) ) ).

% sum_pos
tff(fact_8709_sum__pos,axiom,
    ! [I5: set_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups7873554091576472773nteger(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8710_sum__pos,axiom,
    ! [I5: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8711_sum__pos,axiom,
    ! [I5: set_o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(Fa,(I3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups7872700643590313910_o_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8712_sum__pos,axiom,
    ! [I5: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(Fa,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),I5)) ) ) ) ).

% sum_pos
tff(fact_8713_sum__pos,axiom,
    ! [I5: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(Fa,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8714_sum__pos,axiom,
    ! [I5: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Fa,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups7237345082560585321er_nat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8715_sum__pos,axiom,
    ! [I5: set_o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(Fa,(I3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8716_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I2)),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(Fa,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(Fa,(I3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8717_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(Fa,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(Fa,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8718_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(Fa,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(Fa,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8719_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(Fa,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(Fa,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3674199335183972705nteger(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8720_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I2)),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(Fa,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(Fa,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8721_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(Fa,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(Fa,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(Fa),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8722_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(Fa,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(Fa,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8723_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Code_integer_rat(Fa,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(Fa,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8724_less__1__prod2,axiom,
    ! [I5: set_o,I2: $o,Fa: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( aa_set_o_o(member_o((I2)),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(Fa,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(Fa,(I3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8725_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I2: code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Code_integer_int(Fa,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_Code_integer_int(Fa,I3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3188404863801439024er_int(Fa,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8726_less__1__prod,axiom,
    ! [I5: set_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(Fa,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3674199335183972705nteger(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8727_less__1__prod,axiom,
    ! [I5: set_o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(Fa,(I3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8728_less__1__prod,axiom,
    ! [I5: set_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(Fa,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),I5)) ) ) ) ).

% less_1_prod
tff(fact_8729_less__1__prod,axiom,
    ! [I5: set_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(Fa,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8730_less__1__prod,axiom,
    ! [I5: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Code_integer_rat(Fa,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8731_less__1__prod,axiom,
    ! [I5: set_o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(Fa,(I3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8732_less__1__prod,axiom,
    ! [I5: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(Fa,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_set_nat_rat(groups73079841787564623at_rat(Fa),I5)) ) ) ) ).

% less_1_prod
tff(fact_8733_less__1__prod,axiom,
    ! [I5: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(Fa,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8734_less__1__prod,axiom,
    ! [I5: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Code_integer_int(Fa,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3188404863801439024er_int(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8735_less__1__prod,axiom,
    ! [I5: set_o,Fa: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(Fa,(I3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(Fa,I5)) ) ) ) ).

% less_1_prod
tff(fact_8736_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_Code_integer(Ga,(X3)) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_Code_integer(Ga,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups4406642042086082107nteger(Ga,T7) = groups4406642042086082107nteger(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8737_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_Code_integer(Ga,X3) = aa_nat_Code_integer(Ha,X3) ) )
           => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),T7) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8738_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(Ga,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups879477027807139574nteger(Ga,T7) = groups879477027807139574nteger(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8739_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_rat,Ha: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_rat(Ga,(X3)) = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_rat(Ga,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups7872700643590313910_o_rat(Ga,T7) = groups7872700643590313910_o_rat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8740_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ga,X3) = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_rat(Ga,X3) = aa_nat_rat(Ha,X3) ) )
           => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),T7) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8741_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ga,X3) = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(Ga,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups6602215022474089585er_rat(Ga,T7) = groups6602215022474089585er_rat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8742_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_nat(Ga,(X3)) = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_nat(Ga,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups8507830703676809646_o_nat(Ga,T7) = groups8507830703676809646_o_nat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8743_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_nat(Ga,X3) = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(Ga,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( groups7237345082560585321er_nat(Ga,T7) = groups7237345082560585321er_nat(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8744_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_int,Ha: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_int(Ga,(X3)) = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_int(Ga,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups8505340233167759370_o_int(Ga,T7) = groups8505340233167759370_o_int(Ha,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8745_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_int(Ga,X3) = zero_zero_int ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_int(Ga,X3) = aa_nat_int(Ha,X3) ) )
           => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),T7) = aa_set_nat_int(groups3539618377306564664at_int(Ha),S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8746_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_Code_integer,Ga: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_Code_integer(Ha,(X3)) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_Code_integer(Ga,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups4406642042086082107nteger(Ga,S) = groups4406642042086082107nteger(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8747_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,Ha: fun_nat_Code_integer,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ha,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_Code_integer(Ga,X3) = aa_nat_Code_integer(Ha,X3) ) )
           => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),S) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8748_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ha,X3) = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(Ga,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups879477027807139574nteger(Ga,S) = groups879477027807139574nteger(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8749_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_rat,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_rat(Ha,(X3)) = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_rat(Ga,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups7872700643590313910_o_rat(Ga,S) = groups7872700643590313910_o_rat(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8750_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,Ha: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ha,X3) = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_rat(Ga,X3) = aa_nat_rat(Ha,X3) ) )
           => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),S) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8751_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ha,X3) = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(Ga,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups6602215022474089585er_rat(Ga,S) = groups6602215022474089585er_rat(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8752_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_nat,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_nat(Ha,(X3)) = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_nat(Ga,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups8507830703676809646_o_nat(Ga,S) = groups8507830703676809646_o_nat(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8753_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_nat(Ha,X3) = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(Ga,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( groups7237345082560585321er_nat(Ga,S) = groups7237345082560585321er_nat(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8754_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_int,Ga: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_int(Ha,(X3)) = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_int(Ga,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups8505340233167759370_o_int(Ga,S) = groups8505340233167759370_o_int(Ha,T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8755_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,Ha: fun_nat_int,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_int(Ha,X3) = zero_zero_int ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_int(Ga,X3) = aa_nat_int(Ha,X3) ) )
           => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),S) = aa_set_nat_int(groups3539618377306564664at_int(Ha),T7) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8756_sum_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),T7) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8757_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( groups879477027807139574nteger(Ga,T7) = groups879477027807139574nteger(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8758_sum_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ga,X3) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),T7) = aa_set_nat_rat(groups2906978787729119204at_rat(Ga),S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8759_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ga,X3) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(Ga,T7) = groups6602215022474089585er_rat(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8760_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_nat(Ga,X3) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(Ga,T7) = groups7237345082560585321er_nat(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8761_sum_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_int(Ga,X3) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),T7) = aa_set_nat_int(groups3539618377306564664at_int(Ga),S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8762_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_int(Ga,X3) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(Ga,T7) = groups7234854612051535045er_int(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8763_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(Ga,T7) = groups7873554091576472773nteger(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8764_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_rat(Ga,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(Ga,T7) = groups3906332499630173760nt_rat(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8765_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_nat(Ga,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(Ga,T7) = groups4541462559716669496nt_nat(Ga,S) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8766_sum_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),S) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8767_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( groups879477027807139574nteger(Ga,S) = groups879477027807139574nteger(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8768_sum_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ga,X3) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),S) = aa_set_nat_rat(groups2906978787729119204at_rat(Ga),T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8769_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ga,X3) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(Ga,S) = groups6602215022474089585er_rat(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8770_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_nat(Ga,X3) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(Ga,S) = groups7237345082560585321er_nat(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8771_sum_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_int(Ga,X3) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),S) = aa_set_nat_int(groups3539618377306564664at_int(Ga),T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8772_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_int(Ga,X3) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(Ga,S) = groups7234854612051535045er_int(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8773_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(Ga,S) = groups7873554091576472773nteger(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8774_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_rat(Ga,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(Ga,S) = groups3906332499630173760nt_rat(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8775_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_nat(Ga,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(Ga,S) = groups4541462559716669496nt_nat(Ga,T7) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8776_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_Code_integer(Ga,(A3)) = zero_z3403309356797280102nteger ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_Code_integer(Ha,(B3)) = zero_z3403309356797280102nteger ) )
             => ( ( groups4406642042086082107nteger(Ga,C3) = groups4406642042086082107nteger(Ha,C3) )
               => ( groups4406642042086082107nteger(Ga,A4) = groups4406642042086082107nteger(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8777_sum_Osame__carrierI,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_Code_integer(Ga,A3) = zero_z3403309356797280102nteger ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_Code_integer(Ha,B3) = zero_z3403309356797280102nteger ) )
             => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),C3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),C3) )
               => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),A4) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8778_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Cod1783974652533396728nteger(Ga,A3) = zero_z3403309356797280102nteger ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Cod1783974652533396728nteger(Ha,B3) = zero_z3403309356797280102nteger ) )
             => ( ( groups879477027807139574nteger(Ga,C3) = groups879477027807139574nteger(Ha,C3) )
               => ( groups879477027807139574nteger(Ga,A4) = groups879477027807139574nteger(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8779_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_rat,Ha: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_rat(Ga,(A3)) = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_rat(Ha,(B3)) = zero_zero_rat ) )
             => ( ( groups7872700643590313910_o_rat(Ga,C3) = groups7872700643590313910_o_rat(Ha,C3) )
               => ( groups7872700643590313910_o_rat(Ga,A4) = groups7872700643590313910_o_rat(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8780_sum_Osame__carrierI,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_rat(Ga,A3) = zero_zero_rat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_rat(Ha,B3) = zero_zero_rat ) )
             => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),C3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),C3) )
               => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8781_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_rat(Ga,A3) = zero_zero_rat ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_rat(Ha,B3) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(Ga,C3) = groups6602215022474089585er_rat(Ha,C3) )
               => ( groups6602215022474089585er_rat(Ga,A4) = groups6602215022474089585er_rat(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8782_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_nat(Ga,(A3)) = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_nat(Ha,(B3)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(Ga,C3) = groups8507830703676809646_o_nat(Ha,C3) )
               => ( groups8507830703676809646_o_nat(Ga,A4) = groups8507830703676809646_o_nat(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8783_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_nat(Ga,A3) = zero_zero_nat ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_nat(Ha,B3) = zero_zero_nat ) )
             => ( ( groups7237345082560585321er_nat(Ga,C3) = groups7237345082560585321er_nat(Ha,C3) )
               => ( groups7237345082560585321er_nat(Ga,A4) = groups7237345082560585321er_nat(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8784_sum_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_int,Ha: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_int(Ga,(A3)) = zero_zero_int ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_int(Ha,(B3)) = zero_zero_int ) )
             => ( ( groups8505340233167759370_o_int(Ga,C3) = groups8505340233167759370_o_int(Ha,C3) )
               => ( groups8505340233167759370_o_int(Ga,A4) = groups8505340233167759370_o_int(Ha,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8785_sum_Osame__carrierI,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_int(Ga,A3) = zero_zero_int ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_int(Ha,B3) = zero_zero_int ) )
             => ( ( aa_set_nat_int(groups3539618377306564664at_int(Ga),C3) = aa_set_nat_int(groups3539618377306564664at_int(Ha),C3) )
               => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),A4) = aa_set_nat_int(groups3539618377306564664at_int(Ha),B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8786_sum_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_Code_integer(Ga,(A3)) = zero_z3403309356797280102nteger ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_Code_integer(Ha,(B3)) = zero_z3403309356797280102nteger ) )
             => ( ( groups4406642042086082107nteger(Ga,A4) = groups4406642042086082107nteger(Ha,B4) )
              <=> ( groups4406642042086082107nteger(Ga,C3) = groups4406642042086082107nteger(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8787_sum_Osame__carrier,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_Code_integer(Ga,A3) = zero_z3403309356797280102nteger ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_Code_integer(Ha,B3) = zero_z3403309356797280102nteger ) )
             => ( ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),A4) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),B4) )
              <=> ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),C3) = aa_set871585252387937313nteger(groups7501900531339628137nteger(Ha),C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8788_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Cod1783974652533396728nteger(Ga,A3) = zero_z3403309356797280102nteger ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Cod1783974652533396728nteger(Ha,B3) = zero_z3403309356797280102nteger ) )
             => ( ( groups879477027807139574nteger(Ga,A4) = groups879477027807139574nteger(Ha,B4) )
              <=> ( groups879477027807139574nteger(Ga,C3) = groups879477027807139574nteger(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8789_sum_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_rat,Ha: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_rat(Ga,(A3)) = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_rat(Ha,(B3)) = zero_zero_rat ) )
             => ( ( groups7872700643590313910_o_rat(Ga,A4) = groups7872700643590313910_o_rat(Ha,B4) )
              <=> ( groups7872700643590313910_o_rat(Ga,C3) = groups7872700643590313910_o_rat(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8790_sum_Osame__carrier,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_rat(Ga,A3) = zero_zero_rat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_rat(Ha,B3) = zero_zero_rat ) )
             => ( ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),B4) )
              <=> ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),C3) = aa_set_nat_rat(groups2906978787729119204at_rat(Ha),C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8791_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_rat(Ga,A3) = zero_zero_rat ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_rat(Ha,B3) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(Ga,A4) = groups6602215022474089585er_rat(Ha,B4) )
              <=> ( groups6602215022474089585er_rat(Ga,C3) = groups6602215022474089585er_rat(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8792_sum_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_nat(Ga,(A3)) = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_nat(Ha,(B3)) = zero_zero_nat ) )
             => ( ( groups8507830703676809646_o_nat(Ga,A4) = groups8507830703676809646_o_nat(Ha,B4) )
              <=> ( groups8507830703676809646_o_nat(Ga,C3) = groups8507830703676809646_o_nat(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8793_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_nat(Ga,A3) = zero_zero_nat ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_nat(Ha,B3) = zero_zero_nat ) )
             => ( ( groups7237345082560585321er_nat(Ga,A4) = groups7237345082560585321er_nat(Ha,B4) )
              <=> ( groups7237345082560585321er_nat(Ga,C3) = groups7237345082560585321er_nat(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8794_sum_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_int,Ha: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_int(Ga,(A3)) = zero_zero_int ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_int(Ha,(B3)) = zero_zero_int ) )
             => ( ( groups8505340233167759370_o_int(Ga,A4) = groups8505340233167759370_o_int(Ha,B4) )
              <=> ( groups8505340233167759370_o_int(Ga,C3) = groups8505340233167759370_o_int(Ha,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8795_sum_Osame__carrier,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_int(Ga,A3) = zero_zero_int ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_int(Ha,B3) = zero_zero_int ) )
             => ( ( aa_set_nat_int(groups3539618377306564664at_int(Ga),A4) = aa_set_nat_int(groups3539618377306564664at_int(Ha),B4) )
              <=> ( aa_set_nat_int(groups3539618377306564664at_int(Ga),C3) = aa_set_nat_int(groups3539618377306564664at_int(Ha),C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8796_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8797_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups6602215022474089585er_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(A4,B4))),groups6602215022474089585er_rat(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8798_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups7237345082560585321er_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(A4,B4))),groups7237345082560585321er_nat(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8799_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_int(groups3539618377306564664at_int(Ga),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8800_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups7234854612051535045er_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(A4,B4))),groups7234854612051535045er_int(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8801_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups3906332499630173760nt_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups3906332499630173760nt_rat(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8802_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4541462559716669496nt_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups4541462559716669496nt_nat(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8803_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8804_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,Ga: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4538972089207619220nt_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups4538972089207619220nt_int(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8805_sum_Osubset__diff,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ( groups342789780944988191at_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))),groups342789780944988191at_rat(Ga,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_8806_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),B4)) ) ) ) ).

% sum_diff
tff(fact_8807_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups6602215022474089585er_rat(Fa,minus_2355218937544613996nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups6602215022474089585er_rat(Fa,A4)),groups6602215022474089585er_rat(Fa,B4)) ) ) ) ).

% sum_diff
tff(fact_8808_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),aa_set_nat_int(groups3539618377306564664at_int(Fa),B4)) ) ) ) ).

% sum_diff
tff(fact_8809_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups7234854612051535045er_int(Fa,minus_2355218937544613996nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups7234854612051535045er_int(Fa,A4)),groups7234854612051535045er_int(Fa,B4)) ) ) ) ).

% sum_diff
tff(fact_8810_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
       => ( groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups342789780944988191at_rat(Fa,A4)),groups342789780944988191at_rat(Fa,B4)) ) ) ) ).

% sum_diff
tff(fact_8811_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
       => ( groups975429370522433651at_int(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups975429370522433651at_int(Fa,A4)),groups975429370522433651at_int(Fa,B4)) ) ) ) ).

% sum_diff
tff(fact_8812_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
       => ( groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3906332499630173760nt_rat(Fa,A4)),groups3906332499630173760nt_rat(Fa,B4)) ) ) ) ).

% sum_diff
tff(fact_8813_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
       => ( groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(Fa,A4)),groups4538972089207619220nt_int(Fa,B4)) ) ) ) ).

% sum_diff
tff(fact_8814_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups6906906614972039071t_assn(Ga,A4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups6906906614972039071t_assn(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8815_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups1304777262505850412r_assn(Ga,A4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ga,minus_2355218937544613996nteger(A4,B4))),groups1304777262505850412r_assn(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8816_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8817_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups2555765274223993564er_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ga,minus_2355218937544613996nteger(A4,B4))),groups2555765274223993564er_rat(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8818_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups3190895334310489300er_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(Ga,minus_2355218937544613996nteger(A4,B4))),groups3190895334310489300er_nat(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8819_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups3188404863801439024er_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(Ga,minus_2355218937544613996nteger(A4,B4))),groups3188404863801439024er_int(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8820_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups7882442080178216443t_assn(Ga,A4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups7882442080178216443t_assn(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8821_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups1072433553688619179nt_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups1072433553688619179nt_rat(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8822_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups1707563613775114915nt_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups1707563613775114915nt_nat(Ga,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8823_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8824_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_Code_integer(Ga,(X3)) = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_Code_integer(Ga,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups7694694392188491536nteger(Ga,T7) = groups7694694392188491536nteger(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8825_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_Code_integer(Ga,X3) = aa_nat_Code_integer(Ha,X3) ) )
           => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),T7) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8826_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(Ga,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups3674199335183972705nteger(Ga,T7) = groups3674199335183972705nteger(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8827_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_assn(Ga,(X3)) = one_one_assn ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_assn(Ga,(X3)) = aa_o_assn(Ha,(X3)) ) )
           => ( groups5301882518646026715o_assn(Ga,T7) = groups5301882518646026715o_assn(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8828_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_assn,Ha: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_assn(Ga,X3) = one_one_assn ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_assn(Ga,X3) = aa_nat_assn(Ha,X3) ) )
           => ( groups6906906614972039071t_assn(Ga,T7) = groups6906906614972039071t_assn(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8829_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_assn(Ga,X3) = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_assn(Ga,X3) = aa_Code_integer_assn(Ha,X3) ) )
           => ( groups1304777262505850412r_assn(Ga,T7) = groups1304777262505850412r_assn(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8830_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_rat,Ha: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_rat(Ga,(X3)) = one_one_rat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_rat(Ga,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups2869687844427037835_o_rat(Ga,T7) = groups2869687844427037835_o_rat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8831_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ga,X3) = one_one_rat ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_rat(Ga,X3) = aa_nat_rat(Ha,X3) ) )
           => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),T7) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8832_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ga,X3) = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(Ga,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups2555765274223993564er_rat(Ga,T7) = groups2555765274223993564er_rat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8833_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,Ga: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_nat(Ga,(X3)) = one_one_nat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_nat(Ga,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups3504817904513533571_o_nat(Ga,T7) = groups3504817904513533571_o_nat(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8834_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_Code_integer,Ga: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_Code_integer(Ha,(X3)) = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_Code_integer(Ga,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
           => ( groups7694694392188491536nteger(Ga,S) = groups7694694392188491536nteger(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8835_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,Ha: fun_nat_Code_integer,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ha,X3) = one_one_Code_integer ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_Code_integer(Ga,X3) = aa_nat_Code_integer(Ha,X3) ) )
           => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8836_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ha,X3) = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Cod1783974652533396728nteger(Ga,X3) = aa_Cod1783974652533396728nteger(Ha,X3) ) )
           => ( groups3674199335183972705nteger(Ga,S) = groups3674199335183972705nteger(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8837_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_assn,Ga: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_assn(Ha,(X3)) = one_one_assn ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_assn(Ga,(X3)) = aa_o_assn(Ha,(X3)) ) )
           => ( groups5301882518646026715o_assn(Ga,S) = groups5301882518646026715o_assn(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8838_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,Ha: fun_nat_assn,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_assn(Ha,X3) = one_one_assn ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_assn(Ga,X3) = aa_nat_assn(Ha,X3) ) )
           => ( groups6906906614972039071t_assn(Ga,S) = groups6906906614972039071t_assn(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8839_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_assn(Ha,X3) = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_assn(Ga,X3) = aa_Code_integer_assn(Ha,X3) ) )
           => ( groups1304777262505850412r_assn(Ga,S) = groups1304777262505850412r_assn(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8840_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_rat,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_rat(Ha,(X3)) = one_one_rat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_rat(Ga,(X3)) = aa_o_rat(Ha,(X3)) ) )
           => ( groups2869687844427037835_o_rat(Ga,S) = groups2869687844427037835_o_rat(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8841_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,Ha: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ha,X3) = one_one_rat ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => ( aa_nat_rat(Ga,X3) = aa_nat_rat(Ha,X3) ) )
           => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),S) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8842_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ha,X3) = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_rat(Ga,X3) = aa_Code_integer_rat(Ha,X3) ) )
           => ( groups2555765274223993564er_rat(Ga,S) = groups2555765274223993564er_rat(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8843_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,Ha: fun_o_nat,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T7)
     => ( aa_set_o_o(ord_less_eq_set_o(S),T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T7,S))
             => ( aa_o_nat(Ha,(X3)) = one_one_nat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_nat(Ga,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( groups3504817904513533571_o_nat(Ga,S) = groups3504817904513533571_o_nat(Ha,T7) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8844_prod_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),T7) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8845_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(Ga,T7) = groups3674199335183972705nteger(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8846_prod_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_assn(Ga,X3) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(Ga,T7) = groups6906906614972039071t_assn(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8847_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_assn(Ga,X3) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(Ga,T7) = groups1304777262505850412r_assn(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8848_prod_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ga,X3) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),T7) = aa_set_nat_rat(groups73079841787564623at_rat(Ga),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8849_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ga,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(Ga,T7) = groups2555765274223993564er_rat(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8850_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_nat(Ga,X3) = one_one_nat ) )
         => ( groups3190895334310489300er_nat(Ga,T7) = groups3190895334310489300er_nat(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8851_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_int(Ga,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(Ga,T7) = groups3188404863801439024er_int(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8852_prod_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(Ga,T7) = groups3827104343326376752nteger(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8853_prod_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_assn(Ga,X3) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(Ga,T7) = groups7882442080178216443t_assn(Ga,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8854_prod_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),S) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8855_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(Ga,S) = groups3674199335183972705nteger(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8856_prod_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_assn(Ga,X3) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(Ga,S) = groups6906906614972039071t_assn(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8857_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_assn(Ga,X3) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(Ga,S) = groups1304777262505850412r_assn(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8858_prod_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T7)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
             => ( aa_nat_rat(Ga,X3) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),S) = aa_set_nat_rat(groups73079841787564623at_rat(Ga),T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8859_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_rat(Ga,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(Ga,S) = groups2555765274223993564er_rat(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8860_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_nat(Ga,X3) = one_one_nat ) )
         => ( groups3190895334310489300er_nat(Ga,S) = groups3190895334310489300er_nat(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8861_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T7,S))
             => ( aa_Code_integer_int(Ga,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(Ga,S) = groups3188404863801439024er_int(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8862_prod_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(Ga,S) = groups3827104343326376752nteger(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8863_prod_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T7)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
             => ( aa_int_assn(Ga,X3) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(Ga,S) = groups7882442080178216443t_assn(Ga,T7) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8864_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_Code_integer(Ga,(A3)) = one_one_Code_integer ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_Code_integer(Ha,(B3)) = one_one_Code_integer ) )
             => ( ( groups7694694392188491536nteger(Ga,C3) = groups7694694392188491536nteger(Ha,C3) )
               => ( groups7694694392188491536nteger(Ga,A4) = groups7694694392188491536nteger(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8865_prod_Osame__carrierI,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_Code_integer(Ga,A3) = one_one_Code_integer ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_Code_integer(Ha,B3) = one_one_Code_integer ) )
             => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),C3) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),C3) )
               => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),A4) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8866_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Cod1783974652533396728nteger(Ga,A3) = one_one_Code_integer ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Cod1783974652533396728nteger(Ha,B3) = one_one_Code_integer ) )
             => ( ( groups3674199335183972705nteger(Ga,C3) = groups3674199335183972705nteger(Ha,C3) )
               => ( groups3674199335183972705nteger(Ga,A4) = groups3674199335183972705nteger(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8867_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_assn(Ga,(A3)) = one_one_assn ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_assn(Ha,(B3)) = one_one_assn ) )
             => ( ( groups5301882518646026715o_assn(Ga,C3) = groups5301882518646026715o_assn(Ha,C3) )
               => ( groups5301882518646026715o_assn(Ga,A4) = groups5301882518646026715o_assn(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8868_prod_Osame__carrierI,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_assn,Ha: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_assn(Ga,A3) = one_one_assn ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_assn(Ha,B3) = one_one_assn ) )
             => ( ( groups6906906614972039071t_assn(Ga,C3) = groups6906906614972039071t_assn(Ha,C3) )
               => ( groups6906906614972039071t_assn(Ga,A4) = groups6906906614972039071t_assn(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8869_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_assn(Ga,A3) = one_one_assn ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_assn(Ha,B3) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(Ga,C3) = groups1304777262505850412r_assn(Ha,C3) )
               => ( groups1304777262505850412r_assn(Ga,A4) = groups1304777262505850412r_assn(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8870_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_rat,Ha: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_rat(Ga,(A3)) = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_rat(Ha,(B3)) = one_one_rat ) )
             => ( ( groups2869687844427037835_o_rat(Ga,C3) = groups2869687844427037835_o_rat(Ha,C3) )
               => ( groups2869687844427037835_o_rat(Ga,A4) = groups2869687844427037835_o_rat(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8871_prod_Osame__carrierI,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_rat(Ga,A3) = one_one_rat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_rat(Ha,B3) = one_one_rat ) )
             => ( ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),C3) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),C3) )
               => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8872_prod_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_rat(Ga,A3) = one_one_rat ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_rat(Ha,B3) = one_one_rat ) )
             => ( ( groups2555765274223993564er_rat(Ga,C3) = groups2555765274223993564er_rat(Ha,C3) )
               => ( groups2555765274223993564er_rat(Ga,A4) = groups2555765274223993564er_rat(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8873_prod_Osame__carrierI,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_nat(Ga,(A3)) = one_one_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_nat(Ha,(B3)) = one_one_nat ) )
             => ( ( groups3504817904513533571_o_nat(Ga,C3) = groups3504817904513533571_o_nat(Ha,C3) )
               => ( groups3504817904513533571_o_nat(Ga,A4) = groups3504817904513533571_o_nat(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8874_prod_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_Code_integer,Ha: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_Code_integer(Ga,(A3)) = one_one_Code_integer ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_Code_integer(Ha,(B3)) = one_one_Code_integer ) )
             => ( ( groups7694694392188491536nteger(Ga,A4) = groups7694694392188491536nteger(Ha,B4) )
              <=> ( groups7694694392188491536nteger(Ga,C3) = groups7694694392188491536nteger(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8875_prod_Osame__carrier,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_Code_integer,Ha: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_Code_integer(Ga,A3) = one_one_Code_integer ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_Code_integer(Ha,B3) = one_one_Code_integer ) )
             => ( ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),A4) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),B4) )
              <=> ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),C3) = aa_set871585252387937313nteger(groups3455450783089532116nteger(Ha),C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8876_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co8499238673658658103nteger,Ha: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Cod1783974652533396728nteger(Ga,A3) = one_one_Code_integer ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Cod1783974652533396728nteger(Ha,B3) = one_one_Code_integer ) )
             => ( ( groups3674199335183972705nteger(Ga,A4) = groups3674199335183972705nteger(Ha,B4) )
              <=> ( groups3674199335183972705nteger(Ga,C3) = groups3674199335183972705nteger(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8877_prod_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_assn(Ga,(A3)) = one_one_assn ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_assn(Ha,(B3)) = one_one_assn ) )
             => ( ( groups5301882518646026715o_assn(Ga,A4) = groups5301882518646026715o_assn(Ha,B4) )
              <=> ( groups5301882518646026715o_assn(Ga,C3) = groups5301882518646026715o_assn(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8878_prod_Osame__carrier,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_assn,Ha: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_assn(Ga,A3) = one_one_assn ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_assn(Ha,B3) = one_one_assn ) )
             => ( ( groups6906906614972039071t_assn(Ga,A4) = groups6906906614972039071t_assn(Ha,B4) )
              <=> ( groups6906906614972039071t_assn(Ga,C3) = groups6906906614972039071t_assn(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8879_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_assn(Ga,A3) = one_one_assn ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_assn(Ha,B3) = one_one_assn ) )
             => ( ( groups1304777262505850412r_assn(Ga,A4) = groups1304777262505850412r_assn(Ha,B4) )
              <=> ( groups1304777262505850412r_assn(Ga,C3) = groups1304777262505850412r_assn(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8880_prod_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_rat,Ha: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_rat(Ga,(A3)) = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_rat(Ha,(B3)) = one_one_rat ) )
             => ( ( groups2869687844427037835_o_rat(Ga,A4) = groups2869687844427037835_o_rat(Ha,B4) )
              <=> ( groups2869687844427037835_o_rat(Ga,C3) = groups2869687844427037835_o_rat(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8881_prod_Osame__carrier,axiom,
    ! [C3: set_nat,A4: set_nat,B4: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),A4))
               => ( aa_nat_rat(Ga,A3) = one_one_rat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C3),B4))
                 => ( aa_nat_rat(Ha,B3) = one_one_rat ) )
             => ( ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),B4) )
              <=> ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),C3) = aa_set_nat_rat(groups73079841787564623at_rat(Ha),C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8882_prod_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_rat(Ga,A3) = one_one_rat ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_rat(Ha,B3) = one_one_rat ) )
             => ( ( groups2555765274223993564er_rat(Ga,A4) = groups2555765274223993564er_rat(Ha,B4) )
              <=> ( groups2555765274223993564er_rat(Ga,C3) = groups2555765274223993564er_rat(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8883_prod_Osame__carrier,axiom,
    ! [C3: set_o,A4: set_o,B4: set_o,Ga: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C3)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),C3)
       => ( aa_set_o_o(ord_less_eq_set_o(B4),C3)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C3,A4))
               => ( aa_o_nat(Ga,(A3)) = one_one_nat ) )
           => ( ! [B3: $o] :
                  ( aa_set_o_o(member_o((B3)),minus_minus_set_o(C3,B4))
                 => ( aa_o_nat(Ha,(B3)) = one_one_nat ) )
             => ( ( groups3504817904513533571_o_nat(Ga,A4) = groups3504817904513533571_o_nat(Ha,B4) )
              <=> ( groups3504817904513533571_o_nat(Ga,C3) = groups3504817904513533571_o_nat(Ha,C3) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8884_sum_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,A4)),groups3906332499630173760nt_rat(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8885_sum_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,sup_su848401254843788991nteger(A4,B4))),groups6602215022474089585er_rat(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,A4)),groups6602215022474089585er_rat(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8886_sum_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,A4)),groups4541462559716669496nt_nat(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8887_sum_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,sup_su848401254843788991nteger(A4,B4))),groups7237345082560585321er_nat(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,A4)),groups7237345082560585321er_nat(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8888_sum_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,sup_su848401254843788991nteger(A4,B4))),groups7234854612051535045er_int(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,A4)),groups7234854612051535045er_int(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8889_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),B4)) ) ) ) ).

% sum.union_inter
tff(fact_8890_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)),aa_set_nat_int(groups3539618377306564664at_int(Ga),B4)) ) ) ) ).

% sum.union_inter
tff(fact_8891_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),B4)) ) ) ) ).

% sum.union_inter
tff(fact_8892_sum_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ga,A4)),groups4538972089207619220nt_int(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8893_sum_Ounion__inter,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4))),groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Ga,A4)),groups342789780944988191at_rat(Ga,B4)) ) ) ) ).

% sum.union_inter
tff(fact_8894_sum_OInt__Diff,axiom,
    ! [A4: set_int,Ga: fun_int_rat,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8895_sum_OInt__Diff,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,inf_in1364745209274528805nteger(A4,B4))),groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8896_sum_OInt__Diff,axiom,
    ! [A4: set_int,Ga: fun_int_nat,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8897_sum_OInt__Diff,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_nat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,inf_in1364745209274528805nteger(A4,B4))),groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8898_sum_OInt__Diff,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_int,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,inf_in1364745209274528805nteger(A4,B4))),groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(A4,B4))) ) ) ).

% sum.Int_Diff
tff(fact_8899_sum_OInt__Diff,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8900_sum_OInt__Diff,axiom,
    ! [A4: set_nat,Ga: fun_nat_int,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8901_sum_OInt__Diff,axiom,
    ! [A4: set_nat,Ga: fun_nat_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8902_sum_OInt__Diff,axiom,
    ! [A4: set_int,Ga: fun_int_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4538972089207619220nt_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))),groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8903_sum_OInt__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( groups342789780944988191at_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))),groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))) ) ) ).

% sum.Int_Diff
tff(fact_8904_prod_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ga,A4)),groups7882442080178216443t_assn(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8905_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ga,sup_su848401254843788991nteger(A4,B4))),groups1304777262505850412r_assn(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ga,A4)),groups1304777262505850412r_assn(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8906_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,A4)),groups6906906614972039071t_assn(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8907_prod_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ga,A4)),groups1072433553688619179nt_rat(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8908_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ga,sup_su848401254843788991nteger(A4,B4))),groups2555765274223993564er_rat(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ga,A4)),groups2555765274223993564er_rat(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8909_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),B4)) ) ) ) ).

% prod.union_inter
tff(fact_8910_prod_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ga,A4)),groups1707563613775114915nt_nat(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8911_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(Ga,sup_su848401254843788991nteger(A4,B4))),groups3190895334310489300er_nat(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(Ga,A4)),groups3190895334310489300er_nat(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8912_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(Ga,sup_su848401254843788991nteger(A4,B4))),groups3188404863801439024er_int(Ga,inf_in1364745209274528805nteger(A4,B4))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(Ga,A4)),groups3188404863801439024er_int(Ga,B4)) ) ) ) ).

% prod.union_inter
tff(fact_8913_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),A4)),aa_set_nat_nat(groups708209901874060359at_nat(Ga),B4)) ) ) ) ).

% prod.union_inter
tff(fact_8914_sum__diff__nat,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups7237345082560585321er_nat(Fa,minus_2355218937544613996nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups7237345082560585321er_nat(Fa,A4)),groups7237345082560585321er_nat(Fa,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8915_sum__diff__nat,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Fa: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B4),A4)
       => ( groups977919841031483927at_nat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(Fa,A4)),groups977919841031483927at_nat(Fa,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8916_sum__diff__nat,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A4)
       => ( groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(Fa,A4)),groups4541462559716669496nt_nat(Fa,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8917_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_8918_push__bit__mask__eq,axiom,
    ! [M: nat,N2: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),aa_nat_int(bit_se2000444600071755411sk_int,N2)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,M))) ).

% push_bit_mask_eq
tff(fact_8919_sum_OIf__cases,axiom,
    ! [A4: set_int,Pa: fun_int_o,Ha: fun_int_rat,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_agz(Pa),Ha),Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8920_sum_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_aha(Pa),Ha),Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups6602215022474089585er_rat(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8921_sum_OIf__cases,axiom,
    ! [A4: set_int,Pa: fun_int_o,Ha: fun_int_nat,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ahb(Pa),Ha),Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8922_sum_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_ahc(Pa),Ha),Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups7237345082560585321er_nat(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8923_sum_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_int,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_ahd(Pa),Ha),Ga),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups7234854612051535045er_int(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8924_sum_OIf__cases,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Ha: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ahe(Pa),Ha),Ga)),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8925_sum_OIf__cases,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Ha: fun_nat_int,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ahf(Pa),Ha),Ga)),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8926_sum_OIf__cases,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Ha: fun_nat_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ahg(Pa),Ha),Ga)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8927_sum_OIf__cases,axiom,
    ! [A4: set_int,Pa: fun_int_o,Ha: fun_int_int,Ga: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_ahh(Pa),Ha),Ga),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))),groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8928_sum_OIf__cases,axiom,
    ! [A4: set_list_nat,Pa: fun_list_nat_o,Ha: fun_list_nat_rat,Ga: fun_list_nat_rat] :
      ( finite8100373058378681591st_nat(A4)
     => ( groups3760926236672600436at_rat(aa_fun1630595088783574696at_rat(aa_fun6629217458047550682at_rat(aTP_Lamm_ahi(Pa),Ha),Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3760926236672600436at_rat(Ha,inf_inf_set_list_nat(A4,collect_list_nat(Pa)))),groups3760926236672600436at_rat(Ga,inf_inf_set_list_nat(A4,uminus3195874150345416415st_nat(collect_list_nat(Pa))))) ) ) ).

% sum.If_cases
tff(fact_8929_prod_OIf__cases,axiom,
    ! [A4: set_int,Pa: fun_int_o,Ha: fun_int_assn,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ahj(Pa),Ha),Ga),A4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))),groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8930_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Co4490034186337852034r_assn,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_ahk(Pa),Ha),Ga),A4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups1304777262505850412r_assn(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8931_prod_OIf__cases,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Ha: fun_nat_assn,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_ahl(Pa),Ha),Ga),A4) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))),groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8932_prod_OIf__cases,axiom,
    ! [A4: set_int,Pa: fun_int_o,Ha: fun_int_rat,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_agz(Pa),Ha),Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))),groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8933_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups2555765274223993564er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_aha(Pa),Ha),Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups2555765274223993564er_rat(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8934_prod_OIf__cases,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Ha: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ahe(Pa),Ha),Ga)),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8935_prod_OIf__cases,axiom,
    ! [A4: set_int,Pa: fun_int_o,Ha: fun_int_nat,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups1707563613775114915nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ahb(Pa),Ha),Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ha,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))),groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),uminus1532241313380277803et_int(collect_int(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8936_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups3190895334310489300er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_ahc(Pa),Ha),Ga),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups3190895334310489300er_nat(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8937_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o,Ha: fun_Code_integer_int,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups3188404863801439024er_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_ahd(Pa),Ha),Ga),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(Ha,inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))),groups3188404863801439024er_int(Ga,inf_in1364745209274528805nteger(A4,uminus804700908173204444nteger(collect_Code_integer(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8938_prod_OIf__cases,axiom,
    ! [A4: set_nat,Pa: fun_nat_o,Ha: fun_nat_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ahg(Pa),Ha),Ga)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),uminus5710092332889474511et_nat(collect_nat(Pa))))) ) ) ).

% prod.If_cases
tff(fact_8939_prod__mono__strict,axiom,
    ! [A4: set_Code_integer,Fa: fun_Co8499238673658658103nteger,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A4)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(Fa,I3)),aa_Cod1783974652533396728nteger(Ga,I3)) ) )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3674199335183972705nteger(Fa,A4)),groups3674199335183972705nteger(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8940_prod__mono__strict,axiom,
    ! [A4: set_o,Fa: fun_o_Code_integer,Ga: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A4)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(I3)))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(Fa,(I3))),aa_o_Code_integer(Ga,(I3))) ) )
       => ( ( A4 != bot_bot_set_o )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7694694392188491536nteger(Fa,A4)),groups7694694392188491536nteger(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8941_prod__mono__strict,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_integer,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A4)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(Fa,I3)),aa_nat_Code_integer(Ga,I3)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),A4)),aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8942_prod__mono__strict,axiom,
    ! [A4: set_int,Fa: fun_int_Code_integer,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A4)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(Fa,I3)),aa_int_Code_integer(Ga,I3)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3827104343326376752nteger(Fa,A4)),groups3827104343326376752nteger(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8943_prod__mono__strict,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_rat,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(Fa,I3)),aa_Code_integer_rat(Ga,I3)) ) )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2555765274223993564er_rat(Fa,A4)),groups2555765274223993564er_rat(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8944_prod__mono__strict,axiom,
    ! [A4: set_o,Fa: fun_o_rat,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(I3)))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(Fa,(I3))),aa_o_rat(Ga,(I3))) ) )
       => ( ( A4 != bot_bot_set_o )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2869687844427037835_o_rat(Fa,A4)),groups2869687844427037835_o_rat(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8945_prod__mono__strict,axiom,
    ! [A4: set_nat,Fa: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,I3)),aa_nat_rat(Ga,I3)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups73079841787564623at_rat(Fa),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8946_prod__mono__strict,axiom,
    ! [A4: set_int,Fa: fun_int_rat,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A4)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,I3)),aa_int_rat(Ga,I3)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(Fa,A4)),groups1072433553688619179nt_rat(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8947_prod__mono__strict,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A4)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(Fa,I3))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(Fa,I3)),aa_Code_integer_nat(Ga,I3)) ) )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups3190895334310489300er_nat(Fa,A4)),groups3190895334310489300er_nat(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8948_prod__mono__strict,axiom,
    ! [A4: set_o,Fa: fun_o_nat,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A4)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(I3)))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(Fa,(I3))),aa_o_nat(Ga,(I3))) ) )
       => ( ( A4 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups3504817904513533571_o_nat(Fa,A4)),groups3504817904513533571_o_nat(Ga,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_8949_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(B3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups4406642042086082107nteger(Fa,A4)),groups4406642042086082107nteger(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8950_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,B3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),A4)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),B4)) ) ) ) ).

% sum_mono2
tff(fact_8951_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,B3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups879477027807139574nteger(Fa,A4)),groups879477027807139574nteger(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8952_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(B3))) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(Fa,A4)),groups7872700643590313910_o_rat(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8953_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),B4)) ) ) ) ).

% sum_mono2
tff(fact_8954_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups6602215022474089585er_rat(Fa,A4)),groups6602215022474089585er_rat(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8955_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(B3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(Fa,A4)),groups8507830703676809646_o_nat(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8956_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(Fa,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups7237345082560585321er_nat(Fa,A4)),groups7237345082560585321er_nat(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8957_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Fa,(B3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups8505340233167759370_o_int(Fa,A4)),groups8505340233167759370_o_int(Fa,B4)) ) ) ) ).

% sum_mono2
tff(fact_8958_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),aa_set_nat_int(groups3539618377306564664at_int(Fa),B4)) ) ) ) ).

% sum_mono2
tff(fact_8959_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( groups7873554091576472773nteger(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups7873554091576472773nteger(Ga,A4)),groups7873554091576472773nteger(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8960_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( groups879477027807139574nteger(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups879477027807139574nteger(Ga,A4)),groups879477027807139574nteger(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8961_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_rat(Ga,X3) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,A4)),groups3906332499630173760nt_rat(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8962_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Code_integer_rat(Ga,X3) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,A4)),groups6602215022474089585er_rat(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8963_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_nat(Ga,X3) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,A4)),groups4541462559716669496nt_nat(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8964_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Code_integer_nat(Ga,X3) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,A4)),groups7237345082560585321er_nat(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8965_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Code_integer_int(Ga,X3) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,A4)),groups7234854612051535045er_int(Ga,B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8966_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
         => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),A4)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Ga),B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8967_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_rat(Ga,X3) = zero_zero_rat ) )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8968_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_int(Ga,X3) = zero_zero_int ) )
         => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)),aa_set_nat_int(groups3539618377306564664at_int(Ga),B4)) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_8969_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_rat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(Ga,insert_Code_integer(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Ga,X)),groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(A4,insert_Code_integer(X,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_8970_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_nat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(Ga,insert_Code_integer(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Ga,X)),groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(A4,insert_Code_integer(X,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_8971_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,Ga: fun_Code_integer_int,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7234854612051535045er_int(Ga,insert_Code_integer(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(Ga,X)),groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(A4,insert_Code_integer(X,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_8972_sum_Oinsert__remove,axiom,
    ! [A4: set_o,Ga: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(Ga,insert_o((X),A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Ga,(X))),groups7872700643590313910_o_rat(Ga,minus_minus_set_o(A4,insert_o((X),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8973_sum_Oinsert__remove,axiom,
    ! [A4: set_o,Ga: fun_o_nat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8507830703676809646_o_nat(Ga,insert_o((X),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Ga,(X))),groups8507830703676809646_o_nat(Ga,minus_minus_set_o(A4,insert_o((X),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8974_sum_Oinsert__remove,axiom,
    ! [A4: set_o,Ga: fun_o_int,X: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8505340233167759370_o_int(Ga,insert_o((X),A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(Ga,(X))),groups8505340233167759370_o_int(Ga,minus_minus_set_o(A4,insert_o((X),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_8975_sum_Oinsert__remove,axiom,
    ! [A4: set_nat,Ga: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),insert_nat(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,X)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ).

% sum.insert_remove
tff(fact_8976_sum_Oinsert__remove,axiom,
    ! [A4: set_nat,Ga: fun_nat_int,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),insert_nat(X,A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,X)),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ).

% sum.insert_remove
tff(fact_8977_sum_Oinsert__remove,axiom,
    ! [A4: set_int,Ga: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(Ga,insert_int(X,A4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Ga,X)),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8978_sum_Oinsert__remove,axiom,
    ! [A4: set_int,Ga: fun_int_nat,X: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(Ga,insert_int(X,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ga,X)),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int)))) ) ) ).

% sum.insert_remove
tff(fact_8979_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
       => ( groups6602215022474089585er_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Ga,X)),groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(A4,insert_Code_integer(X,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_8980_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
       => ( groups7237345082560585321er_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Ga,X)),groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(A4,insert_Code_integer(X,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_8981_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
       => ( groups7234854612051535045er_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(Ga,X)),groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(A4,insert_Code_integer(X,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_8982_sum_Oremove,axiom,
    ! [A4: set_o,X: $o,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X)),A4)
       => ( groups7872700643590313910_o_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Ga,(X))),groups7872700643590313910_o_rat(Ga,minus_minus_set_o(A4,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8983_sum_Oremove,axiom,
    ! [A4: set_o,X: $o,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X)),A4)
       => ( groups8507830703676809646_o_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Ga,(X))),groups8507830703676809646_o_nat(Ga,minus_minus_set_o(A4,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8984_sum_Oremove,axiom,
    ! [A4: set_o,X: $o,Ga: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((X)),A4)
       => ( groups8505340233167759370_o_int(Ga,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(Ga,(X))),groups8505340233167759370_o_int(Ga,minus_minus_set_o(A4,insert_o((X),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_8985_sum_Oremove,axiom,
    ! [A4: set_nat,X: nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(X),A4)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Ga,X)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% sum.remove
tff(fact_8986_sum_Oremove,axiom,
    ! [A4: set_nat,X: nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(X),A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Ga,X)),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(X,bot_bot_set_nat)))) ) ) ) ).

% sum.remove
tff(fact_8987_sum_Oremove,axiom,
    ! [A4: set_int,X: int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(X),A4)
       => ( groups3906332499630173760nt_rat(Ga,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Ga,X)),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8988_sum_Oremove,axiom,
    ! [A4: set_int,X: int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(X),A4)
       => ( groups4541462559716669496nt_nat(Ga,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Ga,X)),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(X,bot_bot_set_int)))) ) ) ) ).

% sum.remove
tff(fact_8989_sum__Un,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Fa,A4)),groups3906332499630173760nt_rat(Fa,B4))),groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8990_sum__Un,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups6602215022474089585er_rat(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Fa,A4)),groups6602215022474089585er_rat(Fa,B4))),groups6602215022474089585er_rat(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8991_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),B4))),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8992_sum__Un,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups7234854612051535045er_int(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Fa,A4)),groups7234854612051535045er_int(Fa,B4))),groups7234854612051535045er_int(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% sum_Un
tff(fact_8993_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),aa_set_nat_int(groups3539618377306564664at_int(Fa),B4))),aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8994_sum__Un,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Fa,A4)),groups4538972089207619220nt_int(Fa,B4))),groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8995_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Fa,A4)),groups342789780944988191at_rat(Fa,B4))),groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8996_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups975429370522433651at_int(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups975429370522433651at_int(Fa,A4)),groups975429370522433651at_int(Fa,B4))),groups975429370522433651at_int(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8997_sum__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,Fa: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A4)
     => ( finite1918287321285529104at_nat(B4)
       => ( groups6424767709179651461at_rat(Fa,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6424767709179651461at_rat(Fa,A4)),groups6424767709179651461at_rat(Fa,B4))),groups6424767709179651461at_rat(Fa,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8998_sum__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,Fa: fun_Pr6968369006614801746at_rat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
       => ( groups3225780264831618053at_rat(Fa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3225780264831618053at_rat(Fa,A4)),groups3225780264831618053at_rat(Fa,B4))),groups3225780264831618053at_rat(Fa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4))) ) ) ) ).

% sum_Un
tff(fact_8999_sum_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A4,B4) = bot_bo3990330152332043303nteger )
         => ( groups6602215022474089585er_rat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,A4)),groups6602215022474089585er_rat(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9000_sum_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A4,B4) = bot_bo3990330152332043303nteger )
         => ( groups7237345082560585321er_nat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,A4)),groups7237345082560585321er_nat(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9001_sum_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A4,B4) = bot_bo3990330152332043303nteger )
         => ( groups7234854612051535045er_int(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,A4)),groups7234854612051535045er_int(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9002_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups7872700643590313910_o_rat(Ga,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups7872700643590313910_o_rat(Ga,A4)),groups7872700643590313910_o_rat(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9003_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups8507830703676809646_o_nat(Ga,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8507830703676809646_o_nat(Ga,A4)),groups8507830703676809646_o_nat(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9004_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,Ga: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups8505340233167759370_o_int(Ga,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups8505340233167759370_o_int(Ga,A4)),groups8505340233167759370_o_int(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9005_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9006_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),A4)),aa_set_nat_int(groups3539618377306564664at_int(Ga),B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9007_sum_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
         => ( groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,A4)),groups3906332499630173760nt_rat(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9008_sum_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
         => ( groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,A4)),groups4541462559716669496nt_nat(Ga,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_9009_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( groups3827104343326376752nteger(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3827104343326376752nteger(Ga,A4)),groups3827104343326376752nteger(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9010_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Cod1783974652533396728nteger(Ga,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(Ga,A4)),groups3674199335183972705nteger(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9011_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_Code_integer(Ga,X3) = one_one_Code_integer ) )
         => ( aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),A4)),aa_set871585252387937313nteger(groups3455450783089532116nteger(Ga),B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9012_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_assn(Ga,X3) = one_one_assn ) )
         => ( groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ga,A4)),groups7882442080178216443t_assn(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9013_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Code_integer_assn(Ga,X3) = one_one_assn ) )
         => ( groups1304777262505850412r_assn(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ga,A4)),groups1304777262505850412r_assn(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9014_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_assn(Ga,X3) = one_one_assn ) )
         => ( groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,A4)),groups6906906614972039071t_assn(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9015_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_rat(Ga,X3) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ga,A4)),groups1072433553688619179nt_rat(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9016_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Code_integer_rat(Ga,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ga,A4)),groups2555765274223993564er_rat(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9017_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_rat(Ga,X3) = one_one_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9018_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_nat(Ga,X3) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ga,A4)),groups1707563613775114915nt_nat(Ga,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_9019_prod_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A4,B4) = bot_bo3990330152332043303nteger )
         => ( groups1304777262505850412r_assn(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ga,A4)),groups1304777262505850412r_assn(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9020_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,Ga: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups5301882518646026715o_assn(Ga,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups5301882518646026715o_assn(Ga,A4)),groups5301882518646026715o_assn(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9021_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,A4)),groups6906906614972039071t_assn(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9022_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
         => ( groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ga,A4)),groups7882442080178216443t_assn(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9023_prod_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A4,B4) = bot_bo3990330152332043303nteger )
         => ( groups2555765274223993564er_rat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ga,A4)),groups2555765274223993564er_rat(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9024_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,Ga: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups2869687844427037835_o_rat(Ga,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2869687844427037835_o_rat(Ga,A4)),groups2869687844427037835_o_rat(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9025_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4) = bot_bot_set_nat )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Ga),B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9026_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4) = bot_bot_set_int )
         => ( groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ga,A4)),groups1072433553688619179nt_rat(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9027_prod_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A4,B4) = bot_bo3990330152332043303nteger )
         => ( groups3190895334310489300er_nat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(Ga,A4)),groups3190895334310489300er_nat(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9028_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,Ga: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A4,B4) = bot_bot_set_o )
         => ( groups3504817904513533571_o_nat(Ga,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3504817904513533571_o_nat(Ga,A4)),groups3504817904513533571_o_nat(Ga,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_9029_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups3906332499630173760nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9030_sum_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups6602215022474089585er_rat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(A4,B4))),groups6602215022474089585er_rat(Ga,minus_2355218937544613996nteger(B4,A4)))),groups6602215022474089585er_rat(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9031_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups4541462559716669496nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9032_sum_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups7237345082560585321er_nat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(A4,B4))),groups7237345082560585321er_nat(Ga,minus_2355218937544613996nteger(B4,A4)))),groups7237345082560585321er_nat(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9033_sum_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups7234854612051535045er_int(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(A4,B4))),groups7234854612051535045er_int(Ga,minus_2355218937544613996nteger(B4,A4)))),groups7234854612051535045er_int(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9034_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_rat(groups2906978787729119204at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9035_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_int(groups3539618377306564664at_int(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9036_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_nat(groups3542108847815614940at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9037_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups4538972089207619220nt_int(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9038_sum_Ounion__diff2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))),groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)))),groups342789780944988191at_rat(Ga,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum.union_diff2
tff(fact_9039_sum__Un2,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))
     => ( groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups3906332499630173760nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9040_sum__Un2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(A4,B4))
     => ( groups6602215022474089585er_rat(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(Fa,minus_2355218937544613996nteger(A4,B4))),groups6602215022474089585er_rat(Fa,minus_2355218937544613996nteger(B4,A4)))),groups6602215022474089585er_rat(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ).

% sum_Un2
tff(fact_9041_sum__Un2,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))
     => ( groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9042_sum__Un2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(A4,B4))
     => ( groups7237345082560585321er_nat(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Fa,minus_2355218937544613996nteger(A4,B4))),groups7237345082560585321er_nat(Fa,minus_2355218937544613996nteger(B4,A4)))),groups7237345082560585321er_nat(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ).

% sum_Un2
tff(fact_9043_sum__Un2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(A4,B4))
     => ( groups7234854612051535045er_int(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(Fa,minus_2355218937544613996nteger(A4,B4))),groups7234854612051535045er_int(Fa,minus_2355218937544613996nteger(B4,A4)))),groups7234854612051535045er_int(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ).

% sum_Un2
tff(fact_9044_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9045_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9046_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9047_sum__Un2,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))
     => ( groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9048_sum__Un2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4))
     => ( groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))),groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4)))),groups342789780944988191at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ).

% sum_Un2
tff(fact_9049_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups7882442080178216443t_assn(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9050_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups1304777262505850412r_assn(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups1304777262505850412r_assn(Ga,minus_2355218937544613996nteger(A4,B4))),groups1304777262505850412r_assn(Ga,minus_2355218937544613996nteger(B4,A4)))),groups1304777262505850412r_assn(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9051_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),groups6906906614972039071t_assn(Ga,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9052_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups1072433553688619179nt_rat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9053_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups2555765274223993564er_rat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Ga,minus_2355218937544613996nteger(A4,B4))),groups2555765274223993564er_rat(Ga,minus_2355218937544613996nteger(B4,A4)))),groups2555765274223993564er_rat(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9054_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_rat(groups73079841787564623at_rat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9055_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4)))),groups1707563613775114915nt_nat(Ga,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9056_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups3190895334310489300er_nat(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups3190895334310489300er_nat(Ga,minus_2355218937544613996nteger(A4,B4))),groups3190895334310489300er_nat(Ga,minus_2355218937544613996nteger(B4,A4)))),groups3190895334310489300er_nat(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9057_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Ga: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups3188404863801439024er_int(Ga,sup_su848401254843788991nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(Ga,minus_2355218937544613996nteger(A4,B4))),groups3188404863801439024er_int(Ga,minus_2355218937544613996nteger(B4,A4)))),groups3188404863801439024er_int(Ga,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9058_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4)))),aa_set_nat_nat(groups708209901874060359at_nat(Ga),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_9059_sum__Un__nat,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Fa,A4)),groups4541462559716669496nt_nat(Fa,B4))),groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_9060_sum__Un__nat,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups7237345082560585321er_nat(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(Fa,A4)),groups7237345082560585321er_nat(Fa,B4))),groups7237345082560585321er_nat(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ).

% sum_Un_nat
tff(fact_9061_sum__Un__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( groups977919841031483927at_nat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups977919841031483927at_nat(Fa,A4)),groups977919841031483927at_nat(Fa,B4))),groups977919841031483927at_nat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_9062_sum__Un__nat,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,Fa: fun_Pr7574179848310716106at_nat] :
      ( finite1918287321285529104at_nat(A4)
     => ( finite1918287321285529104at_nat(B4)
       => ( groups7059897769266147197at_nat(Fa,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7059897769266147197at_nat(Fa,A4)),groups7059897769266147197at_nat(Fa,B4))),groups7059897769266147197at_nat(Fa,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_9063_sum__Un__nat,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,Fa: fun_Pr1253225528153422666at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
       => ( groups3860910324918113789at_nat(Fa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups3860910324918113789at_nat(Fa,A4)),groups3860910324918113789at_nat(Fa,B4))),groups3860910324918113789at_nat(Fa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_9064_sum__Un__nat,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),B4))),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_9065_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_rat,C2: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ahm(A),B),C2),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(B,A)),groups6602215022474089585er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups6602215022474089585er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_9066_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_nat,C2: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_ahn(A),B),C2),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(B,A)),groups7237345082560585321er_nat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups7237345082560585321er_nat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_9067_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_int,C2: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups7234854612051535045er_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_aho(A),B),C2),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(B,A)),groups7234854612051535045er_int(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups7234854612051535045er_int(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_9068_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat,C2: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups7872700643590313910_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_ahp((A)),B),C2),S) = $ite(aa_set_o_o(member_o((A)),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(B,(A))),groups7872700643590313910_o_rat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups7872700643590313910_o_rat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_9069_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat,C2: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_ahq((A)),B),C2),S) = $ite(aa_set_o_o(member_o((A)),S),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(B,(A))),groups8507830703676809646_o_nat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups8507830703676809646_o_nat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_9070_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_int,C2: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups8505340233167759370_o_int(aa_fun4727766026823762812_o_int(aa_fun5648456818326528798_o_int(aTP_Lamm_ahr((A)),B),C2),S) = $ite(aa_set_o_o(member_o((A)),S),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(B,(A))),groups8505340233167759370_o_int(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups8505340233167759370_o_int(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% sum.delta_remove
tff(fact_9071_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat,C2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ahs(A),B),C2)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(B,A)),aa_set_nat_rat(groups2906978787729119204at_rat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))),aa_set_nat_rat(groups2906978787729119204at_rat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))) ) ) ).

% sum.delta_remove
tff(fact_9072_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_int,C2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aht(A),B),C2)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(B,A)),aa_set_nat_int(groups3539618377306564664at_int(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))),aa_set_nat_int(groups3539618377306564664at_int(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))) ) ) ).

% sum.delta_remove
tff(fact_9073_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_rat,C2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ahu(A),B),C2),S) = $ite(aa_set_int_o(member_int(A),S),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(B,A)),groups3906332499630173760nt_rat(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))),groups3906332499630173760nt_rat(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_9074_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_nat,C2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ahv(A),B),C2),S) = $ite(aa_set_int_o(member_int(A),S),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(B,A)),groups4541462559716669496nt_nat(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))),groups4541462559716669496nt_nat(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))) ) ) ).

% sum.delta_remove
tff(fact_9075_sum__div__partition,axiom,
    ! [A4: set_int,Fa: fun_int_nat,B: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(Fa,A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ahw(Fa),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(aa_nat_fun_int_o(aTP_Lamm_ahx(Fa),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4541462559716669496nt_nat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(aa_nat_fun_int_o(aTP_Lamm_ahy(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9076_sum__div__partition,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat,B: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups7237345082560585321er_nat(Fa,A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_ahz(Fa),B),inf_in1364745209274528805nteger(A4,collect_Code_integer(aa_nat7420681472272698950eger_o(aTP_Lamm_aia(Fa),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups7237345082560585321er_nat(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(aa_nat7420681472272698950eger_o(aTP_Lamm_aib(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9077_sum__div__partition,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_int,B: int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups7234854612051535045er_int(Fa,A4)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(aa_int1390634949194299888er_int(aTP_Lamm_aic(Fa),B),inf_in1364745209274528805nteger(A4,collect_Code_integer(aa_int5993532643906289898eger_o(aTP_Lamm_aid(Fa),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups7234854612051535045er_int(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(aa_int5993532643906289898eger_o(aTP_Lamm_aie(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9078_sum__div__partition,axiom,
    ! [A4: set_nat,Fa: fun_nat_int,B: int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_aif(Fa),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_int_fun_nat_o(aTP_Lamm_aig(Fa),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_int_fun_nat_o(aTP_Lamm_aih(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9079_sum__div__partition,axiom,
    ! [A4: set_int,Fa: fun_int_Code_natural,B: code_natural] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( divide5121882707175180666atural(groups6697149243333190288atural(Fa,A4),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,groups6697149243333190288atural(aa_Cod5692187177200718263atural(aTP_Lamm_aii(Fa),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(aa_Cod7299285730463426143_int_o(aTP_Lamm_aij(Fa),B))))),divide5121882707175180666atural(groups6697149243333190288atural(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(aa_Cod7299285730463426143_int_o(aTP_Lamm_aik(Fa),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_9080_sum__div__partition,axiom,
    ! [A4: set_Code_integer,Fa: fun_Co3734316457315071362atural,B: code_natural] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( divide5121882707175180666atural(groups8926444216418632897atural(Fa,A4),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,groups8926444216418632897atural(aa_Cod1476545362447692776atural(aTP_Lamm_ail(Fa),B),inf_in1364745209274528805nteger(A4,collect_Code_integer(aa_Cod403035634466734574eger_o(aTP_Lamm_aim(Fa),B))))),divide5121882707175180666atural(groups8926444216418632897atural(Fa,inf_in1364745209274528805nteger(A4,collect_Code_integer(aa_Cod403035634466734574eger_o(aTP_Lamm_ain(Fa),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_9081_sum__div__partition,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_natural,B: code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( divide5121882707175180666atural(groups6325495683096345652atural(Fa,A4),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,groups6325495683096345652atural(aa_Cod5320533616963873627atural(aTP_Lamm_aio(Fa),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_Cod9120328075896114107_nat_o(aTP_Lamm_aip(Fa),B))))),divide5121882707175180666atural(groups6325495683096345652atural(Fa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_Cod9120328075896114107_nat_o(aTP_Lamm_aiq(Fa),B)))),B)) ) ) ).

% sum_div_partition
tff(fact_9082_sum__div__partition,axiom,
    ! [A4: set_nat,Fa: fun_nat_nat,B: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_air(Fa),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ais(Fa),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ait(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9083_sum__div__partition,axiom,
    ! [A4: set_int,Fa: fun_int_int,B: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,groups4538972089207619220nt_int(Fa,A4)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_aiu(Fa),B),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(aa_int_fun_int_o(aTP_Lamm_aiv(Fa),B))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,groups4538972089207619220nt_int(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(aa_int_fun_int_o(aTP_Lamm_aiw(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9084_sum__div__partition,axiom,
    ! [A4: set_list_nat,Fa: fun_list_nat_nat,B: nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4396056296759096172at_nat(Fa,A4)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(aa_nat3122681260899759163at_nat(aTP_Lamm_aix(Fa),B),inf_inf_set_list_nat(A4,collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aiy(Fa),B))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups4396056296759096172at_nat(Fa,inf_inf_set_list_nat(A4,collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aiz(Fa),B))))),B)) ) ) ).

% sum_div_partition
tff(fact_9085_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Co4490034186337852034r_assn,C2: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups1304777262505850412r_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_aja(A),B),C2),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Code_integer_assn(B,A)),groups1304777262505850412r_assn(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups1304777262505850412r_assn(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.delta_remove
tff(fact_9086_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_assn,C2: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups5301882518646026715o_assn(aa_fun4536517337817438332o_assn(aa_fun3110273112593859354o_assn(aTP_Lamm_ajb((A)),B),C2),S) = $ite(aa_set_o_o(member_o((A)),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(B,(A))),groups5301882518646026715o_assn(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups5301882518646026715o_assn(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_9087_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_assn,C2: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( groups6906906614972039071t_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_ajc(A),B),C2),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(B,A)),groups6906906614972039071t_assn(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))),groups6906906614972039071t_assn(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_9088_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_assn,C2: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups7882442080178216443t_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ajd(A),B),C2),S) = $ite(aa_set_int_o(member_int(A),S),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(B,A)),groups7882442080178216443t_assn(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))),groups7882442080178216443t_assn(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_9089_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_rat,C2: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups2555765274223993564er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ahm(A),B),C2),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(B,A)),groups2555765274223993564er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups2555765274223993564er_rat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.delta_remove
tff(fact_9090_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_rat,C2: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups2869687844427037835_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_ahp((A)),B),C2),S) = $ite(aa_set_o_o(member_o((A)),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(B,(A))),groups2869687844427037835_o_rat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups2869687844427037835_o_rat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_9091_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: fun_nat_rat,C2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_rat(groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ahs(A),B),C2)),S) = $ite(aa_set_nat_o(member_nat(A),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(B,A)),aa_set_nat_rat(groups73079841787564623at_rat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))),aa_set_nat_rat(groups73079841787564623at_rat(C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat(A,bot_bot_set_nat)))) ) ) ).

% prod.delta_remove
tff(fact_9092_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: fun_int_rat,C2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ahu(A),B),C2),S) = $ite(aa_set_int_o(member_int(A),S),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(B,A)),groups1072433553688619179nt_rat(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))),groups1072433553688619179nt_rat(C2,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),insert_int(A,bot_bot_set_int)))) ) ) ).

% prod.delta_remove
tff(fact_9093_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: fun_Code_integer_nat,C2: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( groups3190895334310489300er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_ahn(A),B),C2),S) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_integer_nat(B,A)),groups3190895334310489300er_nat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups3190895334310489300er_nat(C2,minus_2355218937544613996nteger(S,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% prod.delta_remove
tff(fact_9094_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: fun_o_nat,C2: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( groups3504817904513533571_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_ahq((A)),B),C2),S) = $ite(aa_set_o_o(member_o((A)),S),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(B,(A))),groups3504817904513533571_o_nat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))),groups3504817904513533571_o_nat(C2,minus_minus_set_o(S,insert_o((A),bot_bot_set_o)))) ) ) ).

% prod.delta_remove
tff(fact_9095_effect__of__listI,axiom,
    ! [A: array_nat,H_a: heap_e7401611519738050253t_unit,Xs: list_nat,Ha: heap_e7401611519738050253t_unit] :
      ( ( produc1210945592254923927t_unit(A,H_a) = array_alloc_nat(Xs,Ha) )
     => aa_nat_o(heap_T1188250621483087889ay_nat(array_of_list_nat(Xs),Ha,H_a,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_list_nat_nat(size_size_list_nat,Xs))) ) ).

% effect_of_listI
tff(fact_9096_effect__of__listI,axiom,
    ! [A: array_o,H_a: heap_e7401611519738050253t_unit,Xs: list_o,Ha: heap_e7401611519738050253t_unit] :
      ( ( produc3742984184817956739t_unit(A,H_a) = array_alloc_o(Xs,Ha) )
     => aa_nat_o(heap_T1873992626244518189rray_o(array_of_list_o(Xs),Ha,H_a,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),size_size_list_o(Xs))) ) ).

% effect_of_listI
tff(fact_9097_effect__of__listI,axiom,
    ! [A: array_int,H_a: heap_e7401611519738050253t_unit,Xs: list_int,Ha: heap_e7401611519738050253t_unit] :
      ( ( produc2888458824036673339t_unit(A,H_a) = array_alloc_int(Xs,Ha) )
     => aa_nat_o(heap_T6233771638828666989ay_int(array_of_list_int(Xs),Ha,H_a,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),size_size_list_int(Xs))) ) ).

% effect_of_listI
tff(fact_9098_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(B)))
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),B4)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(X3))) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups4406642042086082107nteger(Fa,A4)),groups4406642042086082107nteger(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9099_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B4)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,X3)) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),A4)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9100_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A4))
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B4)
                 => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,X3)) )
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups879477027807139574nteger(Fa,A4)),groups879477027807139574nteger(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9101_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_o_rat(Fa,(B)))
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(X3))) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(Fa,A4)),groups7872700643590313910_o_rat(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9102_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(Fa,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9103_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(Fa,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,X3)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(Fa,A4)),groups6602215022474089585er_rat(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9104_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(Fa,(B)))
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(X3))) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(Fa,A4)),groups8507830703676809646_o_nat(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9105_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Fa,B))
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(Fa,X3)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(Fa,A4)),groups7237345082560585321er_nat(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9106_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,Fa: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( aa_set_o_o(member_o((B)),minus_minus_set_o(B4,A4))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(Fa,(B)))
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Fa,(X3))) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(Fa,A4)),groups8505340233167759370_o_int(Fa,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9107_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(Fa,B))
           => ( ! [X3: nat] :
                  ( aa_set_nat_o(member_nat(X3),B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,X3)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(Fa),A4)),aa_set_nat_int(groups3539618377306564664at_int(Fa),B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_9108_member__le__sum,axiom,
    ! [I2: code_integer,A4: set_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(A4,insert_Code_integer(I2,bot_bo3990330152332043303nteger)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,X3)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(Fa,I2)),groups879477027807139574nteger(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9109_member__le__sum,axiom,
    ! [I2: $o,A4: set_o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(member_o((I2)),A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A4,insert_o((I2),bot_bot_set_o)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(Fa,(I2))),groups4406642042086082107nteger(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9110_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(member_nat(I2),A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(I2,bot_bot_set_nat)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,X3)) )
       => ( aa_set_nat_o(finite_finite_nat,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(Fa,I2)),aa_set871585252387937313nteger(groups7501900531339628137nteger(Fa),A4)) ) ) ) ).

% member_le_sum
tff(fact_9111_member__le__sum,axiom,
    ! [I2: int,A4: set_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(member_int(I2),A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(I2,bot_bot_set_int)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,X3)) )
       => ( aa_set_int_o(finite_finite_int,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(Fa,I2)),groups7873554091576472773nteger(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9112_member__le__sum,axiom,
    ! [I2: code_integer,A4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(A4,insert_Code_integer(I2,bot_bo3990330152332043303nteger)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,X3)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,I2)),groups6602215022474089585er_rat(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9113_member__le__sum,axiom,
    ! [I2: $o,A4: set_o,Fa: fun_o_rat] :
      ( aa_set_o_o(member_o((I2)),A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A4,insert_o((I2),bot_bot_set_o)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(Fa,(I2))),groups7872700643590313910_o_rat(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9114_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(I2),A4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),insert_nat(I2,bot_bot_set_nat)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X3)) )
       => ( aa_set_nat_o(finite_finite_nat,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I2)),aa_set_nat_rat(groups2906978787729119204at_rat(Fa),A4)) ) ) ) ).

% member_le_sum
tff(fact_9115_member__le__sum,axiom,
    ! [I2: int,A4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(member_int(I2),A4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),insert_int(I2,bot_bot_set_int)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X3)) )
       => ( aa_set_int_o(finite_finite_int,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I2)),groups3906332499630173760nt_rat(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9116_member__le__sum,axiom,
    ! [I2: code_integer,A4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),A4)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(A4,insert_Code_integer(I2,bot_bo3990330152332043303nteger)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_Code_integer_nat(Fa,X3)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(Fa,I2)),groups7237345082560585321er_nat(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9117_member__le__sum,axiom,
    ! [I2: $o,A4: set_o,Fa: fun_o_nat] :
      ( aa_set_o_o(member_o((I2)),A4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A4,insert_o((I2),bot_bot_set_o)))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(Fa,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(I2))),groups8507830703676809646_o_nat(Fa,A4)) ) ) ) ).

% member_le_sum
tff(fact_9118_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(Fa,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(Fa,(A3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(Fa,A4)),groups7694694392188491536nteger(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9119_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(Fa,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Fa,A3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),A4)),aa_set871585252387937313nteger(groups3455450783089532116nteger(Fa),B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9120_prod__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(Fa,B3)) )
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(Fa,A3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3674199335183972705nteger(Fa,A4)),groups3674199335183972705nteger(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9121_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(Fa,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(Fa,(A3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(Fa,A4)),groups2869687844427037835_o_rat(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9122_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(Fa,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_set_nat_rat(groups73079841787564623at_rat(Fa),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Fa),B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9123_prod__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(Fa,B3)) )
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2555765274223993564er_rat(Fa,A4)),groups2555765274223993564er_rat(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9124_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,Fa: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
       => ( ! [B3: $o] :
              ( aa_set_o_o(member_o((B3)),minus_minus_set_o(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(Fa,(B3))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(Fa,(A3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(Fa,A4)),groups3502327434004483295_o_int(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9125_prod__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_Code_integer_int(Fa,B3)) )
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_Code_integer_int(Fa,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3188404863801439024er_int(Fa,A4)),groups3188404863801439024er_int(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9126_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(Fa,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Fa,A3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(Fa,A4)),groups3827104343326376752nteger(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9127_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => ( ! [B3: int] :
              ( aa_set_int_o(member_int(B3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(Fa,B3)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(Fa,A4)),groups1072433553688619179nt_rat(Fa,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_9128_prod__Un,axiom,
    ! [A4: set_int,B4: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))
             => ( aa_int_rat(Fa,X3) != zero_zero_rat ) )
         => ( groups1072433553688619179nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(Fa,A4)),groups1072433553688619179nt_rat(Fa,B4))),groups1072433553688619179nt_rat(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),B4))) ) ) ) ) ).

% prod_Un
tff(fact_9129_prod__Un,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A4,B4))
             => ( aa_Code_integer_rat(Fa,X3) != zero_zero_rat ) )
         => ( groups2555765274223993564er_rat(Fa,sup_su848401254843788991nteger(A4,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(Fa,A4)),groups2555765274223993564er_rat(Fa,B4))),groups2555765274223993564er_rat(Fa,inf_in1364745209274528805nteger(A4,B4))) ) ) ) ) ).

% prod_Un
tff(fact_9130_prod__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Fa: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( ! [X3: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))
             => ( aa_Pro3608043709957902237at_rat(Fa,X3) != zero_zero_rat ) )
         => ( groups3442636767675653108at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups3442636767675653108at_rat(Fa,A4)),groups3442636767675653108at_rat(Fa,B4))),groups3442636767675653108at_rat(Fa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A4),B4))) ) ) ) ) ).

% prod_Un
tff(fact_9131_prod__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat,Fa: fun_Pr4065951289917319378at_rat] :
      ( finite1918287321285529104at_nat(A4)
     => ( finite1918287321285529104at_nat(B4)
       => ( ! [X3: produc4166570645942440679at_nat] :
              ( member6689249552917799696at_nat(X3,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4))
             => ( aa_Pro6772482835110228483at_rat(Fa,X3) != zero_zero_rat ) )
         => ( groups5938585286922990810at_rat(Fa,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups5938585286922990810at_rat(Fa,A4)),groups5938585286922990810at_rat(Fa,B4))),groups5938585286922990810at_rat(Fa,aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(inf_in1697001100524423349at_nat,A4),B4))) ) ) ) ) ).

% prod_Un
tff(fact_9132_prod__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,Fa: fun_Pr6968369006614801746at_rat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
       => ( ! [X3: produc3843707927480180839at_nat] :
              ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4))
             => ( aa_Pro5677984708781969027at_rat(Fa,X3) != zero_zero_rat ) )
         => ( groups8874911130973611098at_rat(Fa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups8874911130973611098at_rat(Fa,A4)),groups8874911130973611098at_rat(Fa,B4))),groups8874911130973611098at_rat(Fa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A4),B4))) ) ) ) ) ).

% prod_Un
tff(fact_9133_prod__Un,axiom,
    ! [A4: set_nat,B4: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))
             => ( aa_nat_rat(Fa,X3) != zero_zero_rat ) )
         => ( aa_set_nat_rat(groups73079841787564623at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_set_nat_rat(groups73079841787564623at_rat(Fa),A4)),aa_set_nat_rat(groups73079841787564623at_rat(Fa),B4))),aa_set_nat_rat(groups73079841787564623at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B4))) ) ) ) ) ).

% prod_Un
tff(fact_9134_or__one__eq,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(A,one_one_Code_natural) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_9135_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),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_9136_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_9137_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_9138_one__or__eq,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(one_one_Code_natural,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_o_Code_natural(zero_n8403883297036319079atural,aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_9139_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),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_9140_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_9141_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_9142_OR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ) ).

% OR_upper
tff(fact_9143_or__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N2))))) ).

% or_not_numerals(5)
tff(fact_9144_take__bit__sum,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(N2),A) = aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_aje(A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% take_bit_sum
tff(fact_9145_take__bit__sum,axiom,
    ! [N2: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N2),A) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ajf,A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% take_bit_sum
tff(fact_9146_take__bit__sum,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),A) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ajg,A)),set_or4665077453230672383an_nat(zero_zero_nat,N2)) ).

% take_bit_sum
tff(fact_9147_or__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N2))))) ).

% or_not_numerals(8)
tff(fact_9148_or__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)))) = 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,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N2))))) ).

% or_not_numerals(9)
tff(fact_9149_of__list__def,axiom,
    ! [Xs: list_nat] : array_of_list_nat(Xs) = heap_T4325564685996367510ay_nat(aTP_Lamm_ajh(Xs)) ).

% of_list_def
tff(fact_9150_of__list__def,axiom,
    ! [Xs: list_o] : array_of_list_o(Xs) = heap_T8367841184088661864rray_o(aTP_Lamm_aji(Xs)) ).

% of_list_def
tff(fact_9151_of__list__def,axiom,
    ! [Xs: list_int] : array_of_list_int(Xs) = heap_T147713666487170802ay_int(aTP_Lamm_ajj(Xs)) ).

% of_list_def
tff(fact_9152_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bx,K2))) ).

% finite_Collect_le_nat
tff(fact_9153_finite__Collect__less__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,K2))) ).

% finite_Collect_less_nat
tff(fact_9154_finite__Collect__subsets,axiom,
    ! [A4: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,A4)
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aTP_Lamm_pu(A4))) ) ).

% finite_Collect_subsets
tff(fact_9155_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bp,A4))) ) ).

% finite_Collect_subsets
tff(fact_9156_finite__Collect__subsets,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_ajk(A4))) ) ).

% finite_Collect_subsets
tff(fact_9157_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => finite9047747110432174090at_nat(collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_eq,A4))) ) ).

% finite_Collect_subsets
tff(fact_9158_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_bt,A4))) ) ).

% finite_Collect_subsets
tff(fact_9159_finite__induct__select,axiom,
    ! [S: set_Pr4329608150637261639at_nat,Pa: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
     => ( aa_set5844364881195198285_nat_o(Pa,bot_bo228742789529271731at_nat)
       => ( ! [T8: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(T8),S)
             => ( aa_set5844364881195198285_nat_o(Pa,T8)
               => ? [X5: produc3843707927480180839at_nat] :
                    ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),T8))
                    & aa_set5844364881195198285_nat_o(Pa,insert9069300056098147895at_nat(X5,T8)) ) ) )
         => aa_set5844364881195198285_nat_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_9160_finite__induct__select,axiom,
    ! [S: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [T8: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(T8),S)
             => ( aa_set5643387332196575111eger_o(Pa,T8)
               => ? [X5: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X5),minus_2355218937544613996nteger(S,T8))
                    & aa_set5643387332196575111eger_o(Pa,insert_Code_integer(X5,T8)) ) ) )
         => aa_set5643387332196575111eger_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_9161_finite__induct__select,axiom,
    ! [S: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [T8: set_o] :
              ( aa_set_o_o(ord_less_set_o(T8),S)
             => ( aa_set_o_o(Pa,T8)
               => ? [X5: $o] :
                    ( aa_set_o_o(member_o((X5)),minus_minus_set_o(S,T8))
                    & aa_set_o_o(Pa,insert_o((X5),T8)) ) ) )
         => aa_set_o_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_9162_finite__induct__select,axiom,
    ! [S: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [T8: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T8),S)
             => ( aa_set_nat_o(Pa,T8)
               => ? [X5: nat] :
                    ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),T8))
                    & aa_set_nat_o(Pa,insert_nat(X5,T8)) ) ) )
         => aa_set_nat_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_9163_finite__induct__select,axiom,
    ! [S: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [T8: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,T8),S)
             => ( aa_set_int_o(Pa,T8)
               => ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,S),T8))
                    & aa_set_int_o(Pa,insert_int(X5,T8)) ) ) )
         => aa_set_int_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_9164_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ( aa_set3702741264012733747_nat_o(Pa,bot_bo2099793752762293965at_nat)
       => ( ! [T8: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,T8),S)
             => ( aa_set3702741264012733747_nat_o(Pa,T8)
               => ? [X5: product_prod_nat_nat] :
                    ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),T8))
                    & aa_set3702741264012733747_nat_o(Pa,insert8211810215607154385at_nat(X5,T8)) ) ) )
         => aa_set3702741264012733747_nat_o(Pa,S) ) ) ) ).

% finite_induct_select
tff(fact_9165_finite__Un,axiom,
    ! [F4: set_int,G2: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F4),G2))
    <=> ( aa_set_int_o(finite_finite_int,F4)
        & aa_set_int_o(finite_finite_int,G2) ) ) ).

% finite_Un
tff(fact_9166_finite__Un,axiom,
    ! [F4: set_Code_integer,G2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(F4,G2))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G2) ) ) ).

% finite_Un
tff(fact_9167_finite__Un,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,F4),G2))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G2) ) ) ).

% finite_Un
tff(fact_9168_finite__Un,axiom,
    ! [F4: set_Pr8551490117392284871at_nat,G2: set_Pr8551490117392284871at_nat] :
      ( finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,F4),G2))
    <=> ( finite1918287321285529104at_nat(F4)
        & finite1918287321285529104at_nat(G2) ) ) ).

% finite_Un
tff(fact_9169_finite__Un,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F4),G2))
    <=> ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
        & aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,G2) ) ) ).

% finite_Un
tff(fact_9170_finite__Un,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F4),G2))
    <=> ( aa_set_nat_o(finite_finite_nat,F4)
        & aa_set_nat_o(finite_finite_nat,G2) ) ) ).

% finite_Un
tff(fact_9171_finite__Collect__conjI,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(Pa))
        | finite8100373058378681591st_nat(collect_list_nat(Qa)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ap(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9172_finite__Collect__conjI,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Pa))
        | aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Qa)) )
     => aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_aq(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9173_finite__Collect__conjI,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Pa))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Qa)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ar(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9174_finite__Collect__conjI,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Qa)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_as(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9175_finite__Collect__conjI,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(Pa))
        | aa_set_int_o(finite_finite_int,collect_int(Qa)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_at(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9176_finite__Collect__conjI,axiom,
    ! [Pa: fun_Code_integer_o,Qa: fun_Code_integer_o] :
      ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Pa))
        | aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Qa)) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_ajl(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9177_finite__Collect__conjI,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
        | aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Qa)) )
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_gf(Pa),Qa))) ) ).

% finite_Collect_conjI
tff(fact_9178_finite__Collect__disjI,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_au(Pa),Qa)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(Pa))
        & finite8100373058378681591st_nat(collect_list_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9179_finite__Collect__disjI,axiom,
    ! [Pa: fun_se7249947736802449410_int_o,Qa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(aa_fun6889641154785167298_int_o(aTP_Lamm_av(Pa),Qa)))
    <=> ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Pa))
        & aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9180_finite__Collect__disjI,axiom,
    ! [Pa: fun_set_nat_o,Qa: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_aw(Pa),Qa)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Pa))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9181_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ba(Pa),Qa)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9182_finite__Collect__disjI,axiom,
    ! [Pa: fun_int_o,Qa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(Pa),Qa)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(Pa))
        & aa_set_int_o(finite_finite_int,collect_int(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9183_finite__Collect__disjI,axiom,
    ! [Pa: fun_Code_integer_o,Qa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_ajm(Pa),Qa)))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Pa))
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9184_finite__Collect__disjI,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_ajn(Pa),Qa)))
    <=> ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Qa)) ) ) ).

% finite_Collect_disjI
tff(fact_9185_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ajo(A),B))) ).

% finite_interval_int1
tff(fact_9186_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ajp(A),B))) ).

% finite_interval_int4
tff(fact_9187_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_9188_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_9189_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ajq(A),B))) ).

% finite_interval_int2
tff(fact_9190_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ajr(A),B))) ).

% finite_interval_int3
tff(fact_9191_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N5: nat] :
        ! [X5: list_nat] :
          ( member_list_nat(X5,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X5)),N5) ) ) ).

% finite_maxlen
tff(fact_9192_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N5: nat] :
        ! [X5: list_o] :
          ( member_list_o(X5,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X5)),N5) ) ) ).

% finite_maxlen
tff(fact_9193_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N5: nat] :
        ! [X5: list_int] :
          ( member_list_int(X5,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X5)),N5) ) ) ).

% finite_maxlen
tff(fact_9194_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_jm,I2))) ) ).

% finite_divisors_int
tff(fact_9195_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_9196_not__finite__existsD,axiom,
    ! [Pa: fun_list_nat_o] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(Pa))
     => ? [X_1: list_nat] : aa_list_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9197_not__finite__existsD,axiom,
    ! [Pa: fun_se7249947736802449410_int_o] :
      ( ~ aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,collec5210948495886036740nt_int(Pa))
     => ? [X_1: set_Pr958786334691620121nt_int] : aa_set6170482117830564987_int_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9198_not__finite__existsD,axiom,
    ! [Pa: fun_set_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Pa))
     => ? [X_1: set_nat] : aa_set_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9199_not__finite__existsD,axiom,
    ! [Pa: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(Pa))
     => ? [X_1: nat] : aa_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9200_not__finite__existsD,axiom,
    ! [Pa: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(Pa))
     => ? [X_1: int] : aa_int_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9201_not__finite__existsD,axiom,
    ! [Pa: fun_Code_integer_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Pa))
     => ? [X_1: code_integer] : aa_Code_integer_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9202_not__finite__existsD,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,collec3392354462482085612at_nat(Pa))
     => ? [X_1: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(Pa,X_1) ) ).

% not_finite_existsD
tff(fact_9203_pigeonhole__infinite__rel,axiom,
    ! [A4: set_o,B4: set_nat,Ra: fun_o_fun_nat_o] :
      ( ~ aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_o_fun_nat_o(Ra,(X3)),Xa2) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set_o_o(finite_finite_o,collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9204_pigeonhole__infinite__rel,axiom,
    ! [A4: set_o,B4: set_int,Ra: fun_o_fun_int_o] :
      ( ~ aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_o_fun_int_o(Ra,(X3)),Xa2) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B4)
              & ~ aa_set_o_o(finite_finite_o,collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_aaa(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9205_pigeonhole__infinite__rel,axiom,
    ! [A4: set_o,B4: set_Code_integer,Ra: fun_o_4021313873839786239eger_o] :
      ( ~ aa_set_o_o(finite_finite_o,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A4)
             => ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),B4)
                  & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Ra,(X3)),Xa2) ) )
         => ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B4)
              & ~ aa_set_o_o(finite_finite_o,collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ajs(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9206_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,Ra: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(Ra,X3),Xa2) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_aba(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9207_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_int,Ra: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_nat_fun_int_o(Ra,X3),Xa2) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_aae(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9208_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_Code_integer,Ra: fun_na6318815019539281885eger_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),B4)
                  & aa_Code_integer_o(aa_nat7420681472272698950eger_o(Ra,X3),Xa2) ) )
         => ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_aaw(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9209_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_nat,Ra: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_int_fun_nat_o(Ra,X3),Xa2) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_zq(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9210_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_int,Ra: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_int_fun_int_o(Ra,X3),Xa2) ) )
         => ? [X3: int] :
              ( aa_set_int_o(member_int(X3),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ajt(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9211_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B4: set_Code_integer,Ra: fun_in4891666191172872833eger_o] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
             => ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),B4)
                  & aa_Code_integer_o(aa_int5993532643906289898eger_o(Ra,X3),Xa2) ) )
         => ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_aju(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9212_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Code_integer,B4: set_nat,Ra: fun_Co1375048237525739357_nat_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_Cod1075288278386115654_nat_o(Ra,X3),Xa2) ) )
         => ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),B4)
              & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_zv(A4),Ra),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_9213_finite__has__minimal2,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((A)),A4)
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(A))
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa2)),(X3))
                 => ( (X3)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9214_finite__has__minimal2,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A4)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
       => ? [X3: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
            & aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(X3),A)
            & ! [Xa2: set_Pr958786334691620121nt_int] :
                ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Xa2),A4)
               => ( aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9215_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(A),A4)
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A4)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),A)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A4)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9216_finite__has__minimal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A4)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),A)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9217_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),A)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9218_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9219_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),A)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9220_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9221_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_9222_finite__has__maximal2,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((A)),A4)
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
            & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(X3))
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(Xa2))
                 => ( (X3)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9223_finite__has__maximal2,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A4)
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A),A4)
       => ? [X3: set_Pr958786334691620121nt_int] :
            ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X3),A4)
            & aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(A),X3)
            & ! [Xa2: set_Pr958786334691620121nt_int] :
                ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Xa2),A4)
               => ( aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9224_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(A),A4)
       => ? [X3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X3),A4)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X3)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A4)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9225_finite__has__maximal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A4)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),X3)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9226_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),X3)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9227_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X3)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9228_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X3)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9229_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9230_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_9231_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_9232_rev__finite__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% rev_finite_subset
tff(fact_9233_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_9234_rev__finite__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% rev_finite_subset
tff(fact_9235_infinite__super,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => ~ aa_set_nat_o(finite_finite_nat,T7) ) ) ).

% infinite_super
tff(fact_9236_infinite__super,axiom,
    ! [S: set_Code_integer,T7: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T7)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7) ) ) ).

% infinite_super
tff(fact_9237_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
       => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T7) ) ) ).

% infinite_super
tff(fact_9238_infinite__super,axiom,
    ! [S: set_int,T7: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ~ aa_set_int_o(finite_finite_int,S)
       => ~ aa_set_int_o(finite_finite_int,T7) ) ) ).

% infinite_super
tff(fact_9239_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_subset
tff(fact_9240_finite__subset,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% finite_subset
tff(fact_9241_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_subset
tff(fact_9242_finite__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_subset
tff(fact_9243_finite__UnI,axiom,
    ! [F4: set_int,G2: set_int] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(finite_finite_int,G2)
       => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,F4),G2)) ) ) ).

% finite_UnI
tff(fact_9244_finite__UnI,axiom,
    ! [F4: set_Code_integer,G2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G2)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(F4,G2)) ) ) ).

% finite_UnI
tff(fact_9245_finite__UnI,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G2)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,F4),G2)) ) ) ).

% finite_UnI
tff(fact_9246_finite__UnI,axiom,
    ! [F4: set_Pr8551490117392284871at_nat,G2: set_Pr8551490117392284871at_nat] :
      ( finite1918287321285529104at_nat(F4)
     => ( finite1918287321285529104at_nat(G2)
       => finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,F4),G2)) ) ) ).

% finite_UnI
tff(fact_9247_finite__UnI,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,G2)
       => aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,F4),G2)) ) ) ).

% finite_UnI
tff(fact_9248_finite__UnI,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(finite_finite_nat,G2)
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,F4),G2)) ) ) ).

% finite_UnI
tff(fact_9249_Un__infinite,axiom,
    ! [S: set_int,T7: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T7)) ) ).

% Un_infinite
tff(fact_9250_Un__infinite,axiom,
    ! [S: set_Code_integer,T7: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(S,T7)) ) ).

% Un_infinite
tff(fact_9251_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
     => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T7)) ) ).

% Un_infinite
tff(fact_9252_Un__infinite,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ~ finite1918287321285529104at_nat(S)
     => ~ finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T7)) ) ).

% Un_infinite
tff(fact_9253_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
     => ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T7)) ) ).

% Un_infinite
tff(fact_9254_Un__infinite,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T7)) ) ).

% Un_infinite
tff(fact_9255_infinite__Un,axiom,
    ! [S: set_int,T7: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),T7))
    <=> ( ~ aa_set_int_o(finite_finite_int,S)
        | ~ aa_set_int_o(finite_finite_int,T7) ) ) ).

% infinite_Un
tff(fact_9256_infinite__Un,axiom,
    ! [S: set_Code_integer,T7: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,sup_su848401254843788991nteger(S,T7))
    <=> ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
        | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7) ) ) ).

% infinite_Un
tff(fact_9257_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T7))
    <=> ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S)
        | ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T7) ) ) ).

% infinite_Un
tff(fact_9258_infinite__Un,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ~ finite1918287321285529104at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,S),T7))
    <=> ( ~ finite1918287321285529104at_nat(S)
        | ~ finite1918287321285529104at_nat(T7) ) ) ).

% infinite_Un
tff(fact_9259_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T7))
    <=> ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,S)
        | ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,T7) ) ) ).

% infinite_Un
tff(fact_9260_infinite__Un,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T7))
    <=> ( ~ aa_set_nat_o(finite_finite_nat,S)
        | ~ aa_set_nat_o(finite_finite_nat,T7) ) ) ).

% infinite_Un
tff(fact_9261_finite__psubset__induct,axiom,
    ! [A4: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A7)
           => ( ! [B8: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B8),A7)
                 => aa_set_nat_o(Pa,B8) )
             => aa_set_nat_o(Pa,A7) ) )
       => aa_set_nat_o(Pa,A4) ) ) ).

% finite_psubset_induct
tff(fact_9262_finite__psubset__induct,axiom,
    ! [A4: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(finite_finite_int,A7)
           => ( ! [B8: set_int] :
                  ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,B8),A7)
                 => aa_set_int_o(Pa,B8) )
             => aa_set_int_o(Pa,A7) ) )
       => aa_set_int_o(Pa,A4) ) ) ).

% finite_psubset_induct
tff(fact_9263_finite__psubset__induct,axiom,
    ! [A4: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [A7: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
           => ( ! [B8: set_Code_integer] :
                  ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(B8),A7)
                 => aa_set5643387332196575111eger_o(Pa,B8) )
             => aa_set5643387332196575111eger_o(Pa,A7) ) )
       => aa_set5643387332196575111eger_o(Pa,A4) ) ) ).

% finite_psubset_induct
tff(fact_9264_finite__psubset__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
           => ( ! [B8: set_Pr1261947904930325089at_nat] :
                  ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B8),A7)
                 => aa_set3702741264012733747_nat_o(Pa,B8) )
             => aa_set3702741264012733747_nat_o(Pa,A7) ) )
       => aa_set3702741264012733747_nat_o(Pa,A4) ) ) ).

% finite_psubset_induct
tff(fact_9265_Suc__0__or__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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))),N2))) ).

% Suc_0_or_eq
tff(fact_9266_or__Suc__0__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N2),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),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))),N2))) ).

% or_Suc_0_eq
tff(fact_9267_or__nat__rec,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N2) = 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))),N2) ))),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9268_finite__has__maximal,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9269_finite__has__maximal,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(Xa2))
                 => ( (X3)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9270_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9271_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A4)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9272_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A4)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9273_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9274_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa2)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_9275_finite__has__minimal,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9276_finite__has__minimal,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A4)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa2)),(X3))
                 => ( (X3)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9277_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( member_set_int(X3,A4)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9278_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A4)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9279_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A4)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9280_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9281_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A4)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X3)
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_9282_finite__subset__induct_H,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,Pa: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),A4)
       => ( aa_set5844364881195198285_nat_o(Pa,bot_bo228742789529271731at_nat)
         => ( ! [A3: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F5)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A4)
                 => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F5),A4)
                   => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F5)
                     => ( aa_set5844364881195198285_nat_o(Pa,F5)
                       => aa_set5844364881195198285_nat_o(Pa,insert9069300056098147895at_nat(A3,F5)) ) ) ) ) )
           => aa_set5844364881195198285_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9283_finite__subset__induct_H,axiom,
    ! [F4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,Pa: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(F4),A4)
       => ( aa_set6820988702716878619_int_o(Pa,bot_bo1488462491386950373nt_int)
         => ( ! [A3: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
                ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F5)
               => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A3),A4)
                 => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(F5),A4)
                   => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A3),F5)
                     => ( aa_set6820988702716878619_int_o(Pa,F5)
                       => aa_set6820988702716878619_int_o(Pa,insert8897473484851387113nt_int(A3,F5)) ) ) ) ) )
           => aa_set6820988702716878619_int_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9284_finite__subset__induct_H,axiom,
    ! [F4: set_set_nat,A4: set_set_nat,Pa: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A4)
       => ( aa_set_set_nat_o(Pa,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F5: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F5)
               => ( aa_set_set_nat_o(member_set_nat(A3),A4)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F5),A4)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A3),F5)
                     => ( aa_set_set_nat_o(Pa,F5)
                       => aa_set_set_nat_o(Pa,insert_set_nat(A3,F5)) ) ) ) ) )
           => aa_set_set_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9285_finite__subset__induct_H,axiom,
    ! [F4: set_Code_integer,A4: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F4),A4)
       => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F5)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
                 => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F5),A4)
                   => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F5)
                     => ( aa_set5643387332196575111eger_o(Pa,F5)
                       => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(A3,F5)) ) ) ) ) )
           => aa_set5643387332196575111eger_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9286_finite__subset__induct_H,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F4),A4)
       => ( aa_set3702741264012733747_nat_o(Pa,bot_bo2099793752762293965at_nat)
         => ( ! [A3: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F5)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A4)
                 => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F5),A4)
                   => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F5)
                     => ( aa_set3702741264012733747_nat_o(Pa,F5)
                       => aa_set3702741264012733747_nat_o(Pa,insert8211810215607154385at_nat(A3,F5)) ) ) ) ) )
           => aa_set3702741264012733747_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9287_finite__subset__induct_H,axiom,
    ! [F4: set_o,A4: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(F4),A4)
       => ( aa_set_o_o(Pa,bot_bot_set_o)
         => ( ! [A3: $o,F5: set_o] :
                ( aa_set_o_o(finite_finite_o,F5)
               => ( aa_set_o_o(member_o((A3)),A4)
                 => ( aa_set_o_o(ord_less_eq_set_o(F5),A4)
                   => ( ~ aa_set_o_o(member_o((A3)),F5)
                     => ( aa_set_o_o(Pa,F5)
                       => aa_set_o_o(Pa,insert_o((A3),F5)) ) ) ) ) )
           => aa_set_o_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9288_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A4: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F4),A4)
       => ( aa_set_nat_o(Pa,bot_bot_set_nat)
         => ( ! [A3: nat,F5: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F5)
               => ( aa_set_nat_o(member_nat(A3),A4)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F5),A4)
                   => ( ~ aa_set_nat_o(member_nat(A3),F5)
                     => ( aa_set_nat_o(Pa,F5)
                       => aa_set_nat_o(Pa,insert_nat(A3,F5)) ) ) ) ) )
           => aa_set_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9289_finite__subset__induct_H,axiom,
    ! [F4: set_int,A4: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F4),A4)
       => ( aa_set_int_o(Pa,bot_bot_set_int)
         => ( ! [A3: int,F5: set_int] :
                ( aa_set_int_o(finite_finite_int,F5)
               => ( aa_set_int_o(member_int(A3),A4)
                 => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F5),A4)
                   => ( ~ aa_set_int_o(member_int(A3),F5)
                     => ( aa_set_int_o(Pa,F5)
                       => aa_set_int_o(Pa,insert_int(A3,F5)) ) ) ) ) )
           => aa_set_int_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_9290_finite__subset__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,Pa: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F4)
     => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(F4),A4)
       => ( aa_set5844364881195198285_nat_o(Pa,bot_bo228742789529271731at_nat)
         => ( ! [A3: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,F5)
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A4)
                 => ( ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),F5)
                   => ( aa_set5844364881195198285_nat_o(Pa,F5)
                     => aa_set5844364881195198285_nat_o(Pa,insert9069300056098147895at_nat(A3,F5)) ) ) ) )
           => aa_set5844364881195198285_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9291_finite__subset__induct,axiom,
    ! [F4: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,Pa: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F4)
     => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(F4),A4)
       => ( aa_set6820988702716878619_int_o(Pa,bot_bo1488462491386950373nt_int)
         => ( ! [A3: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
                ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,F5)
               => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A3),A4)
                 => ( ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(A3),F5)
                   => ( aa_set6820988702716878619_int_o(Pa,F5)
                     => aa_set6820988702716878619_int_o(Pa,insert8897473484851387113nt_int(A3,F5)) ) ) ) )
           => aa_set6820988702716878619_int_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9292_finite__subset__induct,axiom,
    ! [F4: set_set_nat,A4: set_set_nat,Pa: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F4),A4)
       => ( aa_set_set_nat_o(Pa,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F5: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F5)
               => ( aa_set_set_nat_o(member_set_nat(A3),A4)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A3),F5)
                   => ( aa_set_set_nat_o(Pa,F5)
                     => aa_set_set_nat_o(Pa,insert_set_nat(A3,F5)) ) ) ) )
           => aa_set_set_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9293_finite__subset__induct,axiom,
    ! [F4: set_Code_integer,A4: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F4),A4)
       => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F5)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F5)
                   => ( aa_set5643387332196575111eger_o(Pa,F5)
                     => aa_set5643387332196575111eger_o(Pa,insert_Code_integer(A3,F5)) ) ) ) )
           => aa_set5643387332196575111eger_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9294_finite__subset__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,F4),A4)
       => ( aa_set3702741264012733747_nat_o(Pa,bot_bo2099793752762293965at_nat)
         => ( ! [A3: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F5)
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A4)
                 => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),F5)
                   => ( aa_set3702741264012733747_nat_o(Pa,F5)
                     => aa_set3702741264012733747_nat_o(Pa,insert8211810215607154385at_nat(A3,F5)) ) ) ) )
           => aa_set3702741264012733747_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9295_finite__subset__induct,axiom,
    ! [F4: set_o,A4: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(F4),A4)
       => ( aa_set_o_o(Pa,bot_bot_set_o)
         => ( ! [A3: $o,F5: set_o] :
                ( aa_set_o_o(finite_finite_o,F5)
               => ( aa_set_o_o(member_o((A3)),A4)
                 => ( ~ aa_set_o_o(member_o((A3)),F5)
                   => ( aa_set_o_o(Pa,F5)
                     => aa_set_o_o(Pa,insert_o((A3),F5)) ) ) ) )
           => aa_set_o_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9296_finite__subset__induct,axiom,
    ! [F4: set_nat,A4: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F4),A4)
       => ( aa_set_nat_o(Pa,bot_bot_set_nat)
         => ( ! [A3: nat,F5: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F5)
               => ( aa_set_nat_o(member_nat(A3),A4)
                 => ( ~ aa_set_nat_o(member_nat(A3),F5)
                   => ( aa_set_nat_o(Pa,F5)
                     => aa_set_nat_o(Pa,insert_nat(A3,F5)) ) ) ) )
           => aa_set_nat_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9297_finite__subset__induct,axiom,
    ! [F4: set_int,A4: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F4),A4)
       => ( aa_set_int_o(Pa,bot_bot_set_int)
         => ( ! [A3: int,F5: set_int] :
                ( aa_set_int_o(finite_finite_int,F5)
               => ( aa_set_int_o(member_int(A3),A4)
                 => ( ~ aa_set_int_o(member_int(A3),F5)
                   => ( aa_set_int_o(Pa,F5)
                     => aa_set_int_o(Pa,insert_int(A3,F5)) ) ) ) )
           => aa_set_int_o(Pa,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_9298_finite__remove__induct,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,Pa: fun_se7788542033685991124_nat_o] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
     => ( aa_set5844364881195198285_nat_o(Pa,bot_bo228742789529271731at_nat)
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7)
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A7),B4)
                 => ( ! [X5: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),A7)
                       => aa_set5844364881195198285_nat_o(Pa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(X5,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(Pa,A7) ) ) ) )
         => aa_set5844364881195198285_nat_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9299_finite__remove__induct,axiom,
    ! [B4: set_se6260736226359567993nt_int,Pa: fun_se9106460625052465826_int_o] :
      ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,B4)
     => ( aa_set6820988702716878619_int_o(Pa,bot_bo1488462491386950373nt_int)
       => ( ! [A7: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A7)
             => ( ( A7 != bot_bo1488462491386950373nt_int )
               => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A7),B4)
                 => ( ! [X5: set_Pr958786334691620121nt_int] :
                        ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X5),A7)
                       => aa_set6820988702716878619_int_o(Pa,minus_2612819937483484256nt_int(A7,insert8897473484851387113nt_int(X5,bot_bo1488462491386950373nt_int))) )
                   => aa_set6820988702716878619_int_o(Pa,A7) ) ) ) )
         => aa_set6820988702716878619_int_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9300_finite__remove__induct,axiom,
    ! [B4: set_set_nat,Pa: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(Pa,bot_bot_set_set_nat)
       => ( ! [A7: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A7)
             => ( ( A7 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A7),B4)
                 => ( ! [X5: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X5),A7)
                       => aa_set_set_nat_o(Pa,minus_2163939370556025621et_nat(A7,insert_set_nat(X5,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(Pa,A7) ) ) ) )
         => aa_set_set_nat_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9301_finite__remove__induct,axiom,
    ! [B4: set_Code_integer,Pa: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
       => ( ! [A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A7),B4)
                 => ( ! [X5: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A7)
                       => aa_set5643387332196575111eger_o(Pa,minus_2355218937544613996nteger(A7,insert_Code_integer(X5,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(Pa,A7) ) ) ) )
         => aa_set5643387332196575111eger_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9302_finite__remove__induct,axiom,
    ! [B4: set_o,Pa: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(Pa,bot_bot_set_o)
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B4)
                 => ( ! [X5: $o] :
                        ( aa_set_o_o(member_o((X5)),A7)
                       => aa_set_o_o(Pa,minus_minus_set_o(A7,insert_o((X5),bot_bot_set_o))) )
                   => aa_set_o_o(Pa,A7) ) ) ) )
         => aa_set_o_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9303_finite__remove__induct,axiom,
    ! [B4: set_nat,Pa: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(Pa,bot_bot_set_nat)
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A7),B4)
                 => ( ! [X5: nat] :
                        ( aa_set_nat_o(member_nat(X5),A7)
                       => aa_set_nat_o(Pa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(X5,bot_bot_set_nat))) )
                   => aa_set_nat_o(Pa,A7) ) ) ) )
         => aa_set_nat_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9304_finite__remove__induct,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,Pa: fun_se4782206882984618170_nat_o] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(Pa,bot_bo2099793752762293965at_nat)
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A7),B4)
                 => ( ! [X5: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A7)
                       => aa_set3702741264012733747_nat_o(Pa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(Pa,A7) ) ) ) )
         => aa_set3702741264012733747_nat_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9305_finite__remove__induct,axiom,
    ! [B4: set_int,Pa: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(Pa,bot_bot_set_int)
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A7),B4)
                 => ( ! [X5: int] :
                        ( aa_set_int_o(member_int(X5),A7)
                       => aa_set_int_o(Pa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A7),insert_int(X5,bot_bot_set_int))) )
                   => aa_set_int_o(Pa,A7) ) ) ) )
         => aa_set_int_o(Pa,B4) ) ) ) ).

% finite_remove_induct
tff(fact_9306_remove__induct,axiom,
    ! [Pa: fun_se7788542033685991124_nat_o,B4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(Pa,bot_bo228742789529271731at_nat)
     => ( ( ~ aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,B4)
         => aa_set5844364881195198285_nat_o(Pa,B4) )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A7)
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(A7),B4)
                 => ( ! [X5: produc3843707927480180839at_nat] :
                        ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),A7)
                       => aa_set5844364881195198285_nat_o(Pa,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A7),insert9069300056098147895at_nat(X5,bot_bo228742789529271731at_nat))) )
                   => aa_set5844364881195198285_nat_o(Pa,A7) ) ) ) )
         => aa_set5844364881195198285_nat_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9307_remove__induct,axiom,
    ! [Pa: fun_se9106460625052465826_int_o,B4: set_se6260736226359567993nt_int] :
      ( aa_set6820988702716878619_int_o(Pa,bot_bo1488462491386950373nt_int)
     => ( ( ~ aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,B4)
         => aa_set6820988702716878619_int_o(Pa,B4) )
       => ( ! [A7: set_se6260736226359567993nt_int] :
              ( aa_set6820988702716878619_int_o(finite8744585540193469122nt_int,A7)
             => ( ( A7 != bot_bo1488462491386950373nt_int )
               => ( aa_set6820988702716878619_int_o(ord_le483042692224249369nt_int(A7),B4)
                 => ( ! [X5: set_Pr958786334691620121nt_int] :
                        ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X5),A7)
                       => aa_set6820988702716878619_int_o(Pa,minus_2612819937483484256nt_int(A7,insert8897473484851387113nt_int(X5,bot_bo1488462491386950373nt_int))) )
                   => aa_set6820988702716878619_int_o(Pa,A7) ) ) ) )
         => aa_set6820988702716878619_int_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9308_remove__induct,axiom,
    ! [Pa: fun_set_set_nat_o,B4: set_set_nat] :
      ( aa_set_set_nat_o(Pa,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
         => aa_set_set_nat_o(Pa,B4) )
       => ( ! [A7: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A7)
             => ( ( A7 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A7),B4)
                 => ( ! [X5: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X5),A7)
                       => aa_set_set_nat_o(Pa,minus_2163939370556025621et_nat(A7,insert_set_nat(X5,bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(Pa,A7) ) ) ) )
         => aa_set_set_nat_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9309_remove__induct,axiom,
    ! [Pa: fun_se7289222521248229150eger_o,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(Pa,bot_bo3990330152332043303nteger)
     => ( ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_set5643387332196575111eger_o(Pa,B4) )
       => ( ! [A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A7),B4)
                 => ( ! [X5: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A7)
                       => aa_set5643387332196575111eger_o(Pa,minus_2355218937544613996nteger(A7,insert_Code_integer(X5,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(Pa,A7) ) ) ) )
         => aa_set5643387332196575111eger_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9310_remove__induct,axiom,
    ! [Pa: fun_set_o_o,B4: set_o] :
      ( aa_set_o_o(Pa,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B4)
         => aa_set_o_o(Pa,B4) )
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B4)
                 => ( ! [X5: $o] :
                        ( aa_set_o_o(member_o((X5)),A7)
                       => aa_set_o_o(Pa,minus_minus_set_o(A7,insert_o((X5),bot_bot_set_o))) )
                   => aa_set_o_o(Pa,A7) ) ) ) )
         => aa_set_o_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9311_remove__induct,axiom,
    ! [Pa: fun_set_nat_o,B4: set_nat] :
      ( aa_set_nat_o(Pa,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B4)
         => aa_set_nat_o(Pa,B4) )
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A7),B4)
                 => ( ! [X5: nat] :
                        ( aa_set_nat_o(member_nat(X5),A7)
                       => aa_set_nat_o(Pa,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A7),insert_nat(X5,bot_bot_set_nat))) )
                   => aa_set_nat_o(Pa,A7) ) ) ) )
         => aa_set_nat_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9312_remove__induct,axiom,
    ! [Pa: fun_se4782206882984618170_nat_o,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(Pa,bot_bo2099793752762293965at_nat)
     => ( ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
         => aa_set3702741264012733747_nat_o(Pa,B4) )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A7)
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A7),B4)
                 => ( ! [X5: product_prod_nat_nat] :
                        ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),A7)
                       => aa_set3702741264012733747_nat_o(Pa,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A7),insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat))) )
                   => aa_set3702741264012733747_nat_o(Pa,A7) ) ) ) )
         => aa_set3702741264012733747_nat_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9313_remove__induct,axiom,
    ! [Pa: fun_set_int_o,B4: set_int] :
      ( aa_set_int_o(Pa,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B4)
         => aa_set_int_o(Pa,B4) )
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A7),B4)
                 => ( ! [X5: int] :
                        ( aa_set_int_o(member_int(X5),A7)
                       => aa_set_int_o(Pa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A7),insert_int(X5,bot_bot_set_int))) )
                   => aa_set_int_o(Pa,A7) ) ) ) )
         => aa_set_int_o(Pa,B4) ) ) ) ).

% remove_induct
tff(fact_9314_set__encode__insert,axiom,
    ! [A4: set_nat,N2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(N2),A4)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N2,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_9315_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ? [K: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K)) ) ).

% finite_nat_bounded
tff(fact_9316_finite__nat__iff__bounded,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_9317_finite__nat__iff__bounded__le,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_9318_in__finite__psubset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A4,B4),finite_psubset_nat)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
        & aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% in_finite_psubset
tff(fact_9319_in__finite__psubset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A4,B4),finite_psubset_int)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
        & aa_set_int_o(finite_finite_int,B4) ) ) ).

% in_finite_psubset
tff(fact_9320_in__finite__psubset,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( member4307123515891402160nteger(produc7443773368509356479nteger(A4,B4),finite2416775604798480986nteger)
    <=> ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A4),B4)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4) ) ) ).

% in_finite_psubset
tff(fact_9321_in__finite__psubset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(A4,B4)),finite469560695537375940at_nat)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),B4)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4) ) ) ).

% in_finite_psubset
tff(fact_9322_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M3: int] :
        ? [N: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N))
          & aa_set_int_o(member_int(N),S) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_9323_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M3: int] :
        ? [N: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M3),aa_int_int(abs_abs_int,N))
          & aa_set_int_o(member_int(N),S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_9324_finite__psubset__def,axiom,
    finite_psubset_nat = collec6662362479098859352et_nat(produc6247414631856714078_nat_o(aTP_Lamm_ajv)) ).

% finite_psubset_def
tff(fact_9325_finite__psubset__def,axiom,
    finite_psubset_int = collec957716948307931664et_int(produc4109468873575309990_int_o(aTP_Lamm_ajw)) ).

% finite_psubset_def
tff(fact_9326_finite__psubset__def,axiom,
    finite2416775604798480986nteger = collec2770208431294612722nteger(produc1658495936642081476eger_o(aTP_Lamm_ajx)) ).

% finite_psubset_def
tff(fact_9327_finite__psubset__def,axiom,
    finite469560695537375940at_nat = collec6321179662152712658at_nat(produc410239310623530412_nat_o(aTP_Lamm_ajy)) ).

% finite_psubset_def
tff(fact_9328_unbounded__k__infinite,axiom,
    ! [K2: nat,S: set_nat] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),M5)
         => ? [N8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N8)
              & aa_set_nat_o(member_nat(N8),S) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S) ) ).

% unbounded_k_infinite
tff(fact_9329_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M3: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
          & aa_set_nat_o(member_nat(N),S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9330_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M3: nat] :
        ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
          & aa_set_nat_o(member_nat(N),S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_9331_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = $ite(
        ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K2 = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9332_or__nat__unfold,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N2) = $ite(
        M = zero_zero_nat,
        N2,
        $ite(N2 = 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,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9333_sum__diff1_H__aux,axiom,
    ! [F4: set_Code_integer,I5: set_Code_integer,Fa: fun_Co8499238673658658103nteger,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ace(I5),Fa))),F4)
       => ( groups910942671188738463nteger(Fa,minus_2355218937544613996nteger(I5,insert_Code_integer(I2,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(I2),I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups910942671188738463nteger(Fa,I5)),aa_Cod1783974652533396728nteger(Fa,I2)),groups910942671188738463nteger(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9334_sum__diff1_H__aux,axiom,
    ! [F4: set_o,I5: set_o,Fa: fun_o_Code_integer,I2: $o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_aby(I5),Fa))),F4)
       => ( groups1402912129352969042nteger(Fa,minus_minus_set_o(I5,insert_o((I2),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((I2)),I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups1402912129352969042nteger(Fa,I5)),aa_o_Code_integer(Fa,(I2))),groups1402912129352969042nteger(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9335_sum__diff1_H__aux,axiom,
    ! [F4: set_nat,I5: set_nat,Fa: fun_nat_Code_integer,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_aca(I5),Fa))),F4)
       => ( aa_set871585252387937313nteger(groups555127423416065298nteger(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),insert_nat(I2,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(I2),I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_set871585252387937313nteger(groups555127423416065298nteger(Fa),I5)),aa_nat_Code_integer(Fa,I2)),aa_set871585252387937313nteger(groups555127423416065298nteger(Fa),I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9336_sum__diff1_H__aux,axiom,
    ! [F4: set_Code_integer,I5: set_Code_integer,Fa: fun_Code_integer_rat,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_acm(I5),Fa))),F4)
       => ( groups8878813951405302554er_rat(Fa,minus_2355218937544613996nteger(I5,insert_Code_integer(I2,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(I2),I5),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups8878813951405302554er_rat(Fa,I5)),aa_Code_integer_rat(Fa,I2)),groups8878813951405302554er_rat(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9337_sum__diff1_H__aux,axiom,
    ! [F4: set_o,I5: set_o,Fa: fun_o_rat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_acg(I5),Fa))),F4)
       => ( groups3921277224699582669_o_rat(Fa,minus_minus_set_o(I5,insert_o((I2),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((I2)),I5),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,groups3921277224699582669_o_rat(Fa,I5)),aa_o_rat(Fa,(I2))),groups3921277224699582669_o_rat(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9338_sum__diff1_H__aux,axiom,
    ! [F4: set_nat,I5: set_nat,Fa: fun_nat_rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_aci(I5),Fa))),F4)
       => ( aa_set_nat_rat(groups1351286907653491341at_rat(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),insert_nat(I2,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(I2),I5),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_set_nat_rat(groups1351286907653491341at_rat(Fa),I5)),aa_nat_rat(Fa,I2)),aa_set_nat_rat(groups1351286907653491341at_rat(Fa),I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9339_sum__diff1_H__aux,axiom,
    ! [F4: set_Code_integer,I5: set_Code_integer,Fa: fun_Code_integer_int,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(collect_Code_integer(aa_fun3864128940607509264eger_o(aTP_Lamm_ajz(I5),Fa))),F4)
       => ( groups288081504127972206er_int(Fa,minus_2355218937544613996nteger(I5,insert_Code_integer(I2,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(I2),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups288081504127972206er_int(Fa,I5)),aa_Code_integer_int(Fa,I2)),groups288081504127972206er_int(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9340_sum__diff1_H__aux,axiom,
    ! [F4: set_o,I5: set_o,Fa: fun_o_int,I2: $o] :
      ( aa_set_o_o(finite_finite_o,F4)
     => ( aa_set_o_o(ord_less_eq_set_o(collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_aka(I5),Fa))),F4)
       => ( groups4553916814277028129_o_int(Fa,minus_minus_set_o(I5,insert_o((I2),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((I2)),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4553916814277028129_o_int(Fa,I5)),aa_o_int(Fa,(I2))),groups4553916814277028129_o_int(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9341_sum__diff1_H__aux,axiom,
    ! [F4: set_nat,I5: set_nat,Fa: fun_nat_int,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun2180201864171524266_nat_o(aTP_Lamm_akb(I5),Fa))),F4)
       => ( aa_set_nat_int(groups1983926497230936801at_int(Fa),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I5),insert_nat(I2,bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(I2),I5),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups1983926497230936801at_int(Fa),I5)),aa_nat_int(Fa,I2)),aa_set_nat_int(groups1983926497230936801at_int(Fa),I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9342_sum__diff1_H__aux,axiom,
    ! [F4: set_int,I5: set_int,Fa: fun_int_Code_integer,I2: int] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_acc(I5),Fa))),F4)
       => ( groups926780983652909934nteger(Fa,aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,I5),insert_int(I2,bot_bot_set_int))) = $ite(aa_set_int_o(member_int(I2),I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups926780983652909934nteger(Fa,I5)),aa_int_Code_integer(Fa,I2)),groups926780983652909934nteger(Fa,I5)) ) ) ) ).

% sum_diff1'_aux
tff(fact_9343_even__sum__iff,axiom,
    ! [A4: set_o,Fa: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups8507830703676809646_o_nat(Fa,A4))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_akc(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9344_even__sum__iff,axiom,
    ! [A4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups4541462559716669496nt_nat(Fa,A4))
      <=> 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_akd(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9345_even__sum__iff,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups7237345082560585321er_nat(Fa,A4))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite4902975817058060853nteger(collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_ake(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9346_even__sum__iff,axiom,
    ! [A4: set_o,Fa: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups8505340233167759370_o_int(Fa,A4))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun_o_int_fun_o_o(aTP_Lamm_akf(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9347_even__sum__iff,axiom,
    ! [A4: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( 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(Fa),A4))
      <=> 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_akg(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9348_even__sum__iff,axiom,
    ! [A4: set_Code_integer,Fa: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups7234854612051535045er_int(Fa,A4))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite4902975817058060853nteger(collect_Code_integer(aa_fun3864128940607509264eger_o(aTP_Lamm_akh(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9349_even__sum__iff,axiom,
    ! [A4: set_o,Fa: fun_o_Code_natural] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups3230237193842799622atural(Fa,A4))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite_card_o(collect_o(aa_fun2872501519293664346un_o_o(aTP_Lamm_aki(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9350_even__sum__iff,axiom,
    ! [A4: set_nat,Fa: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups6325495683096345652atural(Fa,A4))
      <=> 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_fun440078198722963374_nat_o(aTP_Lamm_akj(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9351_even__sum__iff,axiom,
    ! [A4: set_int,Fa: fun_int_Code_natural] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups6697149243333190288atural(Fa,A4))
      <=> 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_fun296549085072024822_int_o(aTP_Lamm_akk(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9352_even__sum__iff,axiom,
    ! [A4: set_Code_integer,Fa: fun_Co3734316457315071362atural] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),groups8926444216418632897atural(Fa,A4))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),finite4902975817058060853nteger(collect_Code_integer(aa_fun1294772355873590804eger_o(aTP_Lamm_akl(A4),Fa)))) ) ) ).

% even_sum_iff
tff(fact_9353_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(bit_and_not_num(X),Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N5: num] : Xa = aa_num_num(bit0,N5)
             => ( Y != aa_num_option_num(some_num,one) ) ) )
         => ( ( ( X = one )
             => ( ? [N5: num] : Xa = aa_num_num(bit1,N5)
               => ( Y != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa_num_num(bit0,M5) )
                 => ( ( Xa = one )
                   => ( Y != aa_num_option_num(some_num,aa_num_num(bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa = aa_num_num(bit0,N5) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa = aa_num_num(bit1,N5) )
                         => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit1,M5) )
                       => ( ( Xa = one )
                         => ( Y != aa_num_option_num(some_num,aa_num_num(bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa = aa_num_num(bit0,N5) )
                             => ( Y != aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_mr),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa = aa_num_num(bit1,N5) )
                               => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9354_card__Collect__less__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,N2))) = N2 ).

% card_Collect_less_nat
tff(fact_9355_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_9356_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.bounded_iff
tff(fact_9357_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_9358_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.bounded_iff
tff(fact_9359_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb2
tff(fact_9360_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_9361_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_9362_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_9363_max_Oabsorb1,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(ord_max_rat,A),B) = A ) ) ).

% max.absorb1
tff(fact_9364_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_9365_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_9366_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_9367_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z3) ) ) ).

% max_less_iff_conj
tff(fact_9368_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)),Z3)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z3) ) ) ).

% max_less_iff_conj
tff(fact_9369_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z3)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z3) ) ) ).

% max_less_iff_conj
tff(fact_9370_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z3) ) ) ).

% max_less_iff_conj
tff(fact_9371_max_Oabsorb4,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(ord_max_rat,A),B) = B ) ) ).

% max.absorb4
tff(fact_9372_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_9373_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_9374_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_9375_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb3
tff(fact_9376_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_9377_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_9378_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_9379_max__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(ord_ma7524802468073614006at_nat,X),bot_bo2099793752762293965at_nat) = X ).

% max_bot2
tff(fact_9380_max__bot2,axiom,
    ! [X: set_o] : ord_max_set_o(X,bot_bot_set_o) = X ).

% max_bot2
tff(fact_9381_max__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,X),bot_bot_set_nat) = X ).

% max_bot2
tff(fact_9382_max__bot2,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X),bot_bot_set_int) = X ).

% max_bot2
tff(fact_9383_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_9384_max__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(ord_ma7524802468073614006at_nat,bot_bo2099793752762293965at_nat),X) = X ).

% max_bot
tff(fact_9385_max__bot,axiom,
    ! [X: set_o] : ord_max_set_o(bot_bot_set_o,X) = X ).

% max_bot
tff(fact_9386_max__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_max_set_nat,bot_bot_set_nat),X) = X ).

% max_bot
tff(fact_9387_max__bot,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,bot_bot_set_int),X) = X ).

% max_bot
tff(fact_9388_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_9389_map__option__eq__Some,axiom,
    ! [Fa: fun_num_num,Xo: option_num,Y: num] :
      ( ( aa_opt5369159329878751634on_num(map_option_num_num(Fa),Xo) = aa_num_option_num(some_num,Y) )
    <=> ? [Z: num] :
          ( ( Xo = aa_num_option_num(some_num,Z) )
          & ( aa_num_num(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9390_map__option__eq__Some,axiom,
    ! [Fa: fun_nu147038076112158225it_nat,Xo: option_num,Y: produc7388388658123137530it_nat] :
      ( ( map_op2490657901765117089it_nat(Fa,Xo) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y) )
    <=> ? [Z: num] :
          ( ( Xo = aa_num_option_num(some_num,Z) )
          & ( aa_num3651137092533555282it_nat(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9391_map__option__eq__Some,axiom,
    ! [Fa: fun_nu8867722426328123536it_nat,Xo: option_num,Y: produc3260487557148687353it_nat] :
      ( ( map_op7586128837645442720it_nat(Fa,Xo) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Y) )
    <=> ? [Z: num] :
          ( ( Xo = aa_num_option_num(some_num,Z) )
          & ( aa_num8746608028413880913it_nat(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9392_map__option__eq__Some,axiom,
    ! [Fa: fun_nu5234077238548837151it_nat,Xo: option_num,Y: produc8664842809031399944it_nat] :
      ( ( map_op8118133509425879471it_nat(Fa,Xo) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Y) )
    <=> ? [Z: num] :
          ( ( Xo = aa_num_option_num(some_num,Z) )
          & ( aa_num7171627272556384352it_nat(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9393_map__option__eq__Some,axiom,
    ! [Fa: fun_Pr6180708418112149265at_num,Xo: option4065278094766928714it_nat,Y: num] :
      ( ( map_op6233829773304596897at_num(Fa,Xo) = aa_num_option_num(some_num,Y) )
    <=> ? [Z: produc7388388658123137530it_nat] :
          ( ( Xo = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Z) )
          & ( aa_Pro7394308964073035090at_num(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9394_map__option__eq__Some,axiom,
    ! [Fa: fun_Pr5112106203349071120at_num,Xo: option3562590408128118217it_nat,Y: num] :
      ( ( map_op5165227558541518752at_num(Fa,Xo) = aa_num_option_num(some_num,Y) )
    <=> ? [Z: produc3260487557148687353it_nat] :
          ( ( Xo = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Z) )
          & ( aa_Pro6325706749309956945at_num(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9395_map__option__eq__Some,axiom,
    ! [Fa: fun_Pr1075626608990047775at_num,Xo: option8956607266484857688it_nat,Y: num] :
      ( ( map_op2731889312448867759at_num(Fa,Xo) = aa_num_option_num(some_num,Y) )
    <=> ? [Z: produc8664842809031399944it_nat] :
          ( ( Xo = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Z) )
          & ( aa_Pro1785383075579372640at_num(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9396_map__option__eq__Some,axiom,
    ! [Fa: fun_Pr2137375630576721649it_nat,Xo: option4065278094766928714it_nat,Y: produc7388388658123137530it_nat] :
      ( ( map_op8581638957091552385it_nat(Fa,Xo) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y) )
    <=> ? [Z: produc7388388658123137530it_nat] :
          ( ( Xo = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Z) )
          & ( aa_Pro88785284265212210it_nat(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9397_map__option__eq__Some,axiom,
    ! [Fa: fun_Pr1155457543210892272it_nat,Xo: option3562590408128118217it_nat,Y: produc7388388658123137530it_nat] :
      ( ( map_op7599720869725723008it_nat(Fa,Xo) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y) )
    <=> ? [Z: produc3260487557148687353it_nat] :
          ( ( Xo = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Z) )
          & ( aa_Pro8330239233754158641it_nat(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9398_map__option__eq__Some,axiom,
    ! [Fa: fun_Pr7227498785783423999it_nat,Xo: option8956607266484857688it_nat,Y: produc7388388658123137530it_nat] :
      ( ( map_op604423251913076367it_nat(Fa,Xo) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Y) )
    <=> ? [Z: produc8664842809031399944it_nat] :
          ( ( Xo = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Z) )
          & ( aa_Pro2657624141182981696it_nat(Fa,Z) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_9399_None__eq__map__option__iff,axiom,
    ! [Fa: fun_num_num,X: option_num] :
      ( ( none_num = aa_opt5369159329878751634on_num(map_option_num_num(Fa),X) )
    <=> ( X = none_num ) ) ).

% None_eq_map_option_iff
tff(fact_9400_map__option__is__None,axiom,
    ! [Fa: fun_num_num,Opt: option_num] :
      ( ( aa_opt5369159329878751634on_num(map_option_num_num(Fa),Opt) = none_num )
    <=> ( Opt = none_num ) ) ).

% map_option_is_None
tff(fact_9401_option_Omap__disc__iff,axiom,
    ! [Fa: fun_num_num,A: option_num] :
      ( ( aa_opt5369159329878751634on_num(map_option_num_num(Fa),A) = none_num )
    <=> ( A = none_num ) ) ).

% option.map_disc_iff
tff(fact_9402_max__0R,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),zero_zero_nat) = N2 ).

% max_0R
tff(fact_9403_max__0L,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N2) = N2 ).

% max_0L
tff(fact_9404_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_9405_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_9406_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_9407_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_9408_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)) ).

% max_Suc_Suc
tff(fact_9409_card__Collect__le__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bx,N2))) = aa_nat_nat(suc,N2) ).

% card_Collect_le_nat
tff(fact_9410_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,U),aa_num_Code_natural(numera5444537566228673987atural,V)) = $ite(aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_num_Code_natural(numera5444537566228673987atural,U)),aa_num_Code_natural(numera5444537566228673987atural,V)),aa_num_Code_natural(numera5444537566228673987atural,V),aa_num_Code_natural(numera5444537566228673987atural,U)) ).

% max_number_of(1)
tff(fact_9411_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(1)
tff(fact_9412_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_9413_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_9414_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_9415_case__map__option,axiom,
    ! [Ga: fun_Pr5876161306168176259nteger,Ha: fun_Co1561183069634336733nteger,Fa: fun_Co8499238673658658103nteger,X: option_Code_integer] : case_o6517734911099809389nteger(Ga,Ha,aa_opt5394215800039610008nteger(map_op3669829223712506439nteger(Fa),X)) = case_o6517734911099809389nteger(Ga,comp_C7848664941760151970nteger(Ha,Fa),X) ).

% case_map_option
tff(fact_9416_case__map__option,axiom,
    ! [Ga: fun_Pr5876161306168176259nteger,Ha: fun_fu4800041728187187127nteger,Fa: fun_Co1981900195452921617nteger,X: option_Code_integer] : case_o4513113946662230855nteger(Ga,Ha,aa_opt6635412093559109234nteger(map_op3919597019539041057nteger(Fa),X)) = case_o6517734911099809389nteger(Ga,comp_f6739083541385432444nteger(Ha,Fa),X) ).

% case_map_option
tff(fact_9417_case__map__option,axiom,
    ! [Ga: nat,Ha: fun_int_nat,Fa: fun_int_int,X: option_int] : case_option_nat_int(Ga,Ha,aa_opt3115652020397860022on_int(map_option_int_int(Fa),X)) = case_option_nat_int(Ga,comp_int_nat_int(Ha,Fa),X) ).

% case_map_option
tff(fact_9418_case__map__option,axiom,
    ! [Ga: int,Ha: fun_num_int,Fa: fun_num_num,X: option_num] : aa_option_num_int(case_option_int_num(Ga,Ha),aa_opt5369159329878751634on_num(map_option_num_num(Fa),X)) = aa_option_num_int(case_option_int_num(Ga,comp_num_int_num(Ha,Fa)),X) ).

% case_map_option
tff(fact_9419_case__map__option,axiom,
    ! [Ga: option_num,Ha: fun_num_option_num,Fa: fun_num_num,X: option_num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(Ga,Ha),aa_opt5369159329878751634on_num(map_option_num_num(Fa),X)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(Ga,comp_n6731957995704128387um_num(Ha,Fa)),X) ).

% case_map_option
tff(fact_9420_case__map__option,axiom,
    ! [Ga: num,Ha: fun_num_num,Fa: fun_num_num,X: option_num] : aa_option_num_num(case_option_num_num(Ga,Ha),aa_opt5369159329878751634on_num(map_option_num_num(Fa),X)) = aa_option_num_num(case_option_num_num(Ga,comp_num_num_num(Ha,Fa)),X) ).

% case_map_option
tff(fact_9421_prod__constant,axiom,
    ! [Y: nat,A4: set_int] : groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ff,Y),A4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_int(A4)) ).

% prod_constant
tff(fact_9422_prod__constant,axiom,
    ! [Y: nat,A4: set_list_nat] : groups2907647131375434839at_nat(aTP_Lamm_akm(Y),A4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_list_nat(A4)) ).

% prod_constant
tff(fact_9423_prod__constant,axiom,
    ! [Y: nat,A4: set_set_nat] : groups4248547760180025341at_nat(aTP_Lamm_akn(Y),A4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),finite_card_set_nat(A4)) ).

% prod_constant
tff(fact_9424_prod__constant,axiom,
    ! [Y: int,A4: set_list_nat] : groups2905156660866384563at_int(aTP_Lamm_ako(Y),A4) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_list_nat(A4)) ).

% prod_constant
tff(fact_9425_prod__constant,axiom,
    ! [Y: int,A4: set_set_nat] : groups4246057289670975065at_int(aTP_Lamm_akp(Y),A4) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_set_nat(A4)) ).

% prod_constant
tff(fact_9426_prod__constant,axiom,
    ! [Y: nat,A4: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_akq,Y)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_set_nat_nat(finite_card_nat,A4)) ).

% prod_constant
tff(fact_9427_prod__constant,axiom,
    ! [Y: int,A4: set_nat] : aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_akr,Y)),A4) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_set_nat_nat(finite_card_nat,A4)) ).

% prod_constant
tff(fact_9428_prod__constant,axiom,
    ! [Y: int,A4: set_int] : groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_aks,Y),A4) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),finite_card_int(A4)) ).

% prod_constant
tff(fact_9429_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(2)
tff(fact_9430_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_9431_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_9432_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(3)
tff(fact_9433_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_9434_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_9435_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(4)
tff(fact_9436_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_9437_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_9438_sum__constant,axiom,
    ! [Y: rat,A4: set_nat] : aa_set_nat_rat(groups2906978787729119204at_rat(aTP_Lamm_akt(Y)),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_9439_sum__constant,axiom,
    ! [Y: rat,A4: set_int] : groups3906332499630173760nt_rat(aTP_Lamm_aku(Y),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_9440_sum__constant,axiom,
    ! [Y: int,A4: set_nat] : aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_akr,Y)),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_9441_sum__constant,axiom,
    ! [Y: nat,A4: set_int] : groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ff,Y),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_9442_sum__constant,axiom,
    ! [Y: code_integer,A4: set_nat] : aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_akv(Y)),A4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_9443_sum__constant,axiom,
    ! [Y: code_integer,A4: set_int] : groups7873554091576472773nteger(aTP_Lamm_akw(Y),A4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_9444_sum__constant,axiom,
    ! [Y: nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_akq,Y)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,A4))),Y) ).

% sum_constant
tff(fact_9445_sum__constant,axiom,
    ! [Y: int,A4: set_int] : groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_aks,Y),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_int(A4))),Y) ).

% sum_constant
tff(fact_9446_sum__constant,axiom,
    ! [Y: rat,A4: set_list_nat] : groups3760926236672600436at_rat(aTP_Lamm_akx(Y),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_list_nat(A4))),Y) ).

% sum_constant
tff(fact_9447_sum__constant,axiom,
    ! [Y: rat,A4: set_set_nat] : groups7659867448343625626at_rat(aTP_Lamm_aky(Y),A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_set_nat(A4))),Y) ).

% sum_constant
tff(fact_9448_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9449_sum_Oinsert_H,axiom,
    ! [I5: set_o,P2: fun_o_Code_integer,I2: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_aby(I5),P2)))
     => ( groups1402912129352969042nteger(P2,insert_o((I2),I5)) = $ite(aa_set_o_o(member_o((I2)),I5),groups1402912129352969042nteger(P2,I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(P2,(I2))),groups1402912129352969042nteger(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9450_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P2: fun_nat_Code_integer,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3480067038916266681_nat_o(aTP_Lamm_aca(I5),P2)))
     => ( aa_set871585252387937313nteger(groups555127423416065298nteger(P2),insert_nat(I2,I5)) = $ite(aa_set_nat_o(member_nat(I2),I5),aa_set871585252387937313nteger(groups555127423416065298nteger(P2),I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(P2,I2)),aa_set871585252387937313nteger(groups555127423416065298nteger(P2),I5))) ) ) ).

% sum.insert'
tff(fact_9451_sum_Oinsert_H,axiom,
    ! [I5: set_int,P2: fun_int_Code_integer,I2: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun3336537925265328129_int_o(aTP_Lamm_acc(I5),P2)))
     => ( groups926780983652909934nteger(P2,insert_int(I2,I5)) = $ite(aa_set_int_o(member_int(I2),I5),groups926780983652909934nteger(P2,I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(P2,I2)),groups926780983652909934nteger(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9452_sum_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P2: fun_Co8499238673658658103nteger,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun7159535342743012639eger_o(aTP_Lamm_ace(I5),P2)))
     => ( groups910942671188738463nteger(P2,insert_Code_integer(I2,I5)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(I2),I5),groups910942671188738463nteger(P2,I5),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(P2,I2)),groups910942671188738463nteger(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9453_sum_Oinsert_H,axiom,
    ! [I5: set_o,P2: fun_o_rat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_acg(I5),P2)))
     => ( groups3921277224699582669_o_rat(P2,insert_o((I2),I5)) = $ite(aa_set_o_o(member_o((I2)),I5),groups3921277224699582669_o_rat(P2,I5),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(P2,(I2))),groups3921277224699582669_o_rat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9454_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P2: fun_nat_rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_aci(I5),P2)))
     => ( aa_set_nat_rat(groups1351286907653491341at_rat(P2),insert_nat(I2,I5)) = $ite(aa_set_nat_o(member_nat(I2),I5),aa_set_nat_rat(groups1351286907653491341at_rat(P2),I5),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(P2,I2)),aa_set_nat_rat(groups1351286907653491341at_rat(P2),I5))) ) ) ).

% sum.insert'
tff(fact_9455_sum_Oinsert_H,axiom,
    ! [I5: set_int,P2: fun_int_rat,I2: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ack(I5),P2)))
     => ( groups2350640619554545897nt_rat(P2,insert_int(I2,I5)) = $ite(aa_set_int_o(member_int(I2),I5),groups2350640619554545897nt_rat(P2,I5),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(P2,I2)),groups2350640619554545897nt_rat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9456_sum_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P2: fun_Code_integer_rat,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_acm(I5),P2)))
     => ( groups8878813951405302554er_rat(P2,insert_Code_integer(I2,I5)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(I2),I5),groups8878813951405302554er_rat(P2,I5),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(P2,I2)),groups8878813951405302554er_rat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9457_sum_Oinsert_H,axiom,
    ! [I5: set_o,P2: fun_o_nat,I2: $o] :
      ( aa_set_o_o(finite_finite_o,collect_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aco(I5),P2)))
     => ( groups4556407284786078405_o_nat(P2,insert_o((I2),I5)) = $ite(aa_set_o_o(member_o((I2)),I5),groups4556407284786078405_o_nat(P2,I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(P2,(I2))),groups4556407284786078405_o_nat(P2,I5))) ) ) ).

% sum.insert'
tff(fact_9458_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P2: fun_nat_nat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_acq(I5),P2)))
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(P2),insert_nat(I2,I5)) = $ite(aa_set_nat_o(member_nat(I2),I5),aa_set_nat_nat(groups1986416967739987077at_nat(P2),I5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(P2,I2)),aa_set_nat_nat(groups1986416967739987077at_nat(P2),I5))) ) ) ).

% sum.insert'
tff(fact_9459_card__doubleton__eq__2__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( finite3771342082235030671at_nat(insert9069300056098147895at_nat(A,insert9069300056098147895at_nat(B,bot_bo228742789529271731at_nat))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9460_card__doubleton__eq__2__iff,axiom,
    ! [A: list_nat,B: list_nat] :
      ( ( finite_card_list_nat(insert_list_nat(A,insert_list_nat(B,bot_bot_set_list_nat))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9461_card__doubleton__eq__2__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_card_set_nat(insert_set_nat(A,insert_set_nat(B,bot_bot_set_set_nat))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9462_card__doubleton__eq__2__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( finite711546835091564841at_nat(insert8211810215607154385at_nat(A,insert8211810215607154385at_nat(B,bot_bo2099793752762293965at_nat))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9463_card__doubleton__eq__2__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( finite_card_o(insert_o((A),insert_o((B),bot_bot_set_o))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ~ ( (A)
        <=> (B) ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9464_card__doubleton__eq__2__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,insert_nat(A,insert_nat(B,bot_bot_set_nat))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9465_card__doubleton__eq__2__iff,axiom,
    ! [A: int,B: int] :
      ( ( finite_card_int(insert_int(A,insert_int(B,bot_bot_set_int))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
    <=> ( A != B ) ) ).

% card_doubleton_eq_2_iff
tff(fact_9466_sum__of__bool__eq,axiom,
    ! [A4: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4541462559716669496nt_nat(aTP_Lamm_akz(Pa),A4) = aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9467_sum__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups7237345082560585321er_nat(aTP_Lamm_ala(Pa),A4) = aa_nat_nat(semiri1316708129612266289at_nat,finite4902975817058060853nteger(inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9468_sum__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups7234854612051535045er_int(aTP_Lamm_alb(Pa),A4) = aa_nat_int(semiri1314217659103216013at_int,finite4902975817058060853nteger(inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9469_sum__of__bool__eq,axiom,
    ! [A4: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_alc(Pa)),A4) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9470_sum__of__bool__eq,axiom,
    ! [A4: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups7873554091576472773nteger(aTP_Lamm_ald(Pa),A4) = aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9471_sum__of__bool__eq,axiom,
    ! [A4: set_Code_integer,Pa: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups879477027807139574nteger(aTP_Lamm_ale(Pa),A4) = aa_nat_Code_integer(semiri4939895301339042750nteger,finite4902975817058060853nteger(inf_in1364745209274528805nteger(A4,collect_Code_integer(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9472_sum__of__bool__eq,axiom,
    ! [A4: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set871585252387937313nteger(groups7501900531339628137nteger(aTP_Lamm_alf(Pa)),A4) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9473_sum__of__bool__eq,axiom,
    ! [A4: set_nat,Pa: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_alg(Pa)),A4) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),collect_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9474_sum__of__bool__eq,axiom,
    ! [A4: set_int,Pa: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( groups4538972089207619220nt_int(aTP_Lamm_alh(Pa),A4) = aa_nat_int(semiri1314217659103216013at_int,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A4),collect_int(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9475_sum__of__bool__eq,axiom,
    ! [A4: set_list_nat,Pa: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite8100373058378681591st_nat(A4)
       => ( groups4396056296759096172at_nat(aTP_Lamm_ali(Pa),A4) = aa_nat_nat(semiri1316708129612266289at_nat,finite_card_list_nat(inf_inf_set_list_nat(A4,collect_list_nat(Pa)))) ) ) ) ).

% sum_of_bool_eq
tff(fact_9476_option_Osimps_I9_J,axiom,
    ! [Fa: fun_num_num,X2: num] : aa_opt5369159329878751634on_num(map_option_num_num(Fa),aa_num_option_num(some_num,X2)) = aa_num_option_num(some_num,aa_num_num(Fa,X2)) ).

% option.simps(9)
tff(fact_9477_option_Osimps_I9_J,axiom,
    ! [Fa: fun_Pr6180708418112149265at_num,X2: produc7388388658123137530it_nat] : map_op6233829773304596897at_num(Fa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2)) = aa_num_option_num(some_num,aa_Pro7394308964073035090at_num(Fa,X2)) ).

% option.simps(9)
tff(fact_9478_option_Osimps_I9_J,axiom,
    ! [Fa: fun_Pr5112106203349071120at_num,X2: produc3260487557148687353it_nat] : map_op5165227558541518752at_num(Fa,aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,X2)) = aa_num_option_num(some_num,aa_Pro6325706749309956945at_num(Fa,X2)) ).

% option.simps(9)
tff(fact_9479_option_Osimps_I9_J,axiom,
    ! [Fa: fun_Pr1075626608990047775at_num,X2: produc8664842809031399944it_nat] : map_op2731889312448867759at_num(Fa,aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,X2)) = aa_num_option_num(some_num,aa_Pro1785383075579372640at_num(Fa,X2)) ).

% option.simps(9)
tff(fact_9480_option_Osimps_I9_J,axiom,
    ! [Fa: fun_nu147038076112158225it_nat,X2: num] : map_op2490657901765117089it_nat(Fa,aa_num_option_num(some_num,X2)) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_num3651137092533555282it_nat(Fa,X2)) ).

% option.simps(9)
tff(fact_9481_option_Osimps_I9_J,axiom,
    ! [Fa: fun_nu8867722426328123536it_nat,X2: num] : map_op7586128837645442720it_nat(Fa,aa_num_option_num(some_num,X2)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_num8746608028413880913it_nat(Fa,X2)) ).

% option.simps(9)
tff(fact_9482_option_Osimps_I9_J,axiom,
    ! [Fa: fun_nu5234077238548837151it_nat,X2: num] : map_op8118133509425879471it_nat(Fa,aa_num_option_num(some_num,X2)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_num7171627272556384352it_nat(Fa,X2)) ).

% option.simps(9)
tff(fact_9483_option_Osimps_I9_J,axiom,
    ! [Fa: fun_Pr2137375630576721649it_nat,X2: produc7388388658123137530it_nat] : map_op8581638957091552385it_nat(Fa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2)) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro88785284265212210it_nat(Fa,X2)) ).

% option.simps(9)
tff(fact_9484_option_Osimps_I9_J,axiom,
    ! [Fa: fun_Pr1634687943937911152it_nat,X2: produc7388388658123137530it_nat] : map_op4453737856117102208it_nat(Fa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2)) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro5184256220145537841it_nat(Fa,X2)) ).

% option.simps(9)
tff(fact_9485_option_Osimps_I9_J,axiom,
    ! [Fa: fun_Pr7387299081059808255it_nat,X2: produc7388388658123137530it_nat] : map_op8924890681001947535it_nat(Fa,aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,X2)) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_Pro1754719533417077056it_nat(Fa,X2)) ).

% option.simps(9)
tff(fact_9486_map__option__cong,axiom,
    ! [X: option_num,Y: option_num,Fa: fun_num_num,Ga: fun_num_num] :
      ( ( X = Y )
     => ( ! [A3: num] :
            ( ( Y = aa_num_option_num(some_num,A3) )
           => ( aa_num_num(Fa,A3) = aa_num_num(Ga,A3) ) )
       => ( aa_opt5369159329878751634on_num(map_option_num_num(Fa),X) = aa_opt5369159329878751634on_num(map_option_num_num(Ga),Y) ) ) ) ).

% map_option_cong
tff(fact_9487_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_9488_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_9489_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,X)),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_9490_map__option_Ocomp,axiom,
    ! [Fa: fun_Co1561183069634336733nteger,Ga: fun_Co8499238673658658103nteger] : comp_o1169693055247830034nteger(map_op3703932585503671661nteger(Fa),map_op3669829223712506439nteger(Ga)) = map_op3703932585503671661nteger(comp_C7848664941760151970nteger(Fa,Ga)) ).

% map_option.comp
tff(fact_9491_map__option_Ocomp,axiom,
    ! [Fa: fun_fu4800041728187187127nteger,Ga: fun_Co1981900195452921617nteger] : comp_o7605500609165425388nteger(map_op7435587552923436871nteger(Fa),map_op3919597019539041057nteger(Ga)) = map_op3703932585503671661nteger(comp_f6739083541385432444nteger(Fa,Ga)) ).

% map_option.comp
tff(fact_9492_map__option_Ocomp,axiom,
    ! [Fa: fun_int_nat,Ga: fun_int_int] : comp_o4824269118391204861on_int(map_option_int_nat(Fa),map_option_int_int(Ga)) = map_option_int_nat(comp_int_nat_int(Fa,Ga)) ).

% map_option.comp
tff(fact_9493_map__option_Ocomp,axiom,
    ! [Fa: fun_num_num,Ga: fun_num_num] : comp_o8931257242830428707on_num(map_option_num_num(Fa),map_option_num_num(Ga)) = map_option_num_num(comp_num_num_num(Fa,Ga)) ).

% map_option.comp
tff(fact_9494_option_Omap__comp,axiom,
    ! [Ga: fun_Co1561183069634336733nteger,Fa: fun_Co8499238673658658103nteger,V: option_Code_integer] : aa_opt8008171047296870846nteger(map_op3703932585503671661nteger(Ga),aa_opt5394215800039610008nteger(map_op3669829223712506439nteger(Fa),V)) = aa_opt8008171047296870846nteger(map_op3703932585503671661nteger(comp_C7848664941760151970nteger(Ga,Fa)),V) ).

% option.map_comp
tff(fact_9495_option_Omap__comp,axiom,
    ! [Ga: fun_fu4800041728187187127nteger,Fa: fun_Co1981900195452921617nteger,V: option_Code_integer] : aa_opt235805188696355736nteger(map_op7435587552923436871nteger(Ga),aa_opt6635412093559109234nteger(map_op3919597019539041057nteger(Fa),V)) = aa_opt8008171047296870846nteger(map_op3703932585503671661nteger(comp_f6739083541385432444nteger(Ga,Fa)),V) ).

% option.map_comp
tff(fact_9496_option_Omap__comp,axiom,
    ! [Ga: fun_int_nat,Fa: fun_int_int,V: option_int] : aa_opt7293503039907056730on_nat(map_option_int_nat(Ga),aa_opt3115652020397860022on_int(map_option_int_int(Fa),V)) = aa_opt7293503039907056730on_nat(map_option_int_nat(comp_int_nat_int(Ga,Fa)),V) ).

% option.map_comp
tff(fact_9497_option_Omap__comp,axiom,
    ! [Ga: fun_num_num,Fa: fun_num_num,V: option_num] : aa_opt5369159329878751634on_num(map_option_num_num(Ga),aa_opt5369159329878751634on_num(map_option_num_num(Fa),V)) = aa_opt5369159329878751634on_num(map_option_num_num(comp_num_num_num(Ga,Fa)),V) ).

% option.map_comp
tff(fact_9498_map__option_Ocompositionality,axiom,
    ! [Fa: fun_Co1561183069634336733nteger,Ga: fun_Co8499238673658658103nteger,Option: option_Code_integer] : aa_opt8008171047296870846nteger(map_op3703932585503671661nteger(Fa),aa_opt5394215800039610008nteger(map_op3669829223712506439nteger(Ga),Option)) = aa_opt8008171047296870846nteger(map_op3703932585503671661nteger(comp_C7848664941760151970nteger(Fa,Ga)),Option) ).

% map_option.compositionality
tff(fact_9499_map__option_Ocompositionality,axiom,
    ! [Fa: fun_fu4800041728187187127nteger,Ga: fun_Co1981900195452921617nteger,Option: option_Code_integer] : aa_opt235805188696355736nteger(map_op7435587552923436871nteger(Fa),aa_opt6635412093559109234nteger(map_op3919597019539041057nteger(Ga),Option)) = aa_opt8008171047296870846nteger(map_op3703932585503671661nteger(comp_f6739083541385432444nteger(Fa,Ga)),Option) ).

% map_option.compositionality
tff(fact_9500_map__option_Ocompositionality,axiom,
    ! [Fa: fun_int_nat,Ga: fun_int_int,Option: option_int] : aa_opt7293503039907056730on_nat(map_option_int_nat(Fa),aa_opt3115652020397860022on_int(map_option_int_int(Ga),Option)) = aa_opt7293503039907056730on_nat(map_option_int_nat(comp_int_nat_int(Fa,Ga)),Option) ).

% map_option.compositionality
tff(fact_9501_map__option_Ocompositionality,axiom,
    ! [Fa: fun_num_num,Ga: fun_num_num,Option: option_num] : aa_opt5369159329878751634on_num(map_option_num_num(Fa),aa_opt5369159329878751634on_num(map_option_num_num(Ga),Option)) = aa_opt5369159329878751634on_num(map_option_num_num(comp_num_num_num(Fa,Ga)),Option) ).

% map_option.compositionality
tff(fact_9502_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ).

% nat_mult_max_left
tff(fact_9503_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_max_right
tff(fact_9504_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_9505_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_9506_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_9507_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_9508_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_9509_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_9510_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_9511_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_9512_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_9513_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_9514_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_9515_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_9516_max_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),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,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_9517_max_Ostrict__boundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_9518_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_9519_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_9520_less__max__iff__disj,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y) ) ) ).

% less_max_iff_disj
tff(fact_9521_less__max__iff__disj,axiom,
    ! [Z3: num,X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),Y) ) ) ).

% less_max_iff_disj
tff(fact_9522_less__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),Y) ) ) ).

% less_max_iff_disj
tff(fact_9523_less__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),Y) ) ) ).

% less_max_iff_disj
tff(fact_9524_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),Q4)) ).

% nat_add_max_left
tff(fact_9525_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ).

% nat_add_max_right
tff(fact_9526_sup__max,axiom,
    sup_sup_int = ord_max_int ).

% sup_max
tff(fact_9527_sup__max,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_max
tff(fact_9528_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9529_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9530_option_Omap__ident,axiom,
    ! [T6: option_num] : aa_opt5369159329878751634on_num(map_option_num_num(aTP_Lamm_alj),T6) = T6 ).

% option.map_ident
tff(fact_9531_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.coboundedI2
tff(fact_9532_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI2
tff(fact_9533_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_9534_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_9535_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.coboundedI1
tff(fact_9536_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI1
tff(fact_9537_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_9538_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_9539_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_9540_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_9541_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_9542_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_9543_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_9544_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_9545_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_9546_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_9547_le__max__iff__disj,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),X)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z3),Y) ) ) ).

% le_max_iff_disj
tff(fact_9548_le__max__iff__disj,axiom,
    ! [Z3: num,X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),X)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z3),Y) ) ) ).

% le_max_iff_disj
tff(fact_9549_le__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),X)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y) ) ) ).

% le_max_iff_disj
tff(fact_9550_le__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),X)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),Y) ) ) ).

% le_max_iff_disj
tff(fact_9551_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ).

% max.cobounded2
tff(fact_9552_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded2
tff(fact_9553_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_9554_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_9555_max_Ocobounded1,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(ord_max_rat,A),B)) ).

% max.cobounded1
tff(fact_9556_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded1
tff(fact_9557_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_9558_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_9559_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.order_iff
tff(fact_9560_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_9561_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_9562_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_9563_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: 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,C2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_9564_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_9565_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_9566_max_OboundedI,axiom,
    ! [B: int,A: int,C2: 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,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_9567_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.boundedE
tff(fact_9568_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.boundedE
tff(fact_9569_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.boundedE
tff(fact_9570_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.boundedE
tff(fact_9571_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_9572_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_9573_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_9574_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_9575_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.orderE
tff(fact_9576_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_9577_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_9578_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_9579_max_Omono,axiom,
    ! [C2: rat,A: rat,D2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,C2),D2)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ) ).

% max.mono
tff(fact_9580_max_Omono,axiom,
    ! [C2: num,A: num,D2: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C2),D2)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ) ).

% max.mono
tff(fact_9581_max_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_9582_max_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C2),D2)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_9583_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X),Y)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_9584_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_9585_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_9586_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_9587_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_9588_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Y),X)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_9589_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_9590_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),X)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y) = X ) ) ).

% max_absorb1
tff(fact_9591_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_9592_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_9593_max__def,axiom,
    ! [A: set_int,B: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,A),B) = $ite(aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B),B,A) ).

% max_def
tff(fact_9594_max__def,axiom,
    ! [A: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ).

% max_def
tff(fact_9595_max__def,axiom,
    ! [A: num,B: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ).

% max_def
tff(fact_9596_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_9597_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_9598_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Y),Z3)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z3)) ).

% max_add_distrib_right
tff(fact_9599_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z3)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z3)) ).

% max_add_distrib_right
tff(fact_9600_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z3)) ).

% max_add_distrib_right
tff(fact_9601_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z3: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z3) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z3)) ).

% max_add_distrib_left
tff(fact_9602_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z3) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z3)) ).

% max_add_distrib_left
tff(fact_9603_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z3)) ).

% max_add_distrib_left
tff(fact_9604_option_Osimps_I8_J,axiom,
    ! [Fa: fun_num_num] : aa_opt5369159329878751634on_num(map_option_num_num(Fa),none_num) = none_num ).

% option.simps(8)
tff(fact_9605_max__def__raw,axiom,
    ! [X5: set_int,Xa2: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(ord_max_set_int,X5),Xa2) = $ite(aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X5),Xa2),Xa2,X5) ).

% max_def_raw
tff(fact_9606_max__def__raw,axiom,
    ! [X5: rat,Xa2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X5),Xa2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Xa2),Xa2,X5) ).

% max_def_raw
tff(fact_9607_max__def__raw,axiom,
    ! [X5: num,Xa2: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X5),Xa2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Xa2),Xa2,X5) ).

% max_def_raw
tff(fact_9608_max__def__raw,axiom,
    ! [X5: nat,Xa2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X5),Xa2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa2),Xa2,X5) ).

% max_def_raw
tff(fact_9609_max__def__raw,axiom,
    ! [X5: int,Xa2: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X5),Xa2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Xa2),Xa2,X5) ).

% max_def_raw
tff(fact_9610_n__subsets,axiom,
    ! [A4: set_Pr958786334691620121nt_int,K2: nat] :
      ( aa_set6170482117830564987_int_o(finite2998713641127702882nt_int,A4)
     => ( finite4053189226111446337nt_int(collec5210948495886036740nt_int(aa_nat4194932087953772816_int_o(aTP_Lamm_alk(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set4753215920113914285nt_nat(finite6756421564338198497nt_int,A4)),K2) ) ) ).

% n_subsets
tff(fact_9611_n__subsets,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite2364142230527598318st_nat(collect_set_list_nat(aa_nat7721107121239158477_nat_o(aTP_Lamm_all(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_list_nat(A4)),K2) ) ) ).

% n_subsets
tff(fact_9612_n__subsets,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( finite1149291290879098388et_nat(collect_set_set_nat(aa_nat5160323483289836711_nat_o(aTP_Lamm_alm(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_set_nat(A4)),K2) ) ) ).

% n_subsets
tff(fact_9613_n__subsets,axiom,
    ! [A4: set_nat,K2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_aln(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_set_nat_nat(finite_card_nat,A4)),K2) ) ) ).

% n_subsets
tff(fact_9614_n__subsets,axiom,
    ! [A4: set_Code_integer,K2: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( finite1340570857867686763nteger(collec574505750873337192nteger(aa_nat2915507816091602832eger_o(aTP_Lamm_alo(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite4902975817058060853nteger(A4)),K2) ) ) ).

% n_subsets
tff(fact_9615_n__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,K2: nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( finite4356350796350151305at_nat(collec5514110066124741708at_nat(aa_nat1727191234135941576_nat_o(aTP_Lamm_alp(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite711546835091564841at_nat(A4)),K2) ) ) ).

% n_subsets
tff(fact_9616_n__subsets,axiom,
    ! [A4: set_int,K2: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( finite_card_set_int(collect_set_int(aa_nat_fun_set_int_o(aTP_Lamm_alq(A4),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,finite_card_int(A4)),K2) ) ) ).

% n_subsets
tff(fact_9617_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_o_Code_integer,I5: set_o] : groups1402912129352969042nteger(Ga,collect_o(aa_set_o_fun_o_o(aTP_Lamm_alr(Ga),I5))) = groups1402912129352969042nteger(Ga,I5) ).

% sum.non_neutral'
tff(fact_9618_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_nat_Code_integer,I5: set_nat] : aa_set871585252387937313nteger(groups555127423416065298nteger(Ga),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_als(Ga),I5))) = aa_set871585252387937313nteger(groups555127423416065298nteger(Ga),I5) ).

% sum.non_neutral'
tff(fact_9619_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_int_Code_integer,I5: set_int] : groups926780983652909934nteger(Ga,collect_int(aa_set_int_fun_int_o(aTP_Lamm_alt(Ga),I5))) = groups926780983652909934nteger(Ga,I5) ).

% sum.non_neutral'
tff(fact_9620_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_o_rat,I5: set_o] : groups3921277224699582669_o_rat(Ga,collect_o(aa_set_o_fun_o_o(aTP_Lamm_alu(Ga),I5))) = groups3921277224699582669_o_rat(Ga,I5) ).

% sum.non_neutral'
tff(fact_9621_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_nat_rat,I5: set_nat] : aa_set_nat_rat(groups1351286907653491341at_rat(Ga),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_alv(Ga),I5))) = aa_set_nat_rat(groups1351286907653491341at_rat(Ga),I5) ).

% sum.non_neutral'
tff(fact_9622_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_int_rat,I5: set_int] : groups2350640619554545897nt_rat(Ga,collect_int(aa_set_int_fun_int_o(aTP_Lamm_alw(Ga),I5))) = groups2350640619554545897nt_rat(Ga,I5) ).

% sum.non_neutral'
tff(fact_9623_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_o_nat,I5: set_o] : groups4556407284786078405_o_nat(Ga,collect_o(aa_set_o_fun_o_o(aTP_Lamm_alx(Ga),I5))) = groups4556407284786078405_o_nat(Ga,I5) ).

% sum.non_neutral'
tff(fact_9624_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_nat_nat,I5: set_nat] : aa_set_nat_nat(groups1986416967739987077at_nat(Ga),collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_aly(Ga),I5))) = aa_set_nat_nat(groups1986416967739987077at_nat(Ga),I5) ).

% sum.non_neutral'
tff(fact_9625_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_int_nat,I5: set_int] : groups2985770679641041633nt_nat(Ga,collect_int(aa_set_int_fun_int_o(aTP_Lamm_alz(Ga),I5))) = groups2985770679641041633nt_nat(Ga,I5) ).

% sum.non_neutral'
tff(fact_9626_sum_Onon__neutral_H,axiom,
    ! [Ga: fun_o_int,I5: set_o] : groups4553916814277028129_o_int(Ga,collect_o(aa_set_o_fun_o_o(aTP_Lamm_ama(Ga),I5))) = groups4553916814277028129_o_int(Ga,I5) ).

% sum.non_neutral'
tff(fact_9627_option_Osize__gen__o__map,axiom,
    ! [Fa: fun_int_nat,Ga: fun_int_int] : comp_o5073597352248849069on_int(size_option_int(Fa),map_option_int_int(Ga)) = size_option_int(comp_int_nat_int(Fa,Ga)) ).

% option.size_gen_o_map
tff(fact_9628_option_Osize__gen__o__map,axiom,
    ! [Fa: fun_num_nat,Ga: fun_num_num] : comp_o6878144249584144265on_num(size_option_num(Fa),map_option_num_num(Ga)) = size_option_num(comp_num_nat_num(Fa,Ga)) ).

% option.size_gen_o_map
tff(fact_9629_card__subset__eq,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => ( ( finite_card_list_nat(A4) = finite_card_list_nat(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_9630_card__subset__eq,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( ( finite_card_set_nat(A4) = finite_card_set_nat(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_9631_card__subset__eq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_set_nat_nat(finite_card_nat,B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_9632_card__subset__eq,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ( finite4902975817058060853nteger(A4) = finite4902975817058060853nteger(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_9633_card__subset__eq,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => ( ( finite711546835091564841at_nat(A4) = finite711546835091564841at_nat(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_9634_card__subset__eq,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => ( ( finite_card_int(A4) = finite_card_int(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_9635_infinite__arbitrarily__large,axiom,
    ! [A4: set_list_nat,N2: nat] :
      ( ~ finite8100373058378681591st_nat(A4)
     => ? [B7: set_list_nat] :
          ( finite8100373058378681591st_nat(B7)
          & ( finite_card_list_nat(B7) = N2 )
          & ord_le6045566169113846134st_nat(B7,A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_9636_infinite__arbitrarily__large,axiom,
    ! [A4: set_set_nat,N2: nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ? [B7: set_set_nat] :
          ( aa_set_set_nat_o(finite1152437895449049373et_nat,B7)
          & ( finite_card_set_nat(B7) = N2 )
          & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_9637_infinite__arbitrarily__large,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ? [B7: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B7)
          & ( aa_set_nat_nat(finite_card_nat,B7) = N2 )
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_9638_infinite__arbitrarily__large,axiom,
    ! [A4: set_Code_integer,N2: nat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ? [B7: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B7)
          & ( finite4902975817058060853nteger(B7) = N2 )
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_9639_infinite__arbitrarily__large,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ? [B7: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B7)
          & ( finite711546835091564841at_nat(B7) = N2 )
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_9640_infinite__arbitrarily__large,axiom,
    ! [A4: set_int,N2: nat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ? [B7: set_int] :
          ( aa_set_int_o(finite_finite_int,B7)
          & ( finite_card_int(B7) = N2 )
          & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_9641_nat__minus__add__max,axiom,
    ! [N2: 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,N2),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),M) ).

% nat_minus_add_max
tff(fact_9642_card__le__if__inj__on__rel,axiom,
    ! [B4: set_o,A4: set_o,R: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A4)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B4)
                & aa_o_o(aa_o_fun_o_o(R,(A3)),(B9)) ) )
       => ( ! [A12: $o,A23: $o,B3: $o] :
              ( aa_set_o_o(member_o((A12)),A4)
             => ( aa_set_o_o(member_o((A23)),A4)
               => ( aa_set_o_o(member_o((B3)),B4)
                 => ( aa_o_o(aa_o_fun_o_o(R,(A12)),(B3))
                   => ( aa_o_o(aa_o_fun_o_o(R,(A23)),(B3))
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A4)),finite_card_o(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9643_card__le__if__inj__on__rel,axiom,
    ! [B4: set_o,A4: set_nat,R: fun_nat_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B4)
                & aa_o_o(aa_nat_fun_o_o(R,A3),(B9)) ) )
       => ( ! [A12: nat,A23: nat,B3: $o] :
              ( aa_set_nat_o(member_nat(A12),A4)
             => ( aa_set_nat_o(member_nat(A23),A4)
               => ( aa_set_o_o(member_o((B3)),B4)
                 => ( aa_o_o(aa_nat_fun_o_o(R,A12),(B3))
                   => ( aa_o_o(aa_nat_fun_o_o(R,A23),(B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_o(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9644_card__le__if__inj__on__rel,axiom,
    ! [B4: set_o,A4: set_int,R: fun_int_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ? [B9: $o] :
                ( aa_set_o_o(member_o((B9)),B4)
                & aa_o_o(aa_int_fun_o_o(R,A3),(B9)) ) )
       => ( ! [A12: int,A23: int,B3: $o] :
              ( aa_set_int_o(member_int(A12),A4)
             => ( aa_set_int_o(member_int(A23),A4)
               => ( aa_set_o_o(member_o((B3)),B4)
                 => ( aa_o_o(aa_int_fun_o_o(R,A12),(B3))
                   => ( aa_o_o(aa_int_fun_o_o(R,A23),(B3))
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_o(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9645_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_o,R: fun_o_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A4)
           => ? [B9: nat] :
                ( aa_set_nat_o(member_nat(B9),B4)
                & aa_nat_o(aa_o_fun_nat_o(R,(A3)),B9) ) )
       => ( ! [A12: $o,A23: $o,B3: nat] :
              ( aa_set_o_o(member_o((A12)),A4)
             => ( aa_set_o_o(member_o((A23)),A4)
               => ( aa_set_nat_o(member_nat(B3),B4)
                 => ( aa_nat_o(aa_o_fun_nat_o(R,(A12)),B3)
                   => ( aa_nat_o(aa_o_fun_nat_o(R,(A23)),B3)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9646_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ? [B9: nat] :
                ( aa_set_nat_o(member_nat(B9),B4)
                & aa_nat_o(aa_nat_fun_nat_o(R,A3),B9) ) )
       => ( ! [A12: nat,A23: nat,B3: nat] :
              ( aa_set_nat_o(member_nat(A12),A4)
             => ( aa_set_nat_o(member_nat(A23),A4)
               => ( aa_set_nat_o(member_nat(B3),B4)
                 => ( aa_nat_o(aa_nat_fun_nat_o(R,A12),B3)
                   => ( aa_nat_o(aa_nat_fun_nat_o(R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9647_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_int,R: fun_int_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ? [B9: nat] :
                ( aa_set_nat_o(member_nat(B9),B4)
                & aa_nat_o(aa_int_fun_nat_o(R,A3),B9) ) )
       => ( ! [A12: int,A23: int,B3: nat] :
              ( aa_set_int_o(member_int(A12),A4)
             => ( aa_set_int_o(member_int(A23),A4)
               => ( aa_set_nat_o(member_nat(B3),B4)
                 => ( aa_nat_o(aa_int_fun_nat_o(R,A12),B3)
                   => ( aa_nat_o(aa_int_fun_nat_o(R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9648_card__le__if__inj__on__rel,axiom,
    ! [B4: set_int,A4: set_o,R: fun_o_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A4)
           => ? [B9: int] :
                ( aa_set_int_o(member_int(B9),B4)
                & aa_int_o(aa_o_fun_int_o(R,(A3)),B9) ) )
       => ( ! [A12: $o,A23: $o,B3: int] :
              ( aa_set_o_o(member_o((A12)),A4)
             => ( aa_set_o_o(member_o((A23)),A4)
               => ( aa_set_int_o(member_int(B3),B4)
                 => ( aa_int_o(aa_o_fun_int_o(R,(A12)),B3)
                   => ( aa_int_o(aa_o_fun_int_o(R,(A23)),B3)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A4)),finite_card_int(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9649_card__le__if__inj__on__rel,axiom,
    ! [B4: set_int,A4: set_nat,R: fun_nat_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ? [B9: int] :
                ( aa_set_int_o(member_int(B9),B4)
                & aa_int_o(aa_nat_fun_int_o(R,A3),B9) ) )
       => ( ! [A12: nat,A23: nat,B3: int] :
              ( aa_set_nat_o(member_nat(A12),A4)
             => ( aa_set_nat_o(member_nat(A23),A4)
               => ( aa_set_int_o(member_int(B3),B4)
                 => ( aa_int_o(aa_nat_fun_int_o(R,A12),B3)
                   => ( aa_int_o(aa_nat_fun_int_o(R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_int(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9650_card__le__if__inj__on__rel,axiom,
    ! [B4: set_int,A4: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ? [B9: int] :
                ( aa_set_int_o(member_int(B9),B4)
                & aa_int_o(aa_int_fun_int_o(R,A3),B9) ) )
       => ( ! [A12: int,A23: int,B3: int] :
              ( aa_set_int_o(member_int(A12),A4)
             => ( aa_set_int_o(member_int(A23),A4)
               => ( aa_set_int_o(member_int(B3),B4)
                 => ( aa_int_o(aa_int_fun_int_o(R,A12),B3)
                   => ( aa_int_o(aa_int_fun_int_o(R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9651_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Code_integer,A4: set_o,R: fun_o_4021313873839786239eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A4)
           => ? [B9: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(B9),B4)
                & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(R,(A3)),B9) ) )
       => ( ! [A12: $o,A23: $o,B3: code_integer] :
              ( aa_set_o_o(member_o((A12)),A4)
             => ( aa_set_o_o(member_o((A23)),A4)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(B3),B4)
                 => ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(R,(A12)),B3)
                   => ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(R,(A23)),B3)
                     => ( (A12)
                      <=> (A23) ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A4)),finite4902975817058060853nteger(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_9652_card__insert__le,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite3771342082235030671at_nat(A4)),finite3771342082235030671at_nat(insert9069300056098147895at_nat(X,A4))) ).

% card_insert_le
tff(fact_9653_card__insert__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(A4)),finite711546835091564841at_nat(insert8211810215607154385at_nat(X,A4))) ).

% card_insert_le
tff(fact_9654_card__insert__le,axiom,
    ! [A4: set_o,X: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A4)),finite_card_o(insert_o((X),A4))) ).

% card_insert_le
tff(fact_9655_card__insert__le,axiom,
    ! [A4: set_nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,insert_nat(X,A4))) ).

% card_insert_le
tff(fact_9656_card__insert__le,axiom,
    ! [A4: set_int,X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(insert_int(X,A4))) ).

% card_insert_le
tff(fact_9657_card__insert__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(insert_list_nat(X,A4))) ).

% card_insert_le
tff(fact_9658_card__insert__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(insert_set_nat(X,A4))) ).

% card_insert_le
tff(fact_9659_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_o,Ra: fun_o_fun_o_o,K2: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(finite_finite_o,T6)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),T6)
             => ( finite_card_o(collect_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_amb(S2),Ra),(X3)))) = aa_o_nat(K2,(X3)) ) )
         => ( groups8507830703676809646_o_nat(aa_fun8807093888949732041_o_nat(aTP_Lamm_amd(T6),Ra),S2) = groups8507830703676809646_o_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9660_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_int,Ra: fun_o_fun_int_o,K2: fun_int_nat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T6)
             => ( finite_card_o(collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_aaa(S2),Ra),X3))) = aa_int_nat(K2,X3) ) )
         => ( groups8507830703676809646_o_nat(aa_fun2417465416094144205_o_nat(aTP_Lamm_ame(T6),Ra),S2) = groups4541462559716669496nt_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9661_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_Code_integer,Ra: fun_o_4021313873839786239eger_o,K2: fun_Code_integer_nat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T6)
             => ( finite_card_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ajs(S2),Ra),X3))) = aa_Code_integer_nat(K2,X3) ) )
         => ( groups8507830703676809646_o_nat(aa_fun5400594822745170686_o_nat(aTP_Lamm_amg(T6),Ra),S2) = groups7237345082560585321er_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9662_sum__multicount__gen,axiom,
    ! [S2: set_int,T6: set_o,Ra: fun_int_fun_o_o,K2: fun_o_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_o_o(finite_finite_o,T6)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),T6)
             => ( finite_card_int(collect_int(aa_o_fun_int_o(aa_fun5351653887499120666_int_o(aTP_Lamm_amh(S2),Ra),(X3)))) = aa_o_nat(K2,(X3)) ) )
         => ( groups4541462559716669496nt_nat(aa_fun7603103345900947235nt_nat(aTP_Lamm_amj(T6),Ra),S2) = groups8507830703676809646_o_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9663_sum__multicount__gen,axiom,
    ! [S2: set_int,T6: set_int,Ra: fun_int_fun_int_o,K2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T6)
             => ( finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ajt(S2),Ra),X3))) = aa_int_nat(K2,X3) ) )
         => ( groups4541462559716669496nt_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_aml(T6),Ra),S2) = groups4541462559716669496nt_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9664_sum__multicount__gen,axiom,
    ! [S2: set_int,T6: set_Code_integer,Ra: fun_in4891666191172872833eger_o,K2: fun_Code_integer_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T6)
             => ( finite_card_int(collect_int(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_aju(S2),Ra),X3))) = aa_Code_integer_nat(K2,X3) ) )
         => ( groups4541462559716669496nt_nat(aa_fun511150112467848510nt_nat(aTP_Lamm_amn(T6),Ra),S2) = groups7237345082560585321er_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9665_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T6: set_o,Ra: fun_Co480733329897759773un_o_o,K2: fun_o_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_o_o(finite_finite_o,T6)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),T6)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_o_f8517747772616379256eger_o(aa_fun3295616927821077850eger_o(aTP_Lamm_amo(S2),Ra),(X3)))) = aa_o_nat(K2,(X3)) ) )
         => ( groups7237345082560585321er_nat(aa_fun7670077909833392133er_nat(aTP_Lamm_amq(T6),Ra),S2) = groups8507830703676809646_o_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9666_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T6: set_int,Ra: fun_Co8353275306710069249_int_o,K2: fun_int_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set_int_o(finite_finite_int,T6)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T6)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_aaj(S2),Ra),X3))) = aa_int_nat(K2,X3) ) )
         => ( groups7237345082560585321er_nat(aa_fun726042532089134319er_nat(aTP_Lamm_amr(T6),Ra),S2) = groups4541462559716669496nt_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9667_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T6: set_Code_integer,Ra: fun_Co2081214437782444048eger_o,K2: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T6)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T6)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_ams(S2),Ra),X3))) = aa_Code_integer_nat(K2,X3) ) )
         => ( groups7237345082560585321er_nat(aa_fun8504338775591962366er_nat(aTP_Lamm_amu(T6),Ra),S2) = groups7237345082560585321er_nat(K2,T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9668_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_nat,Ra: fun_o_fun_nat_o,K2: fun_nat_nat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_nat_o(finite_finite_nat,T6)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T6)
             => ( finite_card_o(collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(S2),Ra),X3))) = aa_nat_nat(K2,X3) ) )
         => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_amv(T6),Ra),S2) = aa_set_nat_nat(groups3542108847815614940at_nat(K2),T6) ) ) ) ) ).

% sum_multicount_gen
tff(fact_9669_card__eq__sum,axiom,
    ! [A4: set_int] : finite_card_int(A4) = groups4541462559716669496nt_nat(aTP_Lamm_amw,A4) ).

% card_eq_sum
tff(fact_9670_card__eq__sum,axiom,
    ! [A4: set_list_nat] : finite_card_list_nat(A4) = groups4396056296759096172at_nat(aTP_Lamm_amx,A4) ).

% card_eq_sum
tff(fact_9671_card__eq__sum,axiom,
    ! [A4: set_set_nat] : finite_card_set_nat(A4) = groups8294997508430121362at_nat(aTP_Lamm_amy,A4) ).

% card_eq_sum
tff(fact_9672_card__eq__sum,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(finite_card_nat,A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_sr),A4) ).

% card_eq_sum
tff(fact_9673_max__Suc2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N2),aa_nat_fun_nat_nat(aTP_Lamm_amz,N2)),M) ).

% max_Suc2
tff(fact_9674_max__Suc1,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N2)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N2),aa_nat_fun_nat_nat(aTP_Lamm_ana,N2)),M) ).

% max_Suc1
tff(fact_9675_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,Ga: fun_nat_rat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_rat(groups1351286907653491341at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_anb(Ga),Ha)),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_set_nat_rat(groups1351286907653491341at_rat(Ga),I5)),aa_set_nat_rat(groups1351286907653491341at_rat(Ha),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9676_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,Ga: fun_int_rat,Ha: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( groups2350640619554545897nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_anc(Ga),Ha),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2350640619554545897nt_rat(Ga,I5)),groups2350640619554545897nt_rat(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9677_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,Ga: fun_Code_integer_rat,Ha: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( groups8878813951405302554er_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_and(Ga),Ha),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups8878813951405302554er_rat(Ga,I5)),groups8878813951405302554er_rat(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9678_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,Ga: fun_nat_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_nat(groups1986416967739987077at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ov(Ga),Ha)),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups1986416967739987077at_nat(Ga),I5)),aa_set_nat_nat(groups1986416967739987077at_nat(Ha),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9679_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,Ga: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( groups2985770679641041633nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ane(Ga),Ha),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2985770679641041633nt_nat(Ga,I5)),groups2985770679641041633nt_nat(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9680_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,Ga: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( groups290571974637022482er_nat(aa_fun3076867170244285586er_nat(aTP_Lamm_anf(Ga),Ha),I5) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups290571974637022482er_nat(Ga,I5)),groups290571974637022482er_nat(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9681_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,Ga: fun_nat_int,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_int(groups1983926497230936801at_int(aa_fun5565043122120104496at_int(aTP_Lamm_ang(Ga),Ha)),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups1983926497230936801at_int(Ga),I5)),aa_set_nat_int(groups1983926497230936801at_int(Ha),I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9682_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,Ga: fun_int_int,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( groups2983280209131991357nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ow(Ga),Ha),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups2983280209131991357nt_int(Ga,I5)),groups2983280209131991357nt_int(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9683_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,Ga: fun_Code_integer_int,Ha: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( groups288081504127972206er_int(aa_fun3291479149765983562er_int(aTP_Lamm_anh(Ga),Ha),I5) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups288081504127972206er_int(Ga,I5)),groups288081504127972206er_int(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9684_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat,Ha: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I5)
     => ( groups3328076802468863542at_rat(aa_fun1488288443938188604at_rat(aTP_Lamm_ani(Ga),Ha),I5) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups3328076802468863542at_rat(Ga,I5)),groups3328076802468863542at_rat(Ha,I5)) ) ) ).

% sum.distrib_triv'
tff(fact_9685_card__ge__0__finite,axiom,
    ! [A4: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A4))
     => finite8100373058378681591st_nat(A4) ) ).

% card_ge_0_finite
tff(fact_9686_card__ge__0__finite,axiom,
    ! [A4: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A4))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A4) ) ).

% card_ge_0_finite
tff(fact_9687_card__ge__0__finite,axiom,
    ! [A4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4))
     => aa_set_nat_o(finite_finite_nat,A4) ) ).

% card_ge_0_finite
tff(fact_9688_card__ge__0__finite,axiom,
    ! [A4: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A4))
     => aa_set_int_o(finite_finite_int,A4) ) ).

% card_ge_0_finite
tff(fact_9689_card__ge__0__finite,axiom,
    ! [A4: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite4902975817058060853nteger(A4))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ).

% card_ge_0_finite
tff(fact_9690_card__ge__0__finite,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite711546835091564841at_nat(A4))
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ).

% card_ge_0_finite
tff(fact_9691_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_list_nat,C3: nat] :
      ( ! [G3: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G3,F4)
         => ( finite8100373058378681591st_nat(G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(G3)),C3) ) )
     => ( finite8100373058378681591st_nat(F4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9692_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_set_nat,C3: nat] :
      ( ! [G3: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(G3),F4)
         => ( aa_set_set_nat_o(finite1152437895449049373et_nat,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(G3)),C3) ) )
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,F4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9693_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_nat,C3: nat] :
      ( ! [G3: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,G3),F4)
         => ( aa_set_nat_o(finite_finite_nat,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,G3)),C3) ) )
     => ( aa_set_nat_o(finite_finite_nat,F4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9694_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_Code_integer,C3: nat] :
      ( ! [G3: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(G3),F4)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(G3)),C3) ) )
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9695_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,C3: nat] :
      ( ! [G3: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,G3),F4)
         => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(G3)),C3) ) )
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9696_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_int,C3: nat] :
      ( ! [G3: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,G3),F4)
         => ( aa_set_int_o(finite_finite_int,G3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(G3)),C3) ) )
     => ( aa_set_int_o(finite_finite_int,F4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(F4)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_9697_card__seteq,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(B4)),finite_card_list_nat(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_9698_card__seteq,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(B4)),finite_card_set_nat(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_9699_card__seteq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B4)),aa_set_nat_nat(finite_card_nat,A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_9700_card__seteq,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(B4)),finite4902975817058060853nteger(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_9701_card__seteq,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(B4)),finite711546835091564841at_nat(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_9702_card__seteq,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B4)),finite_card_int(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_9703_card__mono,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4)) ) ) ).

% card_mono
tff(fact_9704_card__mono,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4)) ) ) ).

% card_mono
tff(fact_9705_card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ).

% card_mono
tff(fact_9706_card__mono,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4)) ) ) ).

% card_mono
tff(fact_9707_card__mono,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(A4)),finite711546835091564841at_nat(B4)) ) ) ).

% card_mono
tff(fact_9708_card__mono,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B4)) ) ) ).

% card_mono
tff(fact_9709_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_list_nat(S))
     => ~ ! [T8: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T8,S)
           => ( ( finite_card_list_nat(T8) = N2 )
             => ~ finite8100373058378681591st_nat(T8) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9710_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_set_nat(S))
     => ~ ! [T8: set_set_nat] :
            ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(T8),S)
           => ( ( finite_card_set_nat(T8) = N2 )
             => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,T8) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9711_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_set_nat_nat(finite_card_nat,S))
     => ~ ! [T8: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,T8),S)
           => ( ( aa_set_nat_nat(finite_card_nat,T8) = N2 )
             => ~ aa_set_nat_o(finite_finite_nat,T8) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9712_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite4902975817058060853nteger(S))
     => ~ ! [T8: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(T8),S)
           => ( ( finite4902975817058060853nteger(T8) = N2 )
             => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T8) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9713_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite711546835091564841at_nat(S))
     => ~ ! [T8: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,T8),S)
           => ( ( finite711546835091564841at_nat(T8) = N2 )
             => ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T8) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9714_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_int(S))
     => ~ ! [T8: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,T8),S)
           => ( ( finite_card_int(T8) = N2 )
             => ~ aa_set_int_o(finite_finite_int,T8) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_9715_card__less__sym__Diff,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite8100373058378681591st_nat(B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4))),finite_card_list_nat(minus_7954133019191499631st_nat(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9716_card__less__sym__Diff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4))),finite_card_set_nat(minus_2163939370556025621et_nat(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9717_card__less__sym__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9718_card__less__sym__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A4)),finite_card_int(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9719_card__less__sym__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,B4))),finite4902975817058060853nteger(minus_2355218937544613996nteger(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9720_card__less__sym__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(A4)),finite711546835091564841at_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_9721_card__le__sym__Diff,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite8100373058378681591st_nat(B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4))),finite_card_list_nat(minus_7954133019191499631st_nat(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9722_card__le__sym__Diff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4))),finite_card_set_nat(minus_2163939370556025621et_nat(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9723_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A4),B4))),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B4),A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9724_card__le__sym__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,A4),B4))),finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,B4),A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9725_card__le__sym__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,B4))),finite4902975817058060853nteger(minus_2355218937544613996nteger(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9726_card__le__sym__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(A4)),finite711546835091564841at_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A4),B4))),finite711546835091564841at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B4),A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_9727_psubset__card__mono,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le1190675801316882794st_nat(A4,B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4)) ) ) ).

% psubset_card_mono
tff(fact_9728_psubset__card__mono,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4)) ) ) ).

% psubset_card_mono
tff(fact_9729_psubset__card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ).

% psubset_card_mono
tff(fact_9730_psubset__card__mono,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A4)),finite_card_int(B4)) ) ) ).

% psubset_card_mono
tff(fact_9731_psubset__card__mono,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4)) ) ) ).

% psubset_card_mono
tff(fact_9732_psubset__card__mono,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite711546835091564841at_nat(A4)),finite711546835091564841at_nat(B4)) ) ) ).

% psubset_card_mono
tff(fact_9733_card__Un__le,axiom,
    ! [A4: set_int,B4: set_int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A4),B4))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_int(A4)),finite_card_int(B4))) ).

% card_Un_le
tff(fact_9734_card__Un__le,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(sup_sup_set_list_nat(A4,B4))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))) ).

% card_Un_le
tff(fact_9735_card__Un__le,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(sup_sup_set_set_nat(A4,B4))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))) ).

% card_Un_le
tff(fact_9736_card__Un__le,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7007076676225009423at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,A4),B4))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite7007076676225009423at_nat(A4)),finite7007076676225009423at_nat(B4))) ).

% card_Un_le
tff(fact_9737_card__Un__le,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite3771342082235030671at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A4),B4))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite3771342082235030671at_nat(A4)),finite3771342082235030671at_nat(B4))) ).

% card_Un_le
tff(fact_9738_card__Un__le,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A4),B4))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))) ).

% card_Un_le
tff(fact_9739_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bx,aTP_Lamm_by) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9740_card__less__Suc2,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_anj(M6),I2))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ank(M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_9741_card__less__Suc,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_anj(M6),I2)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ank(M6),I2))) ) ) ).

% card_less_Suc
tff(fact_9742_card__less,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ank(M6),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9743_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),set_or4665077453230672383an_nat(K2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))))
     => ( A4 = set_or4665077453230672383an_nat(K2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9744_Max_Osemilattice__order__set__axioms,axiom,
    lattic5374021519246970238et_rat(ord_max_rat,aTP_Lamm_bv,aTP_Lamm_bw) ).

% Max.semilattice_order_set_axioms
tff(fact_9745_Max_Osemilattice__order__set__axioms,axiom,
    lattic2566483365489244608et_num(ord_max_num,aTP_Lamm_pv,aTP_Lamm_dr) ).

% Max.semilattice_order_set_axioms
tff(fact_9746_Max_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat(ord_max_nat,aTP_Lamm_bx,aTP_Lamm_by) ).

% Max.semilattice_order_set_axioms
tff(fact_9747_Max_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int(ord_max_int,aTP_Lamm_bz,aTP_Lamm_ca) ).

% Max.semilattice_order_set_axioms
tff(fact_9748_sum__Suc,axiom,
    ! [Fa: fun_int_nat,A4: set_int] : groups4541462559716669496nt_nat(aTP_Lamm_anl(Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(Fa,A4)),finite_card_int(A4)) ).

% sum_Suc
tff(fact_9749_sum__Suc,axiom,
    ! [Fa: fun_list_nat_nat,A4: set_list_nat] : groups4396056296759096172at_nat(aTP_Lamm_anm(Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(Fa,A4)),finite_card_list_nat(A4)) ).

% sum_Suc
tff(fact_9750_sum__Suc,axiom,
    ! [Fa: fun_set_nat_nat,A4: set_set_nat] : groups8294997508430121362at_nat(aTP_Lamm_ann(Fa),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(Fa,A4)),finite_card_set_nat(A4)) ).

% sum_Suc
tff(fact_9751_sum__Suc,axiom,
    ! [Fa: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ano(Fa)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A4)),aa_set_nat_nat(finite_card_nat,A4)) ).

% sum_Suc
tff(fact_9752_sum__multicount,axiom,
    ! [S: set_o,T7: set_o,Ra: fun_o_fun_o_o,K2: nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(finite_finite_o,T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),T7)
             => ( finite_card_o(collect_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_amb(S),Ra),(X3)))) = K2 ) )
         => ( groups8507830703676809646_o_nat(aa_fun8807093888949732041_o_nat(aTP_Lamm_amd(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_o(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9753_sum__multicount,axiom,
    ! [S: set_o,T7: set_nat,Ra: fun_o_fun_nat_o,K2: nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_nat_o(finite_finite_nat,T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T7)
             => ( finite_card_o(collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(S),Ra),X3))) = K2 ) )
         => ( groups8507830703676809646_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_amv(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_set_nat_nat(finite_card_nat,T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9754_sum__multicount,axiom,
    ! [S: set_o,T7: set_int,Ra: fun_o_fun_int_o,K2: nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_int_o(finite_finite_int,T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T7)
             => ( finite_card_o(collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_aaa(S),Ra),X3))) = K2 ) )
         => ( groups8507830703676809646_o_nat(aa_fun2417465416094144205_o_nat(aTP_Lamm_ame(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_int(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9755_sum__multicount,axiom,
    ! [S: set_o,T7: set_Code_integer,Ra: fun_o_4021313873839786239eger_o,K2: nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T7)
             => ( finite_card_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ajs(S),Ra),X3))) = K2 ) )
         => ( groups8507830703676809646_o_nat(aa_fun5400594822745170686_o_nat(aTP_Lamm_amg(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite4902975817058060853nteger(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9756_sum__multicount,axiom,
    ! [S: set_int,T7: set_o,Ra: fun_int_fun_o_o,K2: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_o_o(finite_finite_o,T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),T7)
             => ( finite_card_int(collect_int(aa_o_fun_int_o(aa_fun5351653887499120666_int_o(aTP_Lamm_amh(S),Ra),(X3)))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun7603103345900947235nt_nat(aTP_Lamm_amj(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_o(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9757_sum__multicount,axiom,
    ! [S: set_int,T7: set_nat,Ra: fun_int_fun_nat_o,K2: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_nat_o(finite_finite_nat,T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T7)
             => ( finite_card_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_zq(S),Ra),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_anp(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_set_nat_nat(finite_card_nat,T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9758_sum__multicount,axiom,
    ! [S: set_int,T7: set_int,Ra: fun_int_fun_int_o,K2: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(finite_finite_int,T7)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),T7)
             => ( finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ajt(S),Ra),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_aml(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_int(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9759_sum__multicount,axiom,
    ! [S: set_int,T7: set_Code_integer,Ra: fun_in4891666191172872833eger_o,K2: nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T7)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),T7)
             => ( finite_card_int(collect_int(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_aju(S),Ra),X3))) = K2 ) )
         => ( groups4541462559716669496nt_nat(aa_fun511150112467848510nt_nat(aTP_Lamm_amn(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite4902975817058060853nteger(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9760_sum__multicount,axiom,
    ! [S: set_Code_integer,T7: set_o,Ra: fun_Co480733329897759773un_o_o,K2: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_o_o(finite_finite_o,T7)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),T7)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_o_f8517747772616379256eger_o(aa_fun3295616927821077850eger_o(aTP_Lamm_amo(S),Ra),(X3)))) = K2 ) )
         => ( groups7237345082560585321er_nat(aa_fun7670077909833392133er_nat(aTP_Lamm_amq(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),finite_card_o(T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9761_sum__multicount,axiom,
    ! [S: set_Code_integer,T7: set_nat,Ra: fun_Co1375048237525739357_nat_o,K2: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_nat_o(finite_finite_nat,T7)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),T7)
             => ( finite4902975817058060853nteger(collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_zv(S),Ra),X3))) = K2 ) )
         => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aTP_Lamm_anq(T7),Ra),S) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_set_nat_nat(finite_card_nat,T7)) ) ) ) ) ).

% sum_multicount
tff(fact_9762_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro4764841431850000930nteger(Ga,X3) = zero_z3403309356797280102nteger ) )
       => ( groups3803682039294397627nteger(Ga,S) = groups3803682039294397627nteger(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9763_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr1029825253981700553at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro6819653077972045762at_nat(Ga,X3) = zero_z1048942125864253310at_nat ) )
       => ( groups8361914993108267113at_nat(Ga,S) = groups8361914993108267113at_nat(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9764_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro3608043709957902237at_rat(Ga,X3) = zero_zero_rat ) )
       => ( groups3328076802468863542at_rat(Ga,S) = groups3328076802468863542at_rat(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9765_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro4243173770044397973at_nat(Ga,X3) = zero_zero_nat ) )
       => ( groups3963206862555359278at_nat(Ga,S) = groups3963206862555359278at_nat(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9766_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro4240683299535347697at_int(Ga,X3) = zero_zero_int ) )
       => ( groups3960716392046309002at_int(Ga,S) = groups3960716392046309002at_int(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9767_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
       => ( groups926780983652909934nteger(Ga,S) = groups926780983652909934nteger(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9768_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_in259009099871059828at_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int161471153780056413at_nat(Ga,X3) = zero_z1048942125864253310at_nat ) )
       => ( groups6906599089918680438at_nat(Ga,S) = groups6906599089918680438at_nat(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9769_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_rat(Ga,X3) = zero_zero_rat ) )
       => ( groups2350640619554545897nt_rat(Ga,S) = groups2350640619554545897nt_rat(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9770_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_nat(Ga,X3) = zero_zero_nat ) )
       => ( groups2985770679641041633nt_nat(Ga,S) = groups2985770679641041633nt_nat(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9771_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_int(Ga,X3) = zero_zero_int ) )
       => ( groups2983280209131991357nt_int(Ga,S) = groups2983280209131991357nt_int(Ga,T7) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_9772_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr6026251430685297nteger] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro4764841431850000930nteger(Ga,X3) = zero_z3403309356797280102nteger ) )
       => ( groups3803682039294397627nteger(Ga,T7) = groups3803682039294397627nteger(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9773_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr1029825253981700553at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro6819653077972045762at_nat(Ga,X3) = zero_z1048942125864253310at_nat ) )
       => ( groups8361914993108267113at_nat(Ga,T7) = groups8361914993108267113at_nat(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9774_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr8317260874932935276at_rat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro3608043709957902237at_rat(Ga,X3) = zero_zero_rat ) )
       => ( groups3328076802468863542at_rat(Ga,T7) = groups3328076802468863542at_rat(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9775_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro4243173770044397973at_nat(Ga,X3) = zero_zero_nat ) )
       => ( groups3963206862555359278at_nat(Ga,T7) = groups3963206862555359278at_nat(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9776_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,Ga: fun_Pr7647638413817135296at_int] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),T7)
     => ( ! [X3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,T7),S))
           => ( aa_Pro4240683299535347697at_int(Ga,X3) = zero_zero_int ) )
       => ( groups3960716392046309002at_int(Ga,T7) = groups3960716392046309002at_int(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9777_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
       => ( groups926780983652909934nteger(Ga,T7) = groups926780983652909934nteger(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9778_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_in259009099871059828at_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int161471153780056413at_nat(Ga,X3) = zero_z1048942125864253310at_nat ) )
       => ( groups6906599089918680438at_nat(Ga,T7) = groups6906599089918680438at_nat(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9779_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_rat(Ga,X3) = zero_zero_rat ) )
       => ( groups2350640619554545897nt_rat(Ga,T7) = groups2350640619554545897nt_rat(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9780_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_nat(Ga,X3) = zero_zero_nat ) )
       => ( groups2985770679641041633nt_nat(Ga,T7) = groups2985770679641041633nt_nat(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9781_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_int(Ga,X3) = zero_zero_int ) )
       => ( groups2983280209131991357nt_int(Ga,T7) = groups2983280209131991357nt_int(Ga,S) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_9782_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,Ha: fun_o_Code_integer,Ga: fun_o_Code_integer] :
      ( aa_set_o_o(ord_less_eq_set_o(S),T7)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T7,S))
           => ( aa_o_Code_integer(Ha,(I3)) = zero_z3403309356797280102nteger ) )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),S)
             => ( aa_o_Code_integer(Ga,(X3)) = aa_o_Code_integer(Ha,(X3)) ) )
         => ( groups1402912129352969042nteger(Ga,S) = groups1402912129352969042nteger(Ha,T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9783_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,Ha: fun_nat_Code_integer,Ga: fun_nat_Code_integer] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
           => ( aa_nat_Code_integer(Ha,I3) = zero_z3403309356797280102nteger ) )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S)
             => ( aa_nat_Code_integer(Ga,X3) = aa_nat_Code_integer(Ha,X3) ) )
         => ( aa_set871585252387937313nteger(groups555127423416065298nteger(Ga),S) = aa_set871585252387937313nteger(groups555127423416065298nteger(Ha),T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9784_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,Ha: fun_o_rat,Ga: fun_o_rat] :
      ( aa_set_o_o(ord_less_eq_set_o(S),T7)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T7,S))
           => ( aa_o_rat(Ha,(I3)) = zero_zero_rat ) )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),S)
             => ( aa_o_rat(Ga,(X3)) = aa_o_rat(Ha,(X3)) ) )
         => ( groups3921277224699582669_o_rat(Ga,S) = groups3921277224699582669_o_rat(Ha,T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9785_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,Ha: fun_nat_rat,Ga: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
           => ( aa_nat_rat(Ha,I3) = zero_zero_rat ) )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S)
             => ( aa_nat_rat(Ga,X3) = aa_nat_rat(Ha,X3) ) )
         => ( aa_set_nat_rat(groups1351286907653491341at_rat(Ga),S) = aa_set_nat_rat(groups1351286907653491341at_rat(Ha),T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9786_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,Ha: fun_o_nat,Ga: fun_o_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(S),T7)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T7,S))
           => ( aa_o_nat(Ha,(I3)) = zero_zero_nat ) )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),S)
             => ( aa_o_nat(Ga,(X3)) = aa_o_nat(Ha,(X3)) ) )
         => ( groups4556407284786078405_o_nat(Ga,S) = groups4556407284786078405_o_nat(Ha,T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9787_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,Ha: fun_nat_nat,Ga: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
           => ( aa_nat_nat(Ha,I3) = zero_zero_nat ) )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S)
             => ( aa_nat_nat(Ga,X3) = aa_nat_nat(Ha,X3) ) )
         => ( aa_set_nat_nat(groups1986416967739987077at_nat(Ga),S) = aa_set_nat_nat(groups1986416967739987077at_nat(Ha),T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9788_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,Ha: fun_o_int,Ga: fun_o_int] :
      ( aa_set_o_o(ord_less_eq_set_o(S),T7)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T7,S))
           => ( aa_o_int(Ha,(I3)) = zero_zero_int ) )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),S)
             => ( aa_o_int(Ga,(X3)) = aa_o_int(Ha,(X3)) ) )
         => ( groups4553916814277028129_o_int(Ga,S) = groups4553916814277028129_o_int(Ha,T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9789_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,Ha: fun_nat_int,Ga: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T7)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,T7),S))
           => ( aa_nat_int(Ha,I3) = zero_zero_int ) )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S)
             => ( aa_nat_int(Ga,X3) = aa_nat_int(Ha,X3) ) )
         => ( aa_set_nat_int(groups1983926497230936801at_int(Ga),S) = aa_set_nat_int(groups1983926497230936801at_int(Ha),T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9790_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_int,T7: set_int,Ha: fun_int_Code_integer,Ga: fun_int_Code_integer] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_Code_integer(Ha,I3) = zero_z3403309356797280102nteger ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int_Code_integer(Ga,X3) = aa_int_Code_integer(Ha,X3) ) )
         => ( groups926780983652909934nteger(Ga,S) = groups926780983652909934nteger(Ha,T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9791_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_int,T7: set_int,Ha: fun_int_rat,Ga: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_rat(Ha,I3) = zero_zero_rat ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int_rat(Ga,X3) = aa_int_rat(Ha,X3) ) )
         => ( groups2350640619554545897nt_rat(Ga,S) = groups2350640619554545897nt_rat(Ha,T7) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_9792_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_Code_integer,Ha: fun_int_Code_integer] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_Code_integer(Ga,X3) = zero_z3403309356797280102nteger ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int_Code_integer(Ga,X3) = aa_int_Code_integer(Ha,X3) ) )
         => ( groups926780983652909934nteger(Ga,T7) = groups926780983652909934nteger(Ha,S) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_9793_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_in259009099871059828at_nat,Ha: fun_in259009099871059828at_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int161471153780056413at_nat(Ga,X3) = zero_z1048942125864253310at_nat ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int161471153780056413at_nat(Ga,X3) = aa_int161471153780056413at_nat(Ha,X3) ) )
         => ( groups6906599089918680438at_nat(Ga,T7) = groups6906599089918680438at_nat(Ha,S) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_9794_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_rat,Ha: fun_int_rat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_rat(Ga,X3) = zero_zero_rat ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int_rat(Ga,X3) = aa_int_rat(Ha,X3) ) )
         => ( groups2350640619554545897nt_rat(Ga,T7) = groups2350640619554545897nt_rat(Ha,S) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_9795_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_nat(Ga,X3) = zero_zero_nat ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int_nat(Ga,X3) = aa_int_nat(Ha,X3) ) )
         => ( groups2985770679641041633nt_nat(Ga,T7) = groups2985770679641041633nt_nat(Ha,S) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_9796_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,Ga: fun_int_int,Ha: fun_int_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T7)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,T7),S))
           => ( aa_int_int(Ga,X3) = zero_zero_int ) )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),S)
             => ( aa_int_int(Ga,X3) = aa_int_int(Ha,X3) ) )
         => ( groups2983280209131991357nt_int(Ga,T7) = groups2983280209131991357nt_int(Ha,S) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_9797_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N7),set_or4665077453230672383an_nat(zero_zero_nat,N2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N7)),N2) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9798_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq),S)) ).

% card_sum_le_nat_sum
tff(fact_9799_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(bit_and_not_num(X),Xa) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa = aa_num_num(bit0,N5) )
                 => ( ( Y = aa_num_option_num(some_num,one) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit0,N5))) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa = aa_num_num(bit1,N5) )
                   => ( ( Y = none_num )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit1,N5))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa = one )
                     => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit0,M5)) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa = aa_num_num(bit0,N5) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa = aa_num_num(bit1,N5) )
                           => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N5))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa = one )
                           => ( ( Y = aa_num_option_num(some_num,aa_num_num(bit0,M5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa = aa_num_num(bit0,N5) )
                               => ( ( Y = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_mr),aa_num_option_num(bit_and_not_num(M5),N5)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N5))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa = aa_num_num(bit1,N5) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_and_not_num(M5),N5)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N5))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9800_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(X),Xa) = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != aa_num_option_num(some_num,one) ) ) )
       => ( ( ( X = one )
           => ( ? [N5: num] : Xa = aa_num_num(bit0,N5)
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N5: num] : Xa = aa_num_num(bit1,N5)
               => ( Y != aa_num_option_num(some_num,one) ) ) )
           => ( ( ? [M5: num] : X = aa_num_num(bit0,M5)
               => ( ( Xa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ! [N5: num] :
                        ( ( Xa = aa_num_num(bit0,N5) )
                       => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa = aa_num_num(bit1,N5) )
                         => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) )
                 => ( ( ? [M5: num] : X = aa_num_num(bit1,M5)
                     => ( ( Xa = one )
                       => ( Y != aa_num_option_num(some_num,one) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ! [N5: num] :
                              ( ( Xa = aa_num_num(bit0,N5) )
                             => ( Y != aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa = aa_num_num(bit1,N5) )
                               => ( Y != aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_mr),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9801_Suc__funpow,axiom,
    ! [N2: nat] : compow_fun_nat_nat(N2,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N2) ).

% Suc_funpow
tff(fact_9802_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] : aa_num_option_num(bit_un7362597486090784418nd_num(aa_num_num(bit1,M)),aa_num_num(bit1,N2)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_mr),aa_num_option_num(bit_un7362597486090784418nd_num(M),N2)) ).

% and_num.simps(9)
tff(fact_9803_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( aa_num_option_num(bit_un7362597486090784418nd_num(X),Xa) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(X,Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = aa_num_option_num(some_num,one) )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( X = one )
             => ! [N5: num] :
                  ( ( Xa = aa_num_num(bit0,N5) )
                 => ( ( Y = none_num )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit0,N5))) ) ) )
           => ( ( ( X = one )
               => ! [N5: num] :
                    ( ( Xa = aa_num_num(bit1,N5) )
                   => ( ( Y = aa_num_option_num(some_num,one) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit1,N5))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa_num_num(bit0,M5) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M5),one)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa_num_num(bit0,M5) )
                     => ! [N5: num] :
                          ( ( Xa = aa_num_num(bit0,N5) )
                         => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit0,N5))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa_num_num(bit0,M5) )
                       => ! [N5: num] :
                            ( ( Xa = aa_num_num(bit1,N5) )
                           => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M5),aa_num_num(bit1,N5))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa_num_num(bit1,M5) )
                         => ( ( Xa = one )
                           => ( ( Y = aa_num_option_num(some_num,one) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M5),one)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa_num_num(bit1,M5) )
                           => ! [N5: num] :
                                ( ( Xa = aa_num_num(bit0,N5) )
                               => ( ( Y = aa_opt5369159329878751634on_num(map_option_num_num(bit0),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit0,N5))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa_num_num(bit1,M5) )
                             => ! [N5: num] :
                                  ( ( Xa = aa_num_num(bit1,N5) )
                                 => ( ( Y = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_mr),aa_num_option_num(bit_un7362597486090784418nd_num(M5),N5)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M5),aa_num_num(bit1,N5))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9804_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9805_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ? [R2: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R2),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S)))
          & ! [N8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N8),aa_set_nat_nat(finite_card_nat,S))
             => aa_set_nat_o(member_nat(aa_nat_nat(R2,N8)),S) ) ) ) ).

% finite_enumerate
tff(fact_9806_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,L),U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9807_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or4266950643985792945nteger(L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_9808_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),J2)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J2)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I2),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9809_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_9810_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),aa_nat_nat(suc,I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J2))),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9811_split__seed__def,axiom,
    ! [S2: produc7822875418678951345atural] : split_seed(S2) = produc8282080750456430313atural(aTP_Lamm_ans(S2),S2) ).

% split_seed_def
tff(fact_9812_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I2))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J2))),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9813_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_anu),Xa),X)) ).

% times_int.abs_eq
tff(fact_9814_mod__pure,axiom,
    ! [B: $o,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),Ha)
    <=> ( ( produc8586169260539613262et_nat(Ha) = bot_bot_set_nat )
        & (B) ) ) ).

% mod_pure
tff(fact_9815_mod__emp,axiom,
    ! [Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),Ha)
    <=> ( produc8586169260539613262et_nat(Ha) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_9816_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_anv),X)) ).

% uminus_int.abs_eq
tff(fact_9817_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I2)),J2)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J2)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I2),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9818_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: 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,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_anx),Xa),X) ) ).

% less_int.abs_eq
tff(fact_9819_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: 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,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_anz),Xa),X) ) ).

% less_eq_int.abs_eq
tff(fact_9820_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aob),Xa),X)) ).

% plus_int.abs_eq
tff(fact_9821_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aod),Xa),X)) ).

% minus_int.abs_eq
tff(fact_9822_rat__minus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aof(Q4)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9823_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or2715278749043346189nteger(L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_9824_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N2),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9825_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,N2),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9826_diff__rat__def,axiom,
    ! [Q4: rat,R: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Q4),R) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q4),aa_rat_rat(uminus_uminus_rat,R)) ).

% diff_rat_def
tff(fact_9827_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(R))) ).

% quotient_of_denom_pos'
tff(fact_9828_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,L),U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9829_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q4: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% quotient_of_denom_pos
tff(fact_9830_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9831_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aog),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9832_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aoh),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9833_rat__less__eq__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q4)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_aoj(Q4)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9834_rat__less__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q4)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_aol(Q4)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9835_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_9836_rat__divide__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aon(Q4)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9837_rat__times__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aop(Q4)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9838_rat__plus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q4)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aor(Q4)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9839_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aos),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9840_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_anz),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_eq_int.rep_eq
tff(fact_9841_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_anx),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_int.rep_eq
tff(fact_9842_abs__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_rat_def
tff(fact_9843_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9844_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9845_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_aot(X)) ).

% floor_rat_def
tff(fact_9846_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_anv)) ).

% uminus_int_def
tff(fact_9847_range,axiom,
    ! [K2: code_natural,S2: produc7822875418678951345atural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),K2)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,produc497848011034438852atural(aa_Pro7165575656943818355atural(aa_Cod6352003912080677724atural(range,K2),S2))),K2) ) ).

% range
tff(fact_9848_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9849_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_anu)) ).

% times_int_def
tff(fact_9850_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_aob)) ).

% plus_int_def
tff(fact_9851_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_aod)) ).

% minus_int_def
tff(fact_9852_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a2: int,
          a2:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a2)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            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,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_9853_gcd__pos__int,axiom,
    ! [M: int,N2: 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),N2))
    <=> ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9854_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ).

% gcd_ge_0_int
tff(fact_9855_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U3),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V3),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% bezout_int
tff(fact_9856_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_9857_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_9858_gcd__cases__int,axiom,
    ! [X: int,Y: int,Pa: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9859_gcd__unique__int,axiom,
    ! [D2: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D2)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),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),D2) ) )
    <=> ( D2 = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9860_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9861_gcd__pos__nat,axiom,
    ! [M: nat,N2: 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),N2))
    <=> ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9862_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_9863_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_9864_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) ) ) ).

% gcd_diff1_nat
tff(fact_9865_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) ) ) ).

% gcd_diff2_nat
tff(fact_9866_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9867_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: 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),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9868_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_jn) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9869_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9870_upto_Opsimps,axiom,
    ! [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))
     => ( upto(I2,J2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J2)),nil_int) ) ) ).

% upto.psimps
tff(fact_9871_upto__Nil,axiom,
    ! [I2: int,J2: int] :
      ( ( upto(I2,J2) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I2) ) ).

% upto_Nil
tff(fact_9872_upto__Nil2,axiom,
    ! [I2: int,J2: int] :
      ( ( nil_int = upto(I2,J2) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I2) ) ).

% upto_Nil2
tff(fact_9873_upto__empty,axiom,
    ! [J2: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I2)
     => ( upto(I2,J2) = nil_int ) ) ).

% upto_empty
tff(fact_9874_nth__upto,axiom,
    ! [I2: int,K2: nat,J2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K2))),J2)
     => ( aa_nat_int(nth_int(upto(I2,J2)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K2)) ) ) ).

% nth_upto
tff(fact_9875_length__upto,axiom,
    ! [I2: int,J2: int] : size_size_list_int(upto(I2,J2)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J2),I2)),one_one_int)) ).

% length_upto
tff(fact_9876_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N2))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9877_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9878_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N2))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9879_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = $ite(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,aa_num_int(numeral_numeral_int,N2))),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9880_sorted__wrt__upto,axiom,
    ! [I2: int,J2: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I2,J2)) ).

% sorted_wrt_upto
tff(fact_9881_sorted__upto,axiom,
    ! [M: int,N2: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(M,N2)) ).

% sorted_upto
tff(fact_9882_greaterThanAtMost__upto,axiom,
    ! [I2: int,J2: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or6656581121297822940st_int,I2),J2) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J2)) ).

% greaterThanAtMost_upto
tff(fact_9883_upto__rec1,axiom,
    ! [I2: int,J2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
     => ( upto(I2,J2) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J2)) ) ) ).

% upto_rec1
tff(fact_9884_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) ) ) ).

% upto.elims
tff(fact_9885_upto_Osimps,axiom,
    ! [I2: int,J2: int] :
      upto(I2,J2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J2)),nil_int) ).

% upto.simps
tff(fact_9886_greaterThanLessThan__upto,axiom,
    ! [I2: int,J2: int] : aa_int_set_int(aa_int8639817059373041269et_int(set_or5832277885323065728an_int,I2),J2) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J2),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9887_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_9888_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2),aa_num_num(inc,aa_nat_num(num_of_nat,N2)),one) ).

% num_of_nat.simps(2)
tff(fact_9889_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N2)) = N2 ) ) ).

% numeral_num_of_nat
tff(fact_9890_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),one_one_nat)
     => ( aa_nat_num(num_of_nat,N2) = one ) ) ).

% num_of_nat_One
tff(fact_9891_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N2)) ) ) ).

% num_of_nat_double
tff(fact_9892_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: 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),N2)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N2)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9893_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aou)) ).

% pred_nat_def
tff(fact_9894_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N2)),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N2) ) ).

% less_eq
tff(fact_9895_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9896_upto__rec2,axiom,
    ! [I2: int,J2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
     => ( upto(I2,J2) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J2),one_one_int)),cons_int(J2,nil_int)) ) ) ).

% upto_rec2
tff(fact_9897_less__rat,axiom,
    ! [B: int,D2: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C2),D2))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))) ) ) ) ).

% less_rat
tff(fact_9898_le__rat,axiom,
    ! [B: int,D2: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C2),D2))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))) ) ) ) ).

% le_rat
tff(fact_9899_add__rat,axiom,
    ! [B: int,D2: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C2),D2)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).

% add_rat
tff(fact_9900_Rat__induct__pos,axiom,
    ! [Pa: fun_rat_o,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_rat_o(Pa,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3)) )
     => aa_rat_o(Pa,Q4) ) ).

% Rat_induct_pos
tff(fact_9901_upto__split2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),K2)
       => ( upto(I2,K2) = append_int(upto(I2,J2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),one_one_int),K2)) ) ) ) ).

% upto_split2
tff(fact_9902_upto__split1,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),K2)
       => ( upto(I2,K2) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J2),one_one_int)),upto(J2,K2)) ) ) ) ).

% upto_split1
tff(fact_9903_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9904_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9905_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% Fract_less_one_iff
tff(fact_9906_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9907_Fract__add__one,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N2)),N2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,M),N2)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9908_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9909_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9910_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9911_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_9912_upto__split3,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),K2)
       => ( upto(I2,K2) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J2),one_one_int)),cons_int(J2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J2),one_one_int),K2))) ) ) ) ).

% upto_split3
tff(fact_9913_sort__upto,axiom,
    ! [I2: int,J2: int] : linord1735203802627413978nt_int(aTP_Lamm_fr,upto(I2,J2)) = upto(I2,J2) ).

% sort_upto
tff(fact_9914_sort__upt,axiom,
    ! [M: nat,N2: nat] : linord738340561235409698at_nat(aTP_Lamm_fq,upt(M,N2)) = upt(M,N2) ).

% sort_upt
tff(fact_9915_upt__0__eq__Nil__conv,axiom,
    ! [J2: nat] :
      ( ( upt(zero_zero_nat,J2) = nil_nat )
    <=> ( J2 = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_9916_upt__merge,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),K2) )
     => ( append_nat(upt(I2,J2),upt(J2,K2)) = upt(I2,K2) ) ) ).

% upt_merge
tff(fact_9917_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2)
     => ( upt(I2,J2) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9918_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( upt(I2,J2) = nil_nat )
    <=> ( ( J2 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I2) ) ) ).

% upt_eq_Nil_conv
tff(fact_9919_nth__upt,axiom,
    ! [I2: nat,K2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J2)
     => ( aa_nat_nat(nth_nat(upt(I2,J2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) ) ) ).

% nth_upt
tff(fact_9920_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)),cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))),nil_nat) ).

% upt_rec_numeral
tff(fact_9921_upt__eq__append__conv,axiom,
    ! [I2: nat,J2: nat,Xs: list_nat,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( ( upt(I2,J2) = append_nat(Xs,Ys2) )
      <=> ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),J2)
            & ( upt(I2,K3) = Xs )
            & ( upt(K3,J2) = Ys2 ) ) ) ) ).

% upt_eq_append_conv
tff(fact_9922_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aov,Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ).

% map_add_upt'
tff(fact_9923_upt__append,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ( append_nat(upt(zero_zero_nat,I2),upt(I2,J2)) = upt(zero_zero_nat,J2) ) ) ).

% upt_append
tff(fact_9924_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2)) = append_nat(upt(I2,J2),upt(J2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K2))) ) ) ).

% upt_add_eq_append
tff(fact_9925_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(M,N2)) ).

% sorted_wrt_upt
tff(fact_9926_map__add__upt,axiom,
    ! [N2: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aov,N2),upt(zero_zero_nat,M)) = upt(N2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% map_add_upt
tff(fact_9927_sorted__upt,axiom,
    ! [M: nat,N2: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(M,N2)) ).

% sorted_upt
tff(fact_9928_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ( upt(I2,J2) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J2)) ) ) ).

% upt_conv_Cons
tff(fact_9929_upt__rec,axiom,
    ! [I2: nat,J2: nat] :
      upt(I2,J2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2),cons_nat(I2,upt(aa_nat_nat(suc,I2),J2)),nil_nat) ).

% upt_rec
tff(fact_9930_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      upt(I2,aa_nat_nat(suc,J2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2),append_nat(upt(I2,J2),cons_nat(J2,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9931_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
     => ( upt(I2,aa_nat_nat(suc,J2)) = append_nat(upt(I2,J2),cons_nat(J2,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9932_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U4)
     => ( ! [I3: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),U4) )
           => ~ aa_nat_o(Pa,I3) )
       => ( filter_nat2(Pa,upt(zero_zero_nat,U)) = filter_nat2(Pa,upt(zero_zero_nat,U4)) ) ) ) ).

% upt_filter_extend
tff(fact_9933_map__decr__upt,axiom,
    ! [M: nat,N2: nat] : map_nat_nat(aTP_Lamm_qh,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = upt(M,N2) ).

% map_decr_upt
tff(fact_9934_upt__eq__lel__conv,axiom,
    ! [L: nat,Ha: nat,Is1: list_nat,I2: nat,Is2: list_nat] :
      ( ( upt(L,Ha) = append_nat(Is1,cons_nat(I2,Is2)) )
    <=> ( ( Is1 = upt(L,I2) )
        & ( Is2 = upt(aa_nat_nat(suc,I2),Ha) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Ha) ) ) ).

% upt_eq_lel_conv
tff(fact_9935_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J2: nat,X: nat,Xs: list_nat] :
      ( ( upt(I2,J2) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
        & ( I2 = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat),J2) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9936_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2)
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,upt(M,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fq),set_or4665077453230672383an_nat(M,N2)) ) ) ).

% sum_list_upt
tff(fact_9937_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: 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_aow(M),N7))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aox(M),N7)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aoy(M),N7)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9938_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aow(M),N7))) = 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,N7),M)),one_one_nat)),N7) ).

% card_length_sum_list
tff(fact_9939_hd__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ( aa_list_nat_nat(hd_nat,upt(I2,J2)) = I2 ) ) ).

% hd_upt
tff(fact_9940_finite__int__iff__bounded__le,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
    <=> ? [K3: int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(abs_abs_int,S)),aa_int_set_int(set_ord_atMost_int,K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9941_finite__int__iff__bounded,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
    <=> ? [K3: int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(abs_abs_int,S)),aa_int_set_int(set_ord_lessThan_int,K3)) ) ).

% finite_int_iff_bounded
tff(fact_9942_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_aoz,L),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U) ).

% image_add_int_atLeastLessThan
tff(fact_9943_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_apa,L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9944_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_9945_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9946_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,X: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apb,C2),set_or4665077453230672383an_nat(X,Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),Y),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C2)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9947_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9948_Sup__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple4687483117567863418t_unit(Uu) = product_Unity ).

% Sup_unit_def
tff(fact_9949_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_9950_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_9951_Inf__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple2584293577114468500t_unit(Uu) = product_Unity ).

% Inf_unit_def
tff(fact_9952_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K5)),K5) ) ).

% Inf_nat_def1
tff(fact_9953_Inf__int__def,axiom,
    ! [X6: set_int] : complete_Inf_Inf_int(X6) = aa_int_int(uminus_uminus_int,complete_Sup_Sup_int(image_int_int(uminus_uminus_int,X6))) ).

% Inf_int_def
tff(fact_9954_range__mod,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apc,N2),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N2) ) ) ).

% range_mod
tff(fact_9955_merge__true__star,axiom,
    aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_9956_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_9957_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_9958_mod__true,axiom,
    ! [Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),Ha)
    <=> aa_Pro653082453696054616_nat_o(in_range,Ha) ) ).

% mod_true
tff(fact_9959_UNIV__unit,axiom,
    top_to1996260823553986621t_unit = insert_Product_unit(product_Unity,bot_bo3957492148770167129t_unit) ).

% UNIV_unit
tff(fact_9960_ent__true,axiom,
    ! [Pa: assn] : entails(Pa,top_top_assn) ).

% ent_true
tff(fact_9961_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
tff(fact_9962_mod__star__trueI,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Pa),Ha)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),Ha) ) ).

% mod_star_trueI
tff(fact_9963_mod__star__trueE,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),Ha)
     => ~ ! [H3: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),H3) ) ).

% mod_star_trueE
tff(fact_9964_top__assn__def,axiom,
    top_top_assn = abs_assn(in_range) ).

% top_assn_def
tff(fact_9965_mod__h__bot__iff_I2_J,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),produc7507926704131184380et_nat(Ha,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_9966_binomial__def,axiom,
    ! [N2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K2) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_apd(N2),K2))) ).

% binomial_def
tff(fact_9967_mod__star__trueE_H,axiom,
    ! [Pa: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),Ha)
     => ~ ! [H3: produc3658429121746597890et_nat] :
            ( ( produc1824681642469235216et_nat(H3) = produc1824681642469235216et_nat(Ha) )
           => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,produc8586169260539613262et_nat(H3)),produc8586169260539613262et_nat(Ha))
             => ~ aa_Pro653082453696054616_nat_o(rep_assn(Pa),H3) ) ) ) ).

% mod_star_trueE'
tff(fact_9968_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_9969_drop__upt,axiom,
    ! [M: nat,I2: nat,J2: nat] : drop_nat(M,upt(I2,J2)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M),J2) ).

% drop_upt
tff(fact_9970_take__upt,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M)),N2)
     => ( take_nat(M,upt(I2,N2)) = upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M)) ) ) ).

% take_upt
tff(fact_9971_last__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
     => ( aa_list_nat_nat(last_nat,upt(I2,J2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),one_one_nat) ) ) ).

% last_upt
tff(fact_9972_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9973_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9974_plus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).

% plus_integer.rsp
tff(fact_9975_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_9976_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_ape,aTP_Lamm_ape) ).

% sub.rsp
tff(fact_9977_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_apf),aTP_Lamm_apf) ).

% dup.rsp
tff(fact_9978_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9979_less__eq__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_9980_butlast__upt,axiom,
    ! [M: nat,N2: nat] : butlast_nat(upt(M,N2)) = upt(M,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ).

% butlast_upt
tff(fact_9981_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_apg,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9982_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_aph,fract) ).

% Fract.transfer
tff(fact_9983_uminus__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_api),uminus_uminus_rat) ).

% uminus_rat.transfer
tff(fact_9984_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_apj,times_times_rat) ).

% times_rat.transfer
tff(fact_9985_inverse__rat_Otransfer,axiom,
    aa_fun_rat_rat_o(aa_fun382765352565706371_rat_o(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_apk),inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_9986_positive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_apl,positive) ).

% positive.transfer
tff(fact_9987_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_anu),times_times_int) ).

% times_int.transfer
tff(fact_9988_positive__add,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(positive,X)
     => ( aa_rat_o(positive,Y)
       => aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)) ) ) ).

% positive_add
tff(fact_9989_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9990_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_apm,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9991_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
    <=> aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Y),X)) ) ).

% less_rat_def
tff(fact_9992_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_anv)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_9993_positive__rat,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),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)) ) ).

% positive_rat
tff(fact_9994_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_anx),ord_less_int) ).

% less_int.transfer
tff(fact_9995_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_anz),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9996_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aob),plus_plus_int) ).

% plus_int.transfer
tff(fact_9997_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aod),minus_minus_int) ).

% minus_int.transfer
tff(fact_9998_positive_Orep__eq,axiom,
    ! [X: rat] :
      ( aa_rat_o(positive,X)
    <=> 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,aa_rat7754553326175492643nt_int(rep_Rat,X))),aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_rat7754553326175492643nt_int(rep_Rat,X)))) ) ).

% positive.rep_eq
tff(fact_9999_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N2)),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N2) ) ).

% pred_nat_trancl_eq_le
tff(fact_10000_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_apn) ).

% prod_encode_def
tff(fact_10001_min__0R,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_10002_min__0L,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N2) = zero_zero_nat ).

% min_0L
tff(fact_10003_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)) ).

% min_Suc_Suc
tff(fact_10004_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_10005_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_10006_min__diff,axiom,
    ! [M: nat,I2: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),I2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)),I2) ).

% min_diff
tff(fact_10007_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_min_right
tff(fact_10008_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ).

% nat_mult_min_left
tff(fact_10009_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_10010_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_10011_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_10012_min__Suc1,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N2)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_apo,N2)),M) ).

% min_Suc1
tff(fact_10013_min__Suc2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_app,N2)),M) ).

% min_Suc2
tff(fact_10014_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K2,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),M) ).

% prod_encode_prod_decode_aux
tff(fact_10015_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_anu),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_anu)) ).

% times_int.rsp
tff(fact_10016_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_10017_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_anv)),produc2626176000494625587at_nat(aTP_Lamm_anv)) ).

% uminus_int.rsp
tff(fact_10018_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_10019_intrel__def,axiom,
    intrel = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_apr) ).

% intrel_def
tff(fact_10020_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_anx),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_anx)) ).

% less_int.rsp
tff(fact_10021_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_anz),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_anz)) ).

% less_eq_int.rsp
tff(fact_10022_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aob),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aob)) ).

% plus_int.rsp
tff(fact_10023_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aod),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_aod)) ).

% minus_int.rsp
tff(fact_10024_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10025_mono__times__nat,axiom,
    ! [N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2)) ) ).

% mono_times_nat
tff(fact_10026_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aps,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_10027_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_10028_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_anx)) ).

% less_int_def
tff(fact_10029_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_anz)) ).

% less_eq_int_def
tff(fact_10030_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_10031_positive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_apl) ).

% positive_def
tff(fact_10032_less__integer__def,axiom,
    ord_le6747313008572928689nteger = map_fu4249723513525335408eger_o(code_int_of_integer,map_fu6957801986076833569nt_o_o(code_int_of_integer,id_o),ord_less_int) ).

% less_integer_def
tff(fact_10033_less__eq__integer__def,axiom,
    ord_le3102999989581377725nteger = map_fu4249723513525335408eger_o(code_int_of_integer,map_fu6957801986076833569nt_o_o(code_int_of_integer,id_o),ord_less_eq_int) ).

% less_eq_integer_def
tff(fact_10034_plus__integer__def,axiom,
    plus_p5714425477246183910nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),plus_plus_int) ).

% plus_integer_def
tff(fact_10035_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_apg) ).

% plus_rat_def
tff(fact_10036_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_apk) ).

% inverse_rat_def
tff(fact_10037_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_apj) ).

% times_rat_def
tff(fact_10038_Fract__def,axiom,
    fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_aph) ).

% Fract_def
tff(fact_10039_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_api) ).

% uminus_rat_def
tff(fact_10040_Code__Numeral_Odup__def,axiom,
    code_dup = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),aTP_Lamm_apf) ).

% Code_Numeral.dup_def
tff(fact_10041_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xa),Xa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = 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,X))),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,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,X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_10042_positive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_o(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X))
      <=> 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,X)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ) ) ).

% positive.abs_eq
tff(fact_10043_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_aph,aTP_Lamm_aph) ).

% Fract.rsp
tff(fact_10044_dup_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(code_dup,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,X)) ).

% dup.rep_eq
tff(fact_10045_dup_Oabs__eq,axiom,
    ! [X: int] : aa_Cod1783974652533396728nteger(code_dup,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)) ).

% dup.abs_eq
tff(fact_10046_ratrel__def,axiom,
    ! [X5: product_prod_int_int,Xa2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X5),Xa2)
    <=> ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X5) != zero_zero_int )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa2) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X5)) ) ) ) ).

% ratrel_def
tff(fact_10047_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_apj,aTP_Lamm_apj) ).

% times_rat.rsp
tff(fact_10048_uminus__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_api),aTP_Lamm_api) ).

% uminus_rat.rsp
tff(fact_10049_positive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_apl,aTP_Lamm_apl) ).

% positive.rsp
tff(fact_10050_inverse__rat_Orsp,axiom,
    aa_fun2650603632653508759_int_o(aa_fun3734617511155415725_int_o(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_apk),aTP_Lamm_apk) ).

% inverse_rat.rsp
tff(fact_10051_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_apg,aTP_Lamm_apg) ).

% plus_rat.rsp
tff(fact_10052_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N2: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit0,N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),N2))),one_one_Code_integer) ).

% Code_Numeral.sub_code(8)
tff(fact_10053_cr__rat__def,axiom,
    ! [X5: product_prod_int_int,Xa2: rat] :
      ( aa_rat_o(cr_rat(X5),Xa2)
    <=> ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X5),X5)
        & ( aa_Pro6166618844887768661nt_rat(abs_Rat,X5) = Xa2 ) ) ) ).

% cr_rat_def
tff(fact_10054_Code__Numeral_Osub__def,axiom,
    code_sub = map_fu8540202848259802501nteger(id_num,map_fu8638147718074629079nteger(id_num,code_integer_of_int),aTP_Lamm_ape) ).

% Code_Numeral.sub_def
tff(fact_10055_less__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(code_natural_of_nat,Xa)),aa_nat_Code_natural(code_natural_of_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X) ) ).

% less_natural.abs_eq
tff(fact_10056_plus__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_nat_Code_natural(code_natural_of_nat,Xa)),aa_nat_Code_natural(code_natural_of_nat,X)) = aa_nat_Code_natural(code_natural_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),X)) ).

% plus_natural.abs_eq
tff(fact_10057_less__eq__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(code_natural_of_nat,Xa)),aa_nat_Code_natural(code_natural_of_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X) ) ).

% less_eq_natural.abs_eq
tff(fact_10058_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jl,N2))) = N2 ) ) ).

% Max_divisors_self_nat
tff(fact_10059_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_jm,N2))) = aa_int_int(abs_abs_int,N2) ) ) ).

% Max_divisors_self_int
tff(fact_10060_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N2) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_apt(N2),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_10061_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_10062_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S))) ) ).

% card_le_Suc_Max
tff(fact_10063_divide__nat__def,axiom,
    ! [M: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = $ite(N2 = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_apu(M),N2)))) ).

% divide_nat_def
tff(fact_10064_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_apv(N2),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_10065_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_apw,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_10066_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_10067_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_10068_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_apx,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_10069_Gcd__int__eq,axiom,
    ! [N7: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N7)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N7)) ).

% Gcd_int_eq
tff(fact_10070_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_10071_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_10072_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K2: nat] :
      ( ! [N5: nat] :
          ( aa_set_nat_o(member_nat(N5),N7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apb,K2)),N7) ) ).

% inj_on_diff_nat
tff(fact_10073_inj__Suc,axiom,
    ! [N7: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N7) ).

% inj_Suc
tff(fact_10074_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_10075_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_apy(X)) ).

% quotient_of_def
tff(fact_10076_normalize__stable,axiom,
    ! [Q4: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q4)
       => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q4) ) ) ) ).

% normalize_stable
tff(fact_10077_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B3) ) ) ).

% Rat_cases
tff(fact_10078_Rat__induct,axiom,
    ! [Pa: fun_rat_o,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B3)
           => aa_rat_o(Pa,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3)) ) )
     => aa_rat_o(Pa,Q4) ) ).

% Rat_induct
tff(fact_10079_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
           => ( ( A3 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A3),B3) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_10080_quotient__of__unique,axiom,
    ! [R: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R = aa_int_rat(aa_int_fun_int_rat(fract,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3))
      & ! [Y5: product_prod_int_int] :
          ( ( ( R = aa_int_rat(aa_int_fun_int_rat(fract,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y5)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y5))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y5)) )
         => ( Y5 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_10081_coprime__diff__one__right__nat,axiom,
    ! [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(algebr934650988132801477me_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_10082_coprime__diff__one__left__nat,axiom,
    ! [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(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),N2) ) ).

% coprime_diff_one_left_nat
tff(fact_10083_Field__natLeq__on,axiom,
    ! [N2: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,N2)) ).

% Field_natLeq_on
tff(fact_10084_wf__less,axiom,
    aa_set3702741264012733747_nat_o(wf_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_10085_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ).

% less_than_iff
tff(fact_10086_le__enumerate,axiom,
    ! [S: set_nat,N2: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(infini8530281810654367211te_nat(S),N2)) ) ).

% le_enumerate
tff(fact_10087_finite__le__enumerate,axiom,
    ! [S: set_nat,N2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_set_nat_nat(finite_card_nat,S))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(infini8530281810654367211te_nat(S),N2)) ) ) ).

% finite_le_enumerate
tff(fact_10088_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_10089_Inf__nat__def,axiom,
    ! [X6: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,X6) = ord_Least_nat(aTP_Lamm_ad(X6)) ).

% Inf_nat_def
tff(fact_10090_Least__Suc2,axiom,
    ! [Pa: fun_nat_o,N2: nat,Qa: fun_nat_o,M: nat] :
      ( aa_nat_o(Pa,N2)
     => ( aa_nat_o(Qa,M)
       => ( ~ aa_nat_o(Pa,zero_zero_nat)
         => ( ! [K: nat] :
                ( aa_nat_o(Pa,aa_nat_nat(suc,K))
              <=> aa_nat_o(Qa,K) )
           => ( ord_Least_nat(Pa) = aa_nat_nat(suc,ord_Least_nat(Qa)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_10091_Least__Suc,axiom,
    ! [Pa: fun_nat_o,N2: nat] :
      ( aa_nat_o(Pa,N2)
     => ( ~ aa_nat_o(Pa,zero_zero_nat)
       => ( ord_Least_nat(Pa) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_aqa(Pa))) ) ) ) ).

% Least_Suc
tff(fact_10092_GreatestI__nat,axiom,
    ! [Pa: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(Pa,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(Pa,order_Greatest_nat(Pa)) ) ) ).

% GreatestI_nat
tff(fact_10093_Greatest__le__nat,axiom,
    ! [Pa: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(Pa,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),order_Greatest_nat(Pa)) ) ) ).

% Greatest_le_nat
tff(fact_10094_GreatestI__ex__nat,axiom,
    ! [Pa: fun_nat_o,B: nat] :
      ( ? [X_13: nat] : aa_nat_o(Pa,X_13)
     => ( ! [Y3: nat] :
            ( aa_nat_o(Pa,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(Pa,order_Greatest_nat(Pa)) ) ) ).

% GreatestI_ex_nat
tff(fact_10095_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_aqb(X6)) ).

% Sup_int_def
tff(fact_10096_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_10097_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_10098_bij__betw__Suc,axiom,
    ! [M6: set_nat,N7: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N7)
    <=> ( image_nat_nat(suc,M6) = N7 ) ) ).

% bij_betw_Suc
tff(fact_10099_sqr_Osimps_I3_J,axiom,
    ! [N2: num] : aa_num_num(sqr,aa_num_num(bit1,N2)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,N2)),N2))) ).

% sqr.simps(3)
tff(fact_10100_Rep__unit,axiom,
    ! [X: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(product_Rep_unit,X)),insert_o($true,bot_bot_set_o)) ).

% Rep_unit
tff(fact_10101_Rep__unit__inject,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Product_unit_o(product_Rep_unit,X)
      <=> aa_Product_unit_o(product_Rep_unit,Y) )
    <=> ( X = Y ) ) ).

% Rep_unit_inject
tff(fact_10102_Rep__unit__induct,axiom,
    ! [Y: $o,Pa: fun_o_o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ( ! [X3: product_unit] : aa_o_o(Pa,aa_Product_unit_o(product_Rep_unit,X3))
       => aa_o_o(Pa,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_10103_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ~ ! [X3: product_unit] :
            ( (Y)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X3) ) ) ).

% Rep_unit_cases
tff(fact_10104_type__definition__unit,axiom,
    aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),insert_o($true,bot_bot_set_o)) ).

% type_definition_unit
tff(fact_10105_Unity__def,axiom,
    product_Unity = aa_o_Product_unit(product_Abs_unit,$true) ).

% Unity_def
tff(fact_10106_Rep__unit__inverse,axiom,
    ! [X: product_unit] : aa_o_Product_unit(product_Abs_unit,aa_Product_unit_o(product_Rep_unit,X)) = X ).

% Rep_unit_inverse
tff(fact_10107_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X = aa_o_Product_unit(product_Abs_unit,(Y3)) )
         => ~ aa_set_o_o(member_o((Y3)),insert_o($true,bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_10108_Abs__unit__induct,axiom,
    ! [Pa: fun_Product_unit_o,X: product_unit] :
      ( ! [Y3: $o] :
          ( aa_set_o_o(member_o((Y3)),insert_o($true,bot_bot_set_o))
         => aa_Product_unit_o(Pa,aa_o_Product_unit(product_Abs_unit,(Y3))) )
     => aa_Product_unit_o(Pa,X) ) ).

% Abs_unit_induct
tff(fact_10109_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( aa_set_o_o(member_o((X)),insert_o($true,bot_bot_set_o))
     => ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
       => ( ( aa_o_Product_unit(product_Abs_unit,(X)) = aa_o_Product_unit(product_Abs_unit,(Y)) )
        <=> ( (X)
          <=> (Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_10110_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),insert_o($true,bot_bot_set_o))
     => ( aa_Product_unit_o(product_Rep_unit,aa_o_Product_unit(product_Abs_unit,(Y)))
      <=> (Y) ) ) ).

% Abs_unit_inverse
tff(fact_10111_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa)) ).

% plus_natural.rep_eq
tff(fact_10112_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,X),Xa)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa)) ) ).

% less_natural.rep_eq
tff(fact_10113_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,X),Xa)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa)) ) ).

% less_eq_natural.rep_eq
tff(fact_10114_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_10115_less__natural__def,axiom,
    ord_le5570908160329646204atural = map_fu2458051687388636536ural_o(code_nat_of_natural,map_fu4892316939951275536at_o_o(code_nat_of_natural,id_o),ord_less_nat) ).

% less_natural_def
tff(fact_10116_less__eq__natural__def,axiom,
    ord_le1926595141338095240atural = map_fu2458051687388636536ural_o(code_nat_of_natural,map_fu4892316939951275536at_o_o(code_nat_of_natural,id_o),ord_less_eq_nat) ).

% less_eq_natural_def
tff(fact_10117_natural__decr,axiom,
    ! [N2: code_natural] :
      ( ( N2 != 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,N2)),aa_nat_nat(suc,zero_zero_nat))),aa_Code_natural_nat(code_nat_of_natural,N2)) ) ).

% natural_decr
tff(fact_10118_eventually__sequentially__Suc,axiom,
    ! [Pa: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_aqa(Pa),at_top_nat)
    <=> eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10119_eventually__sequentially__seg,axiom,
    ! [Pa: fun_nat_o,K2: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_aqc(Pa),K2),at_top_nat)
    <=> eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10120_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_eo,at_top_nat) ).

% eventually_False_sequentially
tff(fact_10121_eventually__sequentially,axiom,
    ! [Pa: fun_nat_o] :
      ( eventually_nat(Pa,at_top_nat)
    <=> ? [N9: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N)
         => aa_nat_o(Pa,N) ) ) ).

% eventually_sequentially
tff(fact_10122_eventually__sequentiallyI,axiom,
    ! [C2: nat,Pa: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X3)
         => aa_nat_o(Pa,X3) )
     => eventually_nat(Pa,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10123_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N9),F4) ) ).

% le_sequentially
tff(fact_10124_natLeq__on__wo__rel,axiom,
    ! [N2: nat] : aa_set3702741264012733747_nat_o(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2)))) ).

% natLeq_on_wo_rel
tff(fact_10125_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,T6: 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,S2),T6)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T6)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_10126_total__pair__less,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : total_3592101749530773125at_nat(A4,fun_pair_less) ).

% total_pair_less
tff(fact_10127_trans__pair__less,axiom,
    trans_4347625901269045472at_nat(fun_pair_less) ).

% trans_pair_less
tff(fact_10128_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z3)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z3) ) ).

% pair_less_iff1
tff(fact_10129_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_10130_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_10131_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,T6: 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),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T6)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_10132_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_10133_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,T6: 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,S2),T6)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T6)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_10134_pair__leq__def,axiom,
    fun_pair_leq = sup_su718114333110466843at_nat(fun_pair_less,id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_10135_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_strict) ) ).

% smin_emptyI
tff(fact_10136_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_10137_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,T6: 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),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T6)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_10138_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),YS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_max_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS)),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_10139_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),XS)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS)),fun_min_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS))),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_10140_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,bot_bo2099793752762293965at_nat)),fun_min_weak) ).

% wmin_emptyI
tff(fact_10141_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X6)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X6)),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_10142_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_10143_min__weak__def,axiom,
    fun_min_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(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_10144_max__weak__def,axiom,
    fun_max_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(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_10145_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),Y6)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(X6,Y6)),fun_max_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X6),Y6)),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_10146_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_10147_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_10148_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y6)
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y6)),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_10149_division__segment__int__def,axiom,
    ! [K2: int] :
      aa_int_int(euclid3395696857347342551nt_int,K2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% division_segment_int_def
tff(fact_10150_smsI,axiom,
    ! [A4: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A4),set_ms8126754132646256062at_nat(B4))),fun_max_strict)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),A4),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),B4)),ms_strict) ) ).

% smsI
tff(fact_10151_wmsI,axiom,
    ! [A4: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] :
      ( ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A4),set_ms8126754132646256062at_nat(B4))),fun_max_strict)
        | ( ( A4 = zero_z1048942125864253310at_nat )
          & ( B4 = zero_z1048942125864253310at_nat ) ) )
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),A4),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),B4)),ms_weak) ) ).

% wmsI
tff(fact_10152_ms__weak__def,axiom,
    ms_weak = aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,ms_strict),id_mul2649389997224486051at_nat) ).

% ms_weak_def
tff(fact_10153_ms__weakI1,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A4: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat] :
      ( pw_leq(Z4,Z7)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A4),set_ms8126754132646256062at_nat(B4))),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),A4),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z7),B4)),ms_weak) ) ) ).

% ms_weakI1
tff(fact_10154_ms__strictI,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat,A4: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat] :
      ( pw_leq(Z4,Z7)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A4),set_ms8126754132646256062at_nat(B4))),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),A4),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z7),B4)),ms_strict) ) ) ).

% ms_strictI
tff(fact_10155_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A22 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,X9: multis2468970476368604999at_nat] :
              ( ( A1 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X3,zero_z1048942125864253310at_nat)),X9) )
             => ! [Y8: multis2468970476368604999at_nat] :
                  ( ( A22 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y3,zero_z1048942125864253310at_nat)),Y8) )
                 => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Y3),fun_pair_leq)
                   => ~ pw_leq(X9,Y8) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_10156_pw__leq_Osimps,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
    <=> ( ( ( A1 = zero_z1048942125864253310at_nat )
          & ( A22 = zero_z1048942125864253310at_nat ) )
        | ? [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,X10: multis2468970476368604999at_nat,Y9: multis2468970476368604999at_nat] :
            ( ( A1 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X4,zero_z1048942125864253310at_nat)),X10) )
            & ( A22 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y4,zero_z1048942125864253310at_nat)),Y9) )
            & member8206827879206165904at_nat(produc6161850002892822231at_nat(X4,Y4),fun_pair_leq)
            & pw_leq(X10,Y9) ) ) ) ).

% pw_leq.simps
tff(fact_10157_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
     => ( pw_leq(X6,Y6)
       => pw_leq(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat)),X6),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)),Y6)) ) ) ).

% pw_leq_step
tff(fact_10158_ms__weakI2,axiom,
    ! [Z4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
      ( pw_leq(Z4,Z7)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z4),zero_z1048942125864253310at_nat),aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(Z7),zero_z1048942125864253310at_nat)),ms_weak) ) ).

% ms_weakI2
tff(fact_10159_pw__leq__split,axiom,
    ! [X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( pw_leq(X6,Y6)
     => ? [A7: multis2468970476368604999at_nat,B7: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat] :
          ( ( X6 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(A7),Z8) )
          & ( Y6 = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(B7),Z8) )
          & ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A7),set_ms8126754132646256062at_nat(B7))),fun_max_strict)
            | ( ( B7 = zero_z1048942125864253310at_nat )
              & ( A7 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
tff(fact_10160_bdd__above__nat,axiom,
    ! [X6: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,X6)
    <=> aa_set_nat_o(finite_finite_nat,X6) ) ).

% bdd_above_nat
tff(fact_10161_natLeq__on__well__order__on,axiom,
    ! [N2: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,N2))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2)))) ).

% natLeq_on_well_order_on
tff(fact_10162_natLeq__on__Well__order,axiom,
    ! [N2: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2))))),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2)))) ).

% natLeq_on_Well_order
tff(fact_10163_natLeq__Card__order,axiom,
    aa_set3702741264012733747_nat_o(bNF_Ca1281551314933786834on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Card_order
tff(fact_10164_natLeq__antisym,axiom,
    aa_set3702741264012733747_nat_o(antisym_nat,bNF_Ca8665028551170535155natLeq) ).

% natLeq_antisym
tff(fact_10165_natLeq__natLess__Id,axiom,
    bNF_Ca8459412986667044542atLess = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bNF_Ca8665028551170535155natLeq),id_nat2) ).

% natLeq_natLess_Id
tff(fact_10166_natLeq__trans,axiom,
    aa_set3702741264012733747_nat_o(trans_nat,bNF_Ca8665028551170535155natLeq) ).

% natLeq_trans
tff(fact_10167_natLeq__underS__less,axiom,
    ! [N2: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,N2) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,N2)) ).

% natLeq_underS_less
tff(fact_10168_natLeq__Total,axiom,
    aa_set3702741264012733747_nat_o(total_on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Total
tff(fact_10169_natLeq__Linear__order,axiom,
    aa_set3702741264012733747_nat_o(order_4473980167227706203on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Linear_order
tff(fact_10170_natLeq__Preorder,axiom,
    aa_set3702741264012733747_nat_o(order_4861654808422542329on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Preorder
tff(fact_10171_natLeq__Refl,axiom,
    aa_set3702741264012733747_nat_o(refl_on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Refl
tff(fact_10172_Field__natLeq,axiom,
    field_nat(bNF_Ca8665028551170535155natLeq) = top_top_set_nat ).

% Field_natLeq
tff(fact_10173_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_10174_natLeq__Well__order,axiom,
    aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Well_order
tff(fact_10175_Restr__natLeq2,axiom,
    ! [N2: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,N2),aTP_Lamm_aqd(N2))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2))) ).

% Restr_natLeq2
tff(fact_10176_Restr__natLeq,axiom,
    ! [N2: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,N2)),aTP_Lamm_aqe(N2))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_apz(N2))) ).

% Restr_natLeq
tff(fact_10177_card__of__nat,axiom,
    aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).

% card_of_nat
tff(fact_10178_card__of__Field__natLeq,axiom,
    aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq)),bNF_We5258908940166488438at_nat) ).

% card_of_Field_natLeq
tff(fact_10179_eventually__prod__sequentially,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(Pa,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),M3)
         => ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N)
             => aa_Pro6150637604445964947_nat_o(Pa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),M3)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10180_prod__decode__triangle__add,axiom,
    ! [K2: nat,M: nat] : nat_prod_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),M)) = nat_prod_decode_aux(K2,M) ).

% prod_decode_triangle_add
tff(fact_10181_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N5: nat] :
              ( ( X = aa_nat_nat(suc,N5) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_aqf,nat_prod_decode(N5)) ) ) ) ) ).

% list_decode.elims
tff(fact_10182_list__decode_Osimps_I2_J,axiom,
    ! [N2: nat] : nat_list_decode(aa_nat_nat(suc,N2)) = produc2761476792215241774st_nat(aTP_Lamm_aqf,nat_prod_decode(N2)) ).

% list_decode.simps(2)
tff(fact_10183_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( nat_list_decode(X) = Y )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N5: nat] :
                ( ( X = aa_nat_nat(suc,N5) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_aqf,nat_prod_decode(N5)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N5)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_10184_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))
     => ( nat_list_decode(aa_nat_nat(suc,N2)) = produc2761476792215241774st_nat(aTP_Lamm_aqf,nat_prod_decode(N2)) ) ) ).

% list_decode.psimps(2)
tff(fact_10185_rat_Odomain,axiom,
    ! [X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(domain9213661015745956397nt_rat(pcr_rat),X5)
    <=> ? [Y4: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),X5),Y4)
          & aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Y4),Y4) ) ) ).

% rat.domain
tff(fact_10186_int_Oid__abs__transfer,axiom,
    bNF_re7934895593101944656at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),pcr_int,aTP_Lamm_aqg,abs_Integ) ).

% int.id_abs_transfer
tff(fact_10187_rat_Odomain__par__left__total,axiom,
    ! [P3: 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,P3,aTP_Lamm_aqh)
       => ( domain9213661015745956397nt_rat(pcr_rat) = P3 ) ) ) ).

% rat.domain_par_left_total
tff(fact_10188_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_aqh)
         => ( domain9213661015745956397nt_rat(pcr_rat) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,basic_1567116559311922317nt_int(DR1,DR2)),P22) ) ) ) ) ).

% rat.domain_par
tff(fact_10189_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y3: set_Pr958786334691620121nt_int] :
          ( ( X = aa_set4118085860027418549nt_rat(abs_rat,Y3) )
         => ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y3),collec5210948495886036740nt_int(aTP_Lamm_aqi)) ) ).

% Abs_rat_cases
tff(fact_10190_Abs__rat__induct,axiom,
    ! [Pa: fun_rat_o,X: rat] :
      ( ! [Y3: set_Pr958786334691620121nt_int] :
          ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y3),collec5210948495886036740nt_int(aTP_Lamm_aqi))
         => aa_rat_o(Pa,aa_set4118085860027418549nt_rat(abs_rat,Y3)) )
     => aa_rat_o(Pa,X) ) ).

% Abs_rat_induct
tff(fact_10191_Abs__rat__inject,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(X),collec5210948495886036740nt_int(aTP_Lamm_aqi))
     => ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aqi))
       => ( ( aa_set4118085860027418549nt_rat(abs_rat,X) = aa_set4118085860027418549nt_rat(abs_rat,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_rat_inject
tff(fact_10192_Abs__rat__inverse,axiom,
    ! [Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aqi))
     => ( aa_rat8624586542854104963nt_int(rep_rat,aa_set4118085860027418549nt_rat(abs_rat,Y)) = Y ) ) ).

% Abs_rat_inverse
tff(fact_10193_type__definition__rat,axiom,
    aa_set6820988702716878619_int_o(type_d8554052265237484179nt_int(rep_rat,abs_rat),collec5210948495886036740nt_int(aTP_Lamm_aqi)) ).

% type_definition_rat
tff(fact_10194_Rep__rat,axiom,
    ! [X: rat] : aa_set6820988702716878619_int_o(member2340774599025711042nt_int(aa_rat8624586542854104963nt_int(rep_rat,X)),collec5210948495886036740nt_int(aTP_Lamm_aqi)) ).

% Rep_rat
tff(fact_10195_Rep__rat__cases,axiom,
    ! [Y: set_Pr958786334691620121nt_int] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aqi))
     => ~ ! [X3: rat] : Y != aa_rat8624586542854104963nt_int(rep_rat,X3) ) ).

% Rep_rat_cases
tff(fact_10196_Rep__rat__induct,axiom,
    ! [Y: set_Pr958786334691620121nt_int,Pa: fun_se7249947736802449410_int_o] :
      ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Y),collec5210948495886036740nt_int(aTP_Lamm_aqi))
     => ( ! [X3: rat] : aa_set6170482117830564987_int_o(Pa,aa_rat8624586542854104963nt_int(rep_rat,X3))
       => aa_set6170482117830564987_int_o(Pa,Y) ) ) ).

% Rep_rat_induct
tff(fact_10197_one__assn__def,axiom,
    one_one_assn = abs_assn(one_assn_raw) ).

% one_assn_def
tff(fact_10198_natLeq__Partial__order,axiom,
    aa_set3702741264012733747_nat_o(order_5251275573222108571on_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Partial_order
tff(fact_10199_one__assn__raw_Osimps,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Asa: set_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,produc7507926704131184380et_nat(Ha,Asa))
    <=> ( Asa = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_10200_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( (Y)
            <=> ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_10201_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_10202_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_10203_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H,As))
               => ( As = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_10204_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H,As))
               => ( As != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_10205_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H,As) )
             => ( ( (Y)
                <=> ( As = bot_bot_set_nat ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H,As)) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_10206_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,(B72))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B62)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B52)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B42)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B32)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B22)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B1)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B0))) ).

% integer_of_char_code
tff(fact_10207_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_apk,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_10208_ATP_Olambda__2,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_af,Uu)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),lim_Product_unit(h2))
        & ~ aa_set_nat_o(member_nat(Uu),hoare_new_addrs(h3,as,h2)) ) ) ).

% ATP.lambda_2
tff(fact_10209_ATP_Olambda__3,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ag,Uu)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),lim_Product_unit(h3))
        & ~ aa_set_nat_o(member_nat(Uu),as) ) ) ).

% ATP.lambda_3
tff(fact_10210_ATP_Olambda__4,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_api,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_4
tff(fact_10211_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_qh,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_5
tff(fact_10212_ATP_Olambda__6,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_apf,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_6
tff(fact_10213_ATP_Olambda__7,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aqh,Uu)
    <=> aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Uu),Uu) ) ).

% ATP.lambda_7
tff(fact_10214_ATP_Olambda__8,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_gz,Uu) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),one_one_Code_integer) ).

% ATP.lambda_8
tff(fact_10215_ATP_Olambda__9,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_gw,Uu) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),one_one_rat) ).

% ATP.lambda_9
tff(fact_10216_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_gy,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_10
tff(fact_10217_ATP_Olambda__11,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_gx,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_11
tff(fact_10218_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_apm,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_12
tff(fact_10219_ATP_Olambda__13,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_apl,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_13
tff(fact_10220_ATP_Olambda__14,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_nb,Uu) = produc5538323210962509403atural(next,aTP_Lamm_na(Uu)) ).

% ATP.lambda_14
tff(fact_10221_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_apw,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jl,Uu)) ).

% ATP.lambda_15
tff(fact_10222_ATP_Olambda__16,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_mr,Uu) = aa_num_option_num(some_num,aa_num_num(bit1,Uu)) ).

% ATP.lambda_16
tff(fact_10223_ATP_Olambda__17,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_jt,Uu) = aa_num_option_num(some_num,aa_num_num(bit0,Uu)) ).

% ATP.lambda_17
tff(fact_10224_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aog,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_18
tff(fact_10225_ATP_Olambda__19,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aoh,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_19
tff(fact_10226_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_js,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_20
tff(fact_10227_ATP_Olambda__21,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_apx,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_21
tff(fact_10228_ATP_Olambda__22,axiom,
    ! [Uu: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_aqi,Uu)
    <=> ? [X4: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X4),X4)
          & ( Uu = collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(ratrel,X4)) ) ) ) ).

% ATP.lambda_22
tff(fact_10229_ATP_Olambda__23,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ng(Uu),Uua) = aa_num_option_num(case_num_option_num(aa_num_option_num(some_num,one),aTP_Lamm_ne(Uua),aTP_Lamm_nf(Uua)),Uu) ).

% ATP.lambda_23
tff(fact_10230_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_ps(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_24
tff(fact_10231_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_pq(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_25
tff(fact_10232_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pr,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_26
tff(fact_10233_ATP_Olambda__27,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aos,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_27
tff(fact_10234_ATP_Olambda__28,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aph,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_28
tff(fact_10235_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_pp(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_29
tff(fact_10236_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_pn(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_30
tff(fact_10237_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_po,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_31
tff(fact_10238_ATP_Olambda__32,axiom,
    ! [Uu: option2108551362046372146_int_o,Uua: fun_Pr3238374608701176162_int_o] : aa_fun205347844938550866_int_o(aTP_Lamm_gp(Uu),Uua) = case_o7864645398209642145_int_o(none_f6270875889056561217_int_o,aTP_Lamm_go(Uua),Uu) ).

% ATP.lambda_32
tff(fact_10239_ATP_Olambda__33,axiom,
    ! [Uu: option_nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_gn(Uu),Uua) = case_o7429725398727453821at_nat(none_nat,aTP_Lamm_gm(Uua),Uu) ).

% ATP.lambda_33
tff(fact_10240_ATP_Olambda__34,axiom,
    ! [Uu: option_set_nat,Uua: set_nat] : aa_set6119337032432945050et_nat(aTP_Lamm_gl(Uu),Uua) = case_o4054078431260844265et_nat(none_set_nat,aTP_Lamm_gk(Uua),Uu) ).

% ATP.lambda_34
tff(fact_10241_ATP_Olambda__35,axiom,
    ! [Uu: option8963830502488799655at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set2934564513904597954at_nat(aTP_Lamm_gj(Uu),Uua) = case_o311359030874850053at_nat(none_s625347054029921090at_nat,aTP_Lamm_gi(Uua),Uu) ).

% ATP.lambda_35
tff(fact_10242_ATP_Olambda__36,axiom,
    ! [Uu: option_assn,Uua: assn] : aa_assn_option_assn(aTP_Lamm_gh(Uu),Uua) = case_o4484465799723439917n_assn(none_assn,aTP_Lamm_gg(Uua),Uu) ).

% ATP.lambda_36
tff(fact_10243_ATP_Olambda__37,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_ds(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($true,aa_num_fun_num_o(aTP_Lamm_dr,Uua)),Uu) ) ).

% ATP.lambda_37
tff(fact_10244_ATP_Olambda__38,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_du(Uu),Uua)
    <=> case_option_o_int($true,aa_int_fun_int_o(aTP_Lamm_ca,Uua),Uu) ) ).

% ATP.lambda_38
tff(fact_10245_ATP_Olambda__39,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_dt(Uu),Uua)
    <=> case_option_o_nat($true,aa_nat_fun_nat_o(aTP_Lamm_by,Uua),Uu) ) ).

% ATP.lambda_39
tff(fact_10246_ATP_Olambda__40,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_dq(Uu),Uua)
    <=> case_option_o_rat($true,aa_rat_fun_rat_o(aTP_Lamm_bw,Uua),Uu) ) ).

% ATP.lambda_40
tff(fact_10247_ATP_Olambda__41,axiom,
    ! [Uu: option_assn,Uua: assn] :
      ( aa_assn_o(aTP_Lamm_dp(Uu),Uua)
    <=> case_option_o_assn($true,aa_assn_fun_assn_o(aTP_Lamm_bs,Uua),Uu) ) ).

% ATP.lambda_41
tff(fact_10248_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_nh,Uu),Uua) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_ng(Uua)),Uu) ).

% ATP.lambda_42
tff(fact_10249_ATP_Olambda__43,axiom,
    ! [Uu: option_set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_dk(Uu),Uua)
    <=> case_o223999843215110191et_int($false,aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uua),Uu) ) ).

% ATP.lambda_43
tff(fact_10250_ATP_Olambda__44,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_dl(Uu),Uua)
    <=> case_option_o_rat($false,aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_44
tff(fact_10251_ATP_Olambda__45,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_dm(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($false,aa_num_fun_num_o(ord_less_eq_num,Uua)),Uu) ) ).

% ATP.lambda_45
tff(fact_10252_ATP_Olambda__46,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_dn(Uu),Uua)
    <=> case_option_o_nat($false,aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_46
tff(fact_10253_ATP_Olambda__47,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_do(Uu),Uua)
    <=> case_option_o_int($false,aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_47
tff(fact_10254_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ne(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_jt),aa_num_option_num(bit_take_bit_num(Uu),Uua)) ).

% ATP.lambda_48
tff(fact_10255_ATP_Olambda__49,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_wa(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_vz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_49
tff(fact_10256_ATP_Olambda__50,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_vy(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_50
tff(fact_10257_ATP_Olambda__51,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qs(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_51
tff(fact_10258_ATP_Olambda__52,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_apg,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_52
tff(fact_10259_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_sd(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_53
tff(fact_10260_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_sb(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_54
tff(fact_10261_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sc,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_55
tff(fact_10262_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_se(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_56
tff(fact_10263_ATP_Olambda__57,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_ns,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_57
tff(fact_10264_ATP_Olambda__58,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_apj,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_58
tff(fact_10265_ATP_Olambda__59,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aot(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_59
tff(fact_10266_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_so(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_60
tff(fact_10267_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_sm(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_61
tff(fact_10268_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sn,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_62
tff(fact_10269_ATP_Olambda__63,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_apy(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_63
tff(fact_10270_ATP_Olambda__64,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qp(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_64
tff(fact_10271_ATP_Olambda__65,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_pt(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_65
tff(fact_10272_ATP_Olambda__66,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rf(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_66
tff(fact_10273_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_jn,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_67
tff(fact_10274_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rm,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_68
tff(fact_10275_ATP_Olambda__69,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aqb(Uu),Uua)
    <=> ( aa_set_int_o(member_int(Uua),Uu)
        & ! [X4: int] :
            ( aa_set_int_o(member_int(X4),Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Uua) ) ) ) ).

% ATP.lambda_69
tff(fact_10276_ATP_Olambda__70,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_ajx,Uu),Uua)
    <=> ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(Uu),Uua)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Uua) ) ) ).

% ATP.lambda_70
tff(fact_10277_ATP_Olambda__71,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_ajy,Uu),Uua)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,Uu),Uua)
        & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Uua) ) ) ).

% ATP.lambda_71
tff(fact_10278_ATP_Olambda__72,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ajv,Uu),Uua)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uu),Uua)
        & aa_set_nat_o(finite_finite_nat,Uua) ) ) ).

% ATP.lambda_72
tff(fact_10279_ATP_Olambda__73,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_ajw,Uu),Uua)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Uu),Uua)
        & aa_set_int_o(finite_finite_int,Uua) ) ) ).

% ATP.lambda_73
tff(fact_10280_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aps,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_74
tff(fact_10281_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qj,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_75
tff(fact_10282_ATP_Olambda__76,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,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_76
tff(fact_10283_ATP_Olambda__77,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_agy,Uu),Uua)
    <=> ( aa_set_rat_o(member_rat(Uua),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Uua)),Uu) ) ) ).

% ATP.lambda_77
tff(fact_10284_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_uy(Uu),Uua) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(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_assn(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_78
tff(fact_10285_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_uz(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_79
tff(fact_10286_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_va(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_80
tff(fact_10287_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_vb(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_81
tff(fact_10288_ATP_Olambda__82,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : aa_nat2240542744625857721at_nat(aTP_Lamm_rb(Uu),Uua) = aa_mul5754855110707352700at_nat(plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_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_nat2240542744625857721at_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_82
tff(fact_10289_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rc(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_83
tff(fact_10290_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_re(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_84
tff(fact_10291_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rd(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_85
tff(fact_10292_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qf(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_86
tff(fact_10293_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qg(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_87
tff(fact_10294_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_tt(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_88
tff(fact_10295_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_tv(Uu),Uua) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua)),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_89
tff(fact_10296_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_un(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),set_or4665077453230672383an_nat(zero_zero_nat,Uua)) ).

% ATP.lambda_90
tff(fact_10297_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_za(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_91
tff(fact_10298_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qy(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_92
tff(fact_10299_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qz(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_93
tff(fact_10300_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_qv(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_94
tff(fact_10301_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qw(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_95
tff(fact_10302_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qx(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_96
tff(fact_10303_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qd(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_97
tff(fact_10304_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qe(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_98
tff(fact_10305_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_wf(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_99
tff(fact_10306_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_wg(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_100
tff(fact_10307_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_we(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_101
tff(fact_10308_ATP_Olambda__102,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_afp(Uu),Uua)
    <=> ( aa_Cod1783974652533396728nteger(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_102
tff(fact_10309_ATP_Olambda__103,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_afs(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_103
tff(fact_10310_ATP_Olambda__104,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_afu(Uu),Uua)
    <=> ( aa_Code_integer_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_104
tff(fact_10311_ATP_Olambda__105,axiom,
    ! [Uu: fun_Code_integer_int,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_afw(Uu),Uua)
    <=> ( aa_Code_integer_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_105
tff(fact_10312_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_afn(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_106
tff(fact_10313_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_afq(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_107
tff(fact_10314_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_int,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_afv(Uu),Uua)
    <=> ( aa_nat_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_108
tff(fact_10315_ATP_Olambda__109,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_afo(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = zero_z3403309356797280102nteger ) ) ).

% ATP.lambda_109
tff(fact_10316_ATP_Olambda__110,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_afr(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_110
tff(fact_10317_ATP_Olambda__111,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aft(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_111
tff(fact_10318_ATP_Olambda__112,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_agq(Uu),Uua)
    <=> ( aa_Cod1783974652533396728nteger(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_112
tff(fact_10319_ATP_Olambda__113,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_agt(Uu),Uua)
    <=> ( aa_Code_integer_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_113
tff(fact_10320_ATP_Olambda__114,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_agw(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_114
tff(fact_10321_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ago(Uu),Uua)
    <=> ( aa_nat_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_115
tff(fact_10322_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_agr(Uu),Uua)
    <=> ( aa_nat_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_116
tff(fact_10323_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_agu(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_117
tff(fact_10324_ATP_Olambda__118,axiom,
    ! [Uu: fun_int_Code_integer,Uua: int] :
      ( aa_int_o(aTP_Lamm_agp(Uu),Uua)
    <=> ( aa_int_Code_integer(Uu,Uua) = one_one_Code_integer ) ) ).

% ATP.lambda_118
tff(fact_10325_ATP_Olambda__119,axiom,
    ! [Uu: fun_int_assn,Uua: int] :
      ( aa_int_o(aTP_Lamm_ags(Uu),Uua)
    <=> ( aa_int_assn(Uu,Uua) = one_one_assn ) ) ).

% ATP.lambda_119
tff(fact_10326_ATP_Olambda__120,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_agv(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_120
tff(fact_10327_ATP_Olambda__121,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_agx(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_121
tff(fact_10328_ATP_Olambda__122,axiom,
    ! [Uu: list_int,Uua: heap_e7401611519738050253t_unit] : aa_hea5056872858427967349it_nat(aTP_Lamm_ajj(Uu),Uua) = produc3016605123915955252it_nat(aTP_Lamm_wr(Uu),array_alloc_int(Uu,Uua)) ).

% ATP.lambda_122
tff(fact_10329_ATP_Olambda__123,axiom,
    ! [Uu: list_o,Uua: heap_e7401611519738050253t_unit] : aa_hea6592385862050124941it_nat(aTP_Lamm_aji(Uu),Uua) = produc5512802776183222702it_nat(aTP_Lamm_wq(Uu),array_alloc_o(Uu,Uua)) ).

% ATP.lambda_123
tff(fact_10330_ATP_Olambda__124,axiom,
    ! [Uu: list_nat,Uua: heap_e7401611519738050253t_unit] : aa_hea4612425101438060313it_nat(aTP_Lamm_ajh(Uu),Uua) = produc8657510640128716596it_nat(aTP_Lamm_wp(Uu),array_alloc_nat(Uu,Uua)) ).

% ATP.lambda_124
tff(fact_10331_ATP_Olambda__125,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apn,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_125
tff(fact_10332_ATP_Olambda__126,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_jr(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_126
tff(fact_10333_ATP_Olambda__127,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ho(Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(rep_assn(Uu),Uua) ) ) ).

% ATP.lambda_127
tff(fact_10334_ATP_Olambda__128,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_ape,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_128
tff(fact_10335_ATP_Olambda__129,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_nr,Uu),Uua) = $let(
        l2: num,
        l2:= aa_Code_integer_num(code_num_of_integer,Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_129
tff(fact_10336_ATP_Olambda__130,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_pm,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_130
tff(fact_10337_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_rz(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_131
tff(fact_10338_ATP_Olambda__132,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ip,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_132
tff(fact_10339_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_io,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_133
tff(fact_10340_ATP_Olambda__134,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_iq,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_134
tff(fact_10341_ATP_Olambda__135,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_is,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_135
tff(fact_10342_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ir,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_136
tff(fact_10343_ATP_Olambda__137,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_it,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_137
tff(fact_10344_ATP_Olambda__138,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sp(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_138
tff(fact_10345_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sk,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_139
tff(fact_10346_ATP_Olambda__140,axiom,
    ! [Uu: product_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea256619718444519449it_nat(aTP_Lamm_fg(Uu),Uua) = produc7133225469290080770it_nat(Uu,produc584006145561248582it_nat(Uua,zero_zero_nat)) ).

% ATP.lambda_140
tff(fact_10347_ATP_Olambda__141,axiom,
    ! [Uu: b,Uua: heap_e7401611519738050253t_unit] : aa_hea550092171636296971it_nat(aTP_Lamm_fh(Uu),Uua) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Uu),produc584006145561248582it_nat(Uua,zero_zero_nat)) ).

% ATP.lambda_141
tff(fact_10348_ATP_Olambda__142,axiom,
    ! [Uu: a,Uua: heap_e7401611519738050253t_unit] : aa_hea5645563107516622602it_nat(aTP_Lamm_fi(Uu),Uua) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Uu),produc584006145561248582it_nat(Uua,zero_zero_nat)) ).

% ATP.lambda_142
tff(fact_10349_ATP_Olambda__143,axiom,
    ! [Uu: product_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea256619718444519449it_nat(aTP_Lamm_fb(Uu),Uua) = produc7133225469290080770it_nat(Uu,produc584006145561248582it_nat(Uua,one_one_nat)) ).

% ATP.lambda_143
tff(fact_10350_ATP_Olambda__144,axiom,
    ! [Uu: b,Uua: heap_e7401611519738050253t_unit] : aa_hea550092171636296971it_nat(aTP_Lamm_ez(Uu),Uua) = aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Uu),produc584006145561248582it_nat(Uua,one_one_nat)) ).

% ATP.lambda_144
tff(fact_10351_ATP_Olambda__145,axiom,
    ! [Uu: a,Uua: heap_e7401611519738050253t_unit] : aa_hea5645563107516622602it_nat(aTP_Lamm_fa(Uu),Uua) = aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Uu),produc584006145561248582it_nat(Uua,one_one_nat)) ).

% ATP.lambda_145
tff(fact_10352_ATP_Olambda__146,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_aje(Uu),Uua) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(Uua),aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,Uu),Uua))) ).

% ATP.lambda_146
tff(fact_10353_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ajg,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_147
tff(fact_10354_ATP_Olambda__148,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ajf,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_148
tff(fact_10355_ATP_Olambda__149,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ms,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_149
tff(fact_10356_ATP_Olambda__150,axiom,
    ! [Uu: code_natural,Uua: nat] : aa_nat_Code_natural(aTP_Lamm_ui(Uu),Uua) = minus_7197305767214868737atural(Uu,aa_nat_Code_natural(semiri3763490453095760265atural,Uua)) ).

% ATP.lambda_150
tff(fact_10357_ATP_Olambda__151,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_uj(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_151
tff(fact_10358_ATP_Olambda__152,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_uh(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_152
tff(fact_10359_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uk,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_153
tff(fact_10360_ATP_Olambda__154,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ul,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_154
tff(fact_10361_ATP_Olambda__155,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_tz(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_155
tff(fact_10362_ATP_Olambda__156,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ty(Uu),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_156
tff(fact_10363_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ua,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_157
tff(fact_10364_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ub,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_158
tff(fact_10365_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_aqf,Uu),Uua) = cons_nat(Uu,nat_list_decode(Uua)) ).

% ATP.lambda_159
tff(fact_10366_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aou,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_160
tff(fact_10367_ATP_Olambda__161,axiom,
    ! [Uu: assn,Uua: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_br,Uu),Uua)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_10368_ATP_Olambda__162,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_ajk(Uu),Uua)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_10369_ATP_Olambda__163,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] :
      ( aa_set192106157127879885_nat_o(aa_set8206400607124912991_nat_o(aTP_Lamm_bl,Uu),Uua)
    <=> ord_le8081472938463900775at_nat(Uua,Uu) ) ).

% ATP.lambda_163
tff(fact_10370_ATP_Olambda__164,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_bn,Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(ord_le1268244103169919719at_nat(Uua),Uu) ) ).

% ATP.lambda_164
tff(fact_10371_ATP_Olambda__165,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_eq,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uua),Uu) ) ).

% ATP.lambda_165
tff(fact_10372_ATP_Olambda__166,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_pu(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(Uua),Uu) ) ).

% ATP.lambda_166
tff(fact_10373_ATP_Olambda__167,axiom,
    ! [Uu: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_es,Uu),Uua)
    <=> aa_set_o_o(ord_less_eq_set_o(Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_10374_ATP_Olambda__168,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_bp,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_10375_ATP_Olambda__169,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_bt,Uu),Uua)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uua),Uu) ) ).

% ATP.lambda_169
tff(fact_10376_ATP_Olambda__170,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_bv,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_10377_ATP_Olambda__171,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_pv,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_10378_ATP_Olambda__172,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bx,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_10379_ATP_Olambda__173,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bz,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_173
tff(fact_10380_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_174
tff(fact_10381_ATP_Olambda__175,axiom,
    ! [Uu: assn,Uua: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_bs,Uu),Uua)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_10382_ATP_Olambda__176,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] :
      ( aa_set192106157127879885_nat_o(aa_set8206400607124912991_nat_o(aTP_Lamm_bm,Uu),Uua)
    <=> ord_le7642048601412989811at_nat(Uua,Uu) ) ).

% ATP.lambda_176
tff(fact_10383_ATP_Olambda__177,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_bo,Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(ord_le2604355607129572851at_nat(Uua),Uu) ) ).

% ATP.lambda_177
tff(fact_10384_ATP_Olambda__178,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_er,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,Uua),Uu) ) ).

% ATP.lambda_178
tff(fact_10385_ATP_Olambda__179,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_vv(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(ord_le7563427860532173253nt_int(Uua),Uu) ) ).

% ATP.lambda_179
tff(fact_10386_ATP_Olambda__180,axiom,
    ! [Uu: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_et,Uu),Uua)
    <=> aa_set_o_o(ord_less_set_o(Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_10387_ATP_Olambda__181,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_bq,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_10388_ATP_Olambda__182,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_bu,Uu),Uua)
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_10389_ATP_Olambda__183,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_bw,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_183
tff(fact_10390_ATP_Olambda__184,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_dr,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_10391_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_by,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_185
tff(fact_10392_ATP_Olambda__186,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ca,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_186
tff(fact_10393_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_187
tff(fact_10394_ATP_Olambda__188,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_apa,Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uua),Uu) ).

% ATP.lambda_188
tff(fact_10395_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aov,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_189
tff(fact_10396_ATP_Olambda__190,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aoz,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_190
tff(fact_10397_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_jl,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_191
tff(fact_10398_ATP_Olambda__192,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jm,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_192
tff(fact_10399_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_anv,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_193
tff(fact_10400_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_194
tff(fact_10401_ATP_Olambda__195,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aTP_Lamm_bj(Uu),Uua)
    <=> member6689249552917799696at_nat(Uua,Uu) ) ).

% ATP.lambda_195
tff(fact_10402_ATP_Olambda__196,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_bk(Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_10403_ATP_Olambda__197,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_gq(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_197
tff(fact_10404_ATP_Olambda__198,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gr(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Uua),Uu) ) ).

% ATP.lambda_198
tff(fact_10405_ATP_Olambda__199,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_aa(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_199
tff(fact_10406_ATP_Olambda__200,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_a(Uu),(Uua))
    <=> aa_set_o_o(member_o((Uua)),Uu) ) ).

% ATP.lambda_200
tff(fact_10407_ATP_Olambda__201,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_10408_ATP_Olambda__202,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_10409_ATP_Olambda__203,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_203
tff(fact_10410_ATP_Olambda__204,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_204
tff(fact_10411_ATP_Olambda__205,axiom,
    ! [Uu: produc8664842809031399944it_nat,Uua: produc8664842809031399944it_nat] :
      ( aa_Pro4744234731172327378_nat_o(aTP_Lamm_cu(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_205
tff(fact_10412_ATP_Olambda__206,axiom,
    ! [Uu: produc4166570645942440679at_nat,Uua: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aTP_Lamm_mq(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_206
tff(fact_10413_ATP_Olambda__207,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_lc(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_207
tff(fact_10414_ATP_Olambda__208,axiom,
    ! [Uu: product_prod_nat_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_lg,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_208
tff(fact_10415_ATP_Olambda__209,axiom,
    ! [Uu: produc7388388658123137530it_nat,Uua: produc7388388658123137530it_nat] :
      ( aa_Pro1537530206668979808_nat_o(aTP_Lamm_cs(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_209
tff(fact_10416_ATP_Olambda__210,axiom,
    ! [Uu: produc3260487557148687353it_nat,Uua: produc3260487557148687353it_nat] :
      ( aa_Pro6426884464393220769_nat_o(aTP_Lamm_ct(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_210
tff(fact_10417_ATP_Olambda__211,axiom,
    ! [Uu: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ld(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_211
tff(fact_10418_ATP_Olambda__212,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_lh,(Uu)),(Uua))
    <=> ( (Uua)
      <=> (Uu) ) ) ).

% ATP.lambda_212
tff(fact_10419_ATP_Olambda__213,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_le(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_213
tff(fact_10420_ATP_Olambda__214,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_lf(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_214
tff(fact_10421_ATP_Olambda__215,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_cv,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_215
tff(fact_10422_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_li,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_216
tff(fact_10423_ATP_Olambda__217,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_lj,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_217
tff(fact_10424_ATP_Olambda__218,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_aeh(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua)) ) ).

% ATP.lambda_218
tff(fact_10425_ATP_Olambda__219,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_aej(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(Uu,Uua)) ) ).

% ATP.lambda_219
tff(fact_10426_ATP_Olambda__220,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_adx(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_220
tff(fact_10427_ATP_Olambda__221,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_adz(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set4753215920113914285nt_nat(Uu,Uua)) ) ).

% ATP.lambda_221
tff(fact_10428_ATP_Olambda__222,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_aeb(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_222
tff(fact_10429_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aed(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_223
tff(fact_10430_ATP_Olambda__224,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aef(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_224
tff(fact_10431_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aqd(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_225
tff(fact_10432_ATP_Olambda__226,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : aa_nat2240542744625857721at_nat(aTP_Lamm_qa(Uu),Uua) = aa_nat2240542744625857721at_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_226
tff(fact_10433_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_assn(aTP_Lamm_um(Uu),Uua) = aa_nat_assn(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_227
tff(fact_10434_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aqa(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_228
tff(fact_10435_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qb(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_229
tff(fact_10436_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pw(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_230
tff(fact_10437_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qc(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_231
tff(fact_10438_ATP_Olambda__232,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_nf(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_232
tff(fact_10439_ATP_Olambda__233,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_apr,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_apq(Uu),Uua)) ).

% ATP.lambda_233
tff(fact_10440_ATP_Olambda__234,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aod,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aoc(Uu),Uua)) ).

% ATP.lambda_234
tff(fact_10441_ATP_Olambda__235,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aob,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aoa(Uu),Uua)) ).

% ATP.lambda_235
tff(fact_10442_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_anz,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_any(Uu),Uua)) ).

% ATP.lambda_236
tff(fact_10443_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_anx,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_anw(Uu),Uua)) ).

% ATP.lambda_237
tff(fact_10444_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_anu,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ant(Uu),Uua)) ).

% ATP.lambda_238
tff(fact_10445_ATP_Olambda__239,axiom,
    ! [Uu: fun_a_2990225006182978344t_unit,Uua: a] : aa_a_f2724909083865345251it_nat(aTP_Lamm_kt(Uu),Uua) = produc7488178964372371019it_nat(aa_a_f112692477493148739it_nat(aTP_Lamm_ks(Uu),Uua)) ).

% ATP.lambda_239
tff(fact_10446_ATP_Olambda__240,axiom,
    ! [Uu: fun_b_6677255003012645673t_unit,Uua: b] : aa_b_f1465579311918814052it_nat(aTP_Lamm_kq(Uu),Uua) = produc7488178964372371019it_nat(aa_b_f7124492953218388226it_nat(aTP_Lamm_kp(Uu),Uua)) ).

% ATP.lambda_240
tff(fact_10447_ATP_Olambda__241,axiom,
    ! [Uu: fun_a_6093291656547882009Heap_a,Uua: a] : aa_a_f2042851359257671380it_nat(aTP_Lamm_kj(Uu),Uua) = produc5645566021302314940it_nat(aa_a_f1761598956578211508it_nat(aTP_Lamm_ki(Uu),Uua)) ).

% ATP.lambda_241
tff(fact_10448_ATP_Olambda__242,axiom,
    ! [Uu: fun_b_8355010938008660250Heap_a,Uua: b] : aa_b_f2716021586561434197it_nat(aTP_Lamm_kh(Uu),Uua) = produc5645566021302314940it_nat(aa_b_f8802815735423186547it_nat(aTP_Lamm_kg(Uu),Uua)) ).

% ATP.lambda_242
tff(fact_10449_ATP_Olambda__243,axiom,
    ! [Uu: fun_a_6164325696594227994Heap_b,Uua: a] : aa_a_f9019654378185243605it_nat(aTP_Lamm_kf(Uu),Uua) = produc6148253707941125437it_nat(aa_a_f5359256761190816565it_nat(aTP_Lamm_ke(Uu),Uua)) ).

% ATP.lambda_243
tff(fact_10450_ATP_Olambda__244,axiom,
    ! [Uu: fun_b_8426044978055006235Heap_b,Uua: b] : aa_b_f469452568634230614it_nat(aTP_Lamm_kd(Uu),Uua) = produc6148253707941125437it_nat(aa_b_f3177101503181015796it_nat(aTP_Lamm_kc(Uu),Uua)) ).

% ATP.lambda_244
tff(fact_10451_ATP_Olambda__245,axiom,
    ! [Uu: fun_he5123621388056232515t_unit,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_ex(Uu),Uua) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(aa_hea4146273921685161788t_unit(Uu,Uua),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_245
tff(fact_10452_ATP_Olambda__246,axiom,
    ! [Uu: fun_he6342020312717187381unit_b,Uua: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(aTP_Lamm_ev(Uu),Uua) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,aa_hea5100639427662539566unit_b(Uu,Uua)),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_246
tff(fact_10453_ATP_Olambda__247,axiom,
    ! [Uu: fun_he6342020308413958580unit_a,Uua: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aTP_Lamm_ew(Uu),Uua) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,aa_hea5100639427662539565unit_a(Uu,Uua)),produc584006145561248582it_nat(Uua,one_one_nat))) ).

% ATP.lambda_247
tff(fact_10454_ATP_Olambda__248,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_mz,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_248
tff(fact_10455_ATP_Olambda__249,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_na(Uu),Uua) = aa_Cod6352003912080677724atural(produc6639722614265839536atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uua),times_2397367101498566445atural(Uu,aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_249
tff(fact_10456_ATP_Olambda__250,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ih(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_250
tff(fact_10457_ATP_Olambda__251,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_hc(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_251
tff(fact_10458_ATP_Olambda__252,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ha(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_252
tff(fact_10459_ATP_Olambda__253,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_hd(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_253
tff(fact_10460_ATP_Olambda__254,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_hb(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_254
tff(fact_10461_ATP_Olambda__255,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat] : aa_set1897275755269116354at_nat(aTP_Lamm_cb(Uu),Uua) = some_s287724117700012716at_nat(aa_set6734554978791863420at_nat(aa_set4622835569371386638at_nat(sup_su3035147773818789531at_nat,Uu),Uua)) ).

% ATP.lambda_255
tff(fact_10462_ATP_Olambda__256,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set7071694670514008770at_nat(aTP_Lamm_cd(Uu),Uua) = some_s5890477192898017836at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Uu),Uua)) ).

% ATP.lambda_256
tff(fact_10463_ATP_Olambda__257,axiom,
    ! [Uu: set_nat,Uua: set_nat] : aa_set6119337032432945050et_nat(aTP_Lamm_cf(Uu),Uua) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Uu),Uua)) ).

% ATP.lambda_257
tff(fact_10464_ATP_Olambda__258,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_ch(Uu),Uua) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Uu),Uua)) ).

% ATP.lambda_258
tff(fact_10465_ATP_Olambda__259,axiom,
    ! [Uu: int,Uua: int] : aa_int_option_int(aTP_Lamm_cj(Uu),Uua) = aa_int_option_int(some_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Uu),Uua)) ).

% ATP.lambda_259
tff(fact_10466_ATP_Olambda__260,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_option_assn(aTP_Lamm_gg(Uu),Uua) = some_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Uu),Uua)) ).

% ATP.lambda_260
tff(fact_10467_ATP_Olambda__261,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set2934564513904597954at_nat(aTP_Lamm_gi(Uu),Uua) = some_s147305329494351046at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Uu),Uua)) ).

% ATP.lambda_261
tff(fact_10468_ATP_Olambda__262,axiom,
    ! [Uu: set_nat,Uua: set_nat] : aa_set6119337032432945050et_nat(aTP_Lamm_gk(Uu),Uua) = some_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Uu),Uua)) ).

% ATP.lambda_262
tff(fact_10469_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_option_nat(aTP_Lamm_gm(Uu),Uua) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Uu),Uua)) ).

% ATP.lambda_263
tff(fact_10470_ATP_Olambda__264,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o] : aa_fun205347844938550866_int_o(aTP_Lamm_go(Uu),Uua) = some_f3270274345126892989_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Uu),Uua)) ).

% ATP.lambda_264
tff(fact_10471_ATP_Olambda__265,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_nc(Uu),Uua) = aa_Cod6352003912080677724atural(produc6639722614265839536atural,modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_265
tff(fact_10472_ATP_Olambda__266,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apo,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_266
tff(fact_10473_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_app,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_267
tff(fact_10474_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ana,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_268
tff(fact_10475_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amz,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_269
tff(fact_10476_ATP_Olambda__270,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_hv(Uu),Uua)
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_270
tff(fact_10477_ATP_Olambda__271,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_hu(Uu),(Uua))
    <=> ~ aa_set_o_o(member_o((Uua)),Uu) ) ).

% ATP.lambda_271
tff(fact_10478_ATP_Olambda__272,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_hw(Uu),Uua)
    <=> ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uua),Uu) ) ).

% ATP.lambda_272
tff(fact_10479_ATP_Olambda__273,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hx(Uu),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu) ) ).

% ATP.lambda_273
tff(fact_10480_ATP_Olambda__274,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_hy(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_274
tff(fact_10481_ATP_Olambda__275,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_hz(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_275
tff(fact_10482_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_gs(Uu),Uua) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(Uua,Uu))) ).

% ATP.lambda_276
tff(fact_10483_ATP_Olambda__277,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ale(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_10484_ATP_Olambda__278,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] : aa_Code_integer_nat(aTP_Lamm_ala(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_Code_integer_o(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_10485_ATP_Olambda__279,axiom,
    ! [Uu: fun_Code_integer_o,Uua: code_integer] : aa_Code_integer_int(aTP_Lamm_alb(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_Code_integer_o(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_10486_ATP_Olambda__280,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_ali(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_list_nat_o(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_10487_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_alf(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(Uu,Uua)) ).

% ATP.lambda_281
tff(fact_10488_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_nat(aTP_Lamm_alg(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(Uu,Uua)) ).

% ATP.lambda_282
tff(fact_10489_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_int(aTP_Lamm_alc(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uu,Uua)) ).

% ATP.lambda_283
tff(fact_10490_ATP_Olambda__284,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_Code_integer(aTP_Lamm_ald(Uu),Uua) = aa_o_Code_integer(zero_n356916108424825756nteger,aa_int_o(Uu,Uua)) ).

% ATP.lambda_284
tff(fact_10491_ATP_Olambda__285,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_nat(aTP_Lamm_akz(Uu),Uua) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uu,Uua)) ).

% ATP.lambda_285
tff(fact_10492_ATP_Olambda__286,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_int(aTP_Lamm_alh(Uu),Uua) = aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(Uu,Uua)) ).

% ATP.lambda_286
tff(fact_10493_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_nx(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_287
tff(fact_10494_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ny(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_288
tff(fact_10495_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_nw(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_289
tff(fact_10496_ATP_Olambda__290,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_nv(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_290
tff(fact_10497_ATP_Olambda__291,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_pg(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_291
tff(fact_10498_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_up(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_292
tff(fact_10499_ATP_Olambda__293,axiom,
    ! [Uu: fun_nat_assn,Uua: nat] : aa_nat_fun_assn_assn(aTP_Lamm_uq(Uu),Uua) = aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uu,Uua)) ).

% ATP.lambda_293
tff(fact_10500_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ur(Uu),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_294
tff(fact_10501_ATP_Olambda__295,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_us(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_295
tff(fact_10502_ATP_Olambda__296,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ut(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_296
tff(fact_10503_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_su(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_297
tff(fact_10504_ATP_Olambda__298,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sv(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_298
tff(fact_10505_ATP_Olambda__299,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_nt(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_299
tff(fact_10506_ATP_Olambda__300,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_nu(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_300
tff(fact_10507_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_qk(Uu),Uua) = aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_301
tff(fact_10508_ATP_Olambda__302,axiom,
    ! [Uu: fun_na2338080690716861136at_nat,Uua: nat] : aa_nat1862996181841379217at_nat(aTP_Lamm_ql(Uu),Uua) = plus_p7104986032573967614at_nat(aa_nat2240542744625857721at_nat(Uu,Uua)) ).

% ATP.lambda_302
tff(fact_10509_ATP_Olambda__303,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_qm(Uu),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_303
tff(fact_10510_ATP_Olambda__304,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_qo(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_304
tff(fact_10511_ATP_Olambda__305,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_qn(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_305
tff(fact_10512_ATP_Olambda__306,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_oe(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_306
tff(fact_10513_ATP_Olambda__307,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_anm(Uu),Uua) = aa_nat_nat(suc,aa_list_nat_nat(Uu,Uua)) ).

% ATP.lambda_307
tff(fact_10514_ATP_Olambda__308,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_ann(Uu),Uua) = aa_nat_nat(suc,aa_set_nat_nat(Uu,Uua)) ).

% ATP.lambda_308
tff(fact_10515_ATP_Olambda__309,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ano(Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_309
tff(fact_10516_ATP_Olambda__310,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_nat(aTP_Lamm_anl(Uu),Uua) = aa_nat_nat(suc,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_310
tff(fact_10517_ATP_Olambda__311,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_hp(Uu),Uua)
    <=> ~ aa_list_nat_o(Uu,Uua) ) ).

% ATP.lambda_311
tff(fact_10518_ATP_Olambda__312,axiom,
    ! [Uu: fun_se7249947736802449410_int_o,Uua: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_hq(Uu),Uua)
    <=> ~ aa_set6170482117830564987_int_o(Uu,Uua) ) ).

% ATP.lambda_312
tff(fact_10519_ATP_Olambda__313,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_hr(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_313
tff(fact_10520_ATP_Olambda__314,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_hs(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_314
tff(fact_10521_ATP_Olambda__315,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_ht(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_315
tff(fact_10522_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aqe(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_by,Uu)) ).

% ATP.lambda_316
tff(fact_10523_ATP_Olambda__317,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_jq(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_317
tff(fact_10524_ATP_Olambda__318,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_jo(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_318
tff(fact_10525_ATP_Olambda__319,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jp(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_319
tff(fact_10526_ATP_Olambda__320,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: set_Code_integer,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_set4294055438810586376nteger(aTP_Lamm_aff(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Cod1783974652533396728nteger(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_320
tff(fact_10527_ATP_Olambda__321,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_rat(aa_set7344733238996822275er_rat(aTP_Lamm_afi(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_321
tff(fact_10528_ATP_Olambda__322,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_nat(aa_set1629589760535443195er_nat(aTP_Lamm_afl(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_322
tff(fact_10529_ATP_Olambda__323,axiom,
    ! [Uu: fun_o_Code_integer,Uua: set_o,Uub: $o] :
      aa_o_Code_integer(aa_set9049949509649958604nteger(aTP_Lamm_afd(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_Code_integer(Uu,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_323
tff(fact_10530_ATP_Olambda__324,axiom,
    ! [Uu: fun_o_rat,Uua: set_o,Uub: $o] :
      aa_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_afg(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_rat(Uu,(Uub)),zero_zero_rat) ).

% ATP.lambda_324
tff(fact_10531_ATP_Olambda__325,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: $o] :
      aa_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_afj(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_nat(Uu,(Uub)),zero_zero_nat) ).

% ATP.lambda_325
tff(fact_10532_ATP_Olambda__326,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: $o] :
      aa_o_int(aa_set_o_fun_o_int(aTP_Lamm_afm(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_int(Uu,(Uub)),zero_zero_int) ).

% ATP.lambda_326
tff(fact_10533_ATP_Olambda__327,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      aa_int_Code_integer(aa_set5517701783735052070nteger(aTP_Lamm_afe(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_327
tff(fact_10534_ATP_Olambda__328,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_afh(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_328
tff(fact_10535_ATP_Olambda__329,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      aa_int_nat(aa_set2264519855582034201nt_nat(aTP_Lamm_afk(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_329
tff(fact_10536_ATP_Olambda__330,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: set_Code_integer,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_set4294055438810586376nteger(aTP_Lamm_agg(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Cod1783974652533396728nteger(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_330
tff(fact_10537_ATP_Olambda__331,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_assn(aa_set2437433236263590995r_assn(aTP_Lamm_agj(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_331
tff(fact_10538_ATP_Olambda__332,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: set_Code_integer,Uub: code_integer] :
      aa_Code_integer_rat(aa_set7344733238996822275er_rat(aTP_Lamm_agm(Uu),Uua),Uub) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uua),aa_Code_integer_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_332
tff(fact_10539_ATP_Olambda__333,axiom,
    ! [Uu: fun_o_Code_integer,Uua: set_o,Uub: $o] :
      aa_o_Code_integer(aa_set9049949509649958604nteger(aTP_Lamm_age(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_Code_integer(Uu,(Uub)),one_one_Code_integer) ).

% ATP.lambda_333
tff(fact_10540_ATP_Olambda__334,axiom,
    ! [Uu: fun_o_assn,Uua: set_o,Uub: $o] :
      aa_o_assn(aa_set_o_fun_o_assn(aTP_Lamm_agh(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_assn(Uu,(Uub)),one_one_assn) ).

% ATP.lambda_334
tff(fact_10541_ATP_Olambda__335,axiom,
    ! [Uu: fun_o_rat,Uua: set_o,Uub: $o] :
      aa_o_rat(aa_set_o_fun_o_rat(aTP_Lamm_agk(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_rat(Uu,(Uub)),one_one_rat) ).

% ATP.lambda_335
tff(fact_10542_ATP_Olambda__336,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: $o] :
      aa_o_nat(aa_set_o_fun_o_nat(aTP_Lamm_agn(Uu),Uua),(Uub)) = $ite(aa_set_o_o(member_o((Uub)),Uua),aa_o_nat(Uu,(Uub)),one_one_nat) ).

% ATP.lambda_336
tff(fact_10543_ATP_Olambda__337,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      aa_int_Code_integer(aa_set5517701783735052070nteger(aTP_Lamm_agf(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_337
tff(fact_10544_ATP_Olambda__338,axiom,
    ! [Uu: fun_int_assn,Uua: set_int,Uub: int] :
      aa_int_assn(aa_set5310648174735401585t_assn(aTP_Lamm_agi(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_338
tff(fact_10545_ATP_Olambda__339,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      aa_int_rat(aa_set7979663334043413281nt_rat(aTP_Lamm_agl(Uu),Uua),Uub) = $ite(aa_set_int_o(member_int(Uub),Uua),aa_int_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_339
tff(fact_10546_ATP_Olambda__340,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_xf(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Cod1783974652533396728nteger(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_340
tff(fact_10547_ATP_Olambda__341,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_xj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_341
tff(fact_10548_ATP_Olambda__342,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_xc((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_Code_integer(Uua,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_342
tff(fact_10549_ATP_Olambda__343,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_xg((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_rat(Uua,(Uub)),zero_zero_rat) ).

% ATP.lambda_343
tff(fact_10550_ATP_Olambda__344,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_xk((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_nat(Uua,(Uub)),zero_zero_nat) ).

% ATP.lambda_344
tff(fact_10551_ATP_Olambda__345,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_xd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_345
tff(fact_10552_ATP_Olambda__346,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_xh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_346
tff(fact_10553_ATP_Olambda__347,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_xe(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_347
tff(fact_10554_ATP_Olambda__348,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_xi(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_348
tff(fact_10555_ATP_Olambda__349,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_xl(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_349
tff(fact_10556_ATP_Olambda__350,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_xp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Cod1783974652533396728nteger(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_350
tff(fact_10557_ATP_Olambda__351,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_xt(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_351
tff(fact_10558_ATP_Olambda__352,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_xm((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_Code_integer(Uua,(Uub)),one_one_Code_integer) ).

% ATP.lambda_352
tff(fact_10559_ATP_Olambda__353,axiom,
    ! [Uu: $o,Uua: fun_o_assn,Uub: $o] :
      aa_o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_xq((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_assn(Uua,(Uub)),one_one_assn) ).

% ATP.lambda_353
tff(fact_10560_ATP_Olambda__354,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_xu((Uu)),Uua),(Uub)) = $ite((Uu) = (Uub),aa_o_rat(Uua,(Uub)),one_one_rat) ).

% ATP.lambda_354
tff(fact_10561_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_xn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_355
tff(fact_10562_ATP_Olambda__356,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_xr(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_356
tff(fact_10563_ATP_Olambda__357,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_xv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_357
tff(fact_10564_ATP_Olambda__358,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_xo(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_358
tff(fact_10565_ATP_Olambda__359,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_xs(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_359
tff(fact_10566_ATP_Olambda__360,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_wv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Cod1783974652533396728nteger(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_360
tff(fact_10567_ATP_Olambda__361,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_wz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_361
tff(fact_10568_ATP_Olambda__362,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_ws((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_Code_integer(Uua,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_362
tff(fact_10569_ATP_Olambda__363,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_ww((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_rat(Uua,(Uub)),zero_zero_rat) ).

% ATP.lambda_363
tff(fact_10570_ATP_Olambda__364,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_xa((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_nat(Uua,(Uub)),zero_zero_nat) ).

% ATP.lambda_364
tff(fact_10571_ATP_Olambda__365,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_wt(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_365
tff(fact_10572_ATP_Olambda__366,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_366
tff(fact_10573_ATP_Olambda__367,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_wu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_367
tff(fact_10574_ATP_Olambda__368,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_wy(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_368
tff(fact_10575_ATP_Olambda__369,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_xb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_369
tff(fact_10576_ATP_Olambda__370,axiom,
    ! [Uu: code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_xz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Cod1783974652533396728nteger(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_370
tff(fact_10577_ATP_Olambda__371,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aTP_Lamm_yd(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_371
tff(fact_10578_ATP_Olambda__372,axiom,
    ! [Uu: $o,Uua: fun_o_Code_integer,Uub: $o] :
      aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_xw((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_Code_integer(Uua,(Uub)),one_one_Code_integer) ).

% ATP.lambda_372
tff(fact_10579_ATP_Olambda__373,axiom,
    ! [Uu: $o,Uua: fun_o_assn,Uub: $o] :
      aa_o_assn(aa_fun4536517337817438332o_assn(aTP_Lamm_ya((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_assn(Uua,(Uub)),one_one_assn) ).

% ATP.lambda_373
tff(fact_10580_ATP_Olambda__374,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_ye((Uu)),Uua),(Uub)) = $ite((Uub) = (Uu),aa_o_rat(Uua,(Uub)),one_one_rat) ).

% ATP.lambda_374
tff(fact_10581_ATP_Olambda__375,axiom,
    ! [Uu: nat,Uua: fun_nat_Code_integer,Uub: nat] :
      aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_xx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_375
tff(fact_10582_ATP_Olambda__376,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aTP_Lamm_yb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_376
tff(fact_10583_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_yf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_377
tff(fact_10584_ATP_Olambda__378,axiom,
    ! [Uu: int,Uua: fun_int_Code_integer,Uub: int] :
      aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_xy(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_Code_integer(Uua,Uub),one_one_Code_integer) ).

% ATP.lambda_378
tff(fact_10585_ATP_Olambda__379,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aTP_Lamm_yc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_assn(Uua,Uub),one_one_assn) ).

% ATP.lambda_379
tff(fact_10586_ATP_Olambda__380,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_vd(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_380
tff(fact_10587_ATP_Olambda__381,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_sq(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_381
tff(fact_10588_ATP_Olambda__382,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_vc(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_382
tff(fact_10589_ATP_Olambda__383,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun791458909507501263nteger(aTP_Lamm_adq(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Cod1783974652533396728nteger(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_383
tff(fact_10590_ATP_Olambda__384,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_rat(aa_fun25512833367352522er_rat(aTP_Lamm_adu(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_384
tff(fact_10591_ATP_Olambda__385,axiom,
    ! [Uu: fun_o_Code_integer,Uua: fun_o_o,Uub: $o] :
      aa_o_Code_integer(aa_fun6542514109734600263nteger(aTP_Lamm_adm(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_Code_integer(Uu,(Uub)),zero_z3403309356797280102nteger) ).

% ATP.lambda_385
tff(fact_10592_ATP_Olambda__386,axiom,
    ! [Uu: fun_o_rat,Uua: fun_o_o,Uub: $o] :
      aa_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_adr(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_rat(Uu,(Uub)),zero_zero_rat) ).

% ATP.lambda_386
tff(fact_10593_ATP_Olambda__387,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_o,Uub: $o] :
      aa_o_nat(aa_fun_o_o_fun_o_nat(aTP_Lamm_adv(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_nat(Uu,(Uub)),zero_zero_nat) ).

% ATP.lambda_387
tff(fact_10594_ATP_Olambda__388,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_adn(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_388
tff(fact_10595_ATP_Olambda__389,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ads(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_389
tff(fact_10596_ATP_Olambda__390,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_ado(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),zero_z3403309356797280102nteger) ).

% ATP.lambda_390
tff(fact_10597_ATP_Olambda__391,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_adt(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_391
tff(fact_10598_ATP_Olambda__392,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_adw(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_392
tff(fact_10599_ATP_Olambda__393,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Cod1783974652533396728nteger(aa_fun791458909507501263nteger(aTP_Lamm_aeo(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Cod1783974652533396728nteger(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_393
tff(fact_10600_ATP_Olambda__394,axiom,
    ! [Uu: fun_Co4490034186337852034r_assn,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_assn(aa_fun6392223008197429786r_assn(aTP_Lamm_aes(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_394
tff(fact_10601_ATP_Olambda__395,axiom,
    ! [Uu: fun_o_Code_integer,Uua: fun_o_o,Uub: $o] :
      aa_o_Code_integer(aa_fun6542514109734600263nteger(aTP_Lamm_ael(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_Code_integer(Uu,(Uub)),one_one_Code_integer) ).

% ATP.lambda_395
tff(fact_10602_ATP_Olambda__396,axiom,
    ! [Uu: fun_o_assn,Uua: fun_o_o,Uub: $o] :
      aa_o_assn(aa_fun8560762093604944786o_assn(aTP_Lamm_aep(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_assn(Uu,(Uub)),one_one_assn) ).

% ATP.lambda_396
tff(fact_10603_ATP_Olambda__397,axiom,
    ! [Uu: fun_o_rat,Uua: fun_o_o,Uub: $o] :
      aa_o_rat(aa_fun_o_o_fun_o_rat(aTP_Lamm_aet(Uu),Uua),(Uub)) = $ite(aa_o_o(Uua,(Uub)),aa_o_rat(Uu,(Uub)),one_one_rat) ).

% ATP.lambda_397
tff(fact_10604_ATP_Olambda__398,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] :
      aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_aem(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_398
tff(fact_10605_ATP_Olambda__399,axiom,
    ! [Uu: fun_nat_assn,Uua: fun_nat_o,Uub: nat] :
      aa_nat_assn(aa_fun3990007403578212698t_assn(aTP_Lamm_aeq(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_399
tff(fact_10606_ATP_Olambda__400,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_aeu(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_400
tff(fact_10607_ATP_Olambda__401,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] :
      aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_aen(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_Code_integer(Uu,Uub),one_one_Code_integer) ).

% ATP.lambda_401
tff(fact_10608_ATP_Olambda__402,axiom,
    ! [Uu: fun_int_assn,Uua: fun_int_o,Uub: int] :
      aa_int_assn(aa_fun9028805181350005338t_assn(aTP_Lamm_aer(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_assn(Uu,Uub),one_one_assn) ).

% ATP.lambda_402
tff(fact_10609_ATP_Olambda__403,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he4944555133437423336it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea684056619475940329it_nat(aa_fun2986507860694160780it_nat(aTP_Lamm_di(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_403
tff(fact_10610_ATP_Olambda__404,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he168995381318928090it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea5227063804513972443it_nat(aa_fun261163217688289804it_nat(aTP_Lamm_dg(Uu),Uua),Uub) = $ite(aa_hea7994211679605476167unit_o(Uu,Uub),aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_hea550092171636296971it_nat(Uua,Uub)),none_P6779099274072355161it_nat) ).

% ATP.lambda_404
tff(fact_10611_ATP_Olambda__405,axiom,
    ! [Uu: fun_he8413527710729349838unit_o,Uua: fun_he8889679731534893401it_nat,Uub: heap_e7401611519738050253t_unit] :
      aa_hea4724376117875161946it_nat(aa_fun3370411987892275404it_nat(aTP_Lamm_dh(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_405
tff(fact_10612_ATP_Olambda__406,axiom,
    ! [Uu: fun_num_fun_num_num,Uua: option_num,Uub: num] : aa_num_option_num(aa_opt8899475237019873193on_num(aTP_Lamm_ed(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,Uub),aa_num6390622462783740633on_num(aTP_Lamm_ec(Uu),Uub)),Uua) ).

% ATP.lambda_406
tff(fact_10613_ATP_Olambda__407,axiom,
    ! [Uu: fun_Pr1003294667014919570it_nat,Uua: option8956607266484857688it_nat,Uub: produc8664842809031399944it_nat] : aa_Pro8996696869555642526it_nat(aa_opt7728438819757478003it_nat(aTP_Lamm_eb(Uu),Uua),Uub) = case_o2963978774867076333it_nat(aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,Uub),aa_Pro5635350972512700067it_nat(aTP_Lamm_ea(Uu),Uub),Uua) ).

% ATP.lambda_407
tff(fact_10614_ATP_Olambda__408,axiom,
    ! [Uu: fun_Pr7750636411365408485it_nat,Uua: option3562590408128118217it_nat,Uub: produc3260487557148687353it_nat] : aa_Pro5302094286594872192it_nat(aa_opt996909609682481350it_nat(aTP_Lamm_dz(Uu),Uua),Uub) = case_o6061097939050036047it_nat(aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,Uub),aa_Pro7038079567744603766it_nat(aTP_Lamm_dy(Uu),Uub),Uua) ).

% ATP.lambda_408
tff(fact_10615_ATP_Olambda__409,axiom,
    ! [Uu: fun_Pr774283325640823912it_nat,Uua: option4065278094766928714it_nat,Uub: produc7388388658123137530it_nat] : aa_Pro1383761575262803586it_nat(aa_opt4401797230738824009it_nat(aTP_Lamm_dx(Uu),Uua),Uub) = case_o5901184372446974929it_nat(aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,Uub),aa_Pro8047546174521504121it_nat(aTP_Lamm_dw(Uu),Uub),Uua) ).

% ATP.lambda_409
tff(fact_10616_ATP_Olambda__410,axiom,
    ! [Uu: option_int,Uua: option_int,Uub: int] : aa_int_option_int(aa_opt334079648442228447on_int(aTP_Lamm_ck(Uu),Uua),Uub) = case_o390784466056649525nt_int(Uu,aTP_Lamm_cj(Uub),Uua) ).

% ATP.lambda_410
tff(fact_10617_ATP_Olambda__411,axiom,
    ! [Uu: option_nat,Uua: option_nat,Uub: nat] : aa_nat_option_nat(aa_opt1558076126722380235on_nat(aTP_Lamm_ci(Uu),Uua),Uub) = case_o7429725398727453821at_nat(Uu,aTP_Lamm_ch(Uub),Uua) ).

% ATP.lambda_411
tff(fact_10618_ATP_Olambda__412,axiom,
    ! [Uu: option_set_nat,Uua: option_set_nat,Uub: set_nat] : aa_set6119337032432945050et_nat(aa_opt5648106117674462701et_nat(aTP_Lamm_cg(Uu),Uua),Uub) = case_o4054078431260844265et_nat(Uu,aTP_Lamm_cf(Uub),Uua) ).

% ATP.lambda_412
tff(fact_10619_ATP_Olambda__413,axiom,
    ! [Uu: option1583680563626158861at_nat,Uua: option1583680563626158861at_nat,Uub: set_Pr4329608150637261639at_nat] : aa_set7071694670514008770at_nat(aa_opt1027706798495913626at_nat(aTP_Lamm_ce(Uu),Uua),Uub) = case_o9080059180780454917at_nat(Uu,aTP_Lamm_cd(Uub),Uua) ).

% ATP.lambda_413
tff(fact_10620_ATP_Olambda__414,axiom,
    ! [Uu: option2498585697089621389at_nat,Uua: option2498585697089621389at_nat,Uub: set_Pr8551490117392284871at_nat] : aa_set1897275755269116354at_nat(aa_opt1741456117285354266at_nat(aTP_Lamm_cc(Uu),Uua),Uub) = case_o8757249484648823045at_nat(Uu,aTP_Lamm_cb(Uub),Uua) ).

% ATP.lambda_414
tff(fact_10621_ATP_Olambda__415,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_2990225006182978344t_unit,Uub: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aa_fun372765568861188684it_nat(aTP_Lamm_ku(Uu),Uua),Uub) = case_o7727800614283616222it_nat(none_P9117596204409417319it_nat,produc5648861294512138360it_nat(aTP_Lamm_kt(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_415
tff(fact_10622_ATP_Olambda__416,axiom,
    ! [Uu: heap_Time_Heap_b,Uua: fun_b_6677255003012645673t_unit,Uub: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aa_fun3115769655970763019it_nat(aTP_Lamm_kr(Uu),Uua),Uub) = case_o2632329678403290591it_nat(none_P9117596204409417319it_nat,produc450363836004716793it_nat(aTP_Lamm_kq(Uua)),aa_hea5227063804513972443it_nat(heap_Time_execute_b(Uu),Uub)) ).

% ATP.lambda_416
tff(fact_10623_ATP_Olambda__417,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_6093291656547882009Heap_a,Uub: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aa_fun6119220767042692876it_nat(aTP_Lamm_ko(Uu),Uua),Uub) = case_o6061097939050036047it_nat(none_P2651198173097904984it_nat,produc7850007024774191849it_nat(aTP_Lamm_kj(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_417
tff(fact_10624_ATP_Olambda__418,axiom,
    ! [Uu: heap_Time_Heap_b,Uua: fun_b_8355010938008660250Heap_a,Uub: heap_e7401611519738050253t_unit] : aa_hea4724376117875161946it_nat(aa_fun3907649205913156555it_nat(aTP_Lamm_kn(Uu),Uua),Uub) = case_o965627003169710416it_nat(none_P2651198173097904984it_nat,produc3428986626803312746it_nat(aTP_Lamm_kh(Uua)),aa_hea5227063804513972443it_nat(heap_Time_execute_b(Uu),Uub)) ).

% ATP.lambda_418
tff(fact_10625_ATP_Olambda__419,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: fun_a_6164325696594227994Heap_b,Uub: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(aa_fun7255524789075031756it_nat(aTP_Lamm_km(Uu),Uua),Uub) = case_o1773283271472524752it_nat(none_P6779099274072355161it_nat,produc8352694711413002346it_nat(aTP_Lamm_kf(Uua)),aa_hea4724376117875161946it_nat(heap_Time_execute_a(Uu),Uub)) ).

% ATP.lambda_419
tff(fact_10626_ATP_Olambda__420,axiom,
    ! [Uu: heap_Time_Heap_b,Uua: fun_b_8426044978055006235Heap_b,Uub: heap_e7401611519738050253t_unit] : aa_hea5227063804513972443it_nat(aa_fun5043953227945495435it_nat(aTP_Lamm_kl(Uu),Uua),Uub) = case_o5901184372446974929it_nat(none_P6779099274072355161it_nat,produc3931674313442123243it_nat(aTP_Lamm_kd(Uua)),aa_hea5227063804513972443it_nat(heap_Time_execute_b(Uu),Uub)) ).

% ATP.lambda_420
tff(fact_10627_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr(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_421
tff(fact_10628_ATP_Olambda__422,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vz(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_422
tff(fact_10629_ATP_Olambda__423,axiom,
    ! [Uu: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_zu(Uu),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu,Uub),Uua) ).

% ATP.lambda_423
tff(fact_10630_ATP_Olambda__424,axiom,
    ! [Uu: fun_Co822469698833053651at_int,Uua: nat,Uub: code_integer] : aa_Code_integer_int(aa_nat1674686023564409748er_int(aTP_Lamm_abm(Uu),Uua),Uub) = aa_nat_int(aa_Cod4492442032739075476at_int(Uu,Uub),Uua) ).

% ATP.lambda_424
tff(fact_10631_ATP_Olambda__425,axiom,
    ! [Uu: fun_Co3009391273526075439nt_int,Uua: int,Uub: code_integer] : aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_aai(Uu),Uua),Uub) = aa_int_int(aa_Cod5491795744640130032nt_int(Uu,Uub),Uua) ).

% ATP.lambda_425
tff(fact_10632_ATP_Olambda__426,axiom,
    ! [Uu: fun_o_fun_nat_nat,Uua: nat,Uub: $o] : aa_o_nat(aa_nat_fun_o_nat(aTP_Lamm_zk(Uu),Uua),(Uub)) = aa_nat_nat(aa_o_fun_nat_nat(Uu,(Uub)),Uua) ).

% ATP.lambda_426
tff(fact_10633_ATP_Olambda__427,axiom,
    ! [Uu: fun_o_fun_nat_int,Uua: nat,Uub: $o] : aa_o_int(aa_nat_fun_o_int(aTP_Lamm_abj(Uu),Uua),(Uub)) = aa_nat_int(aa_o_fun_nat_int(Uu,(Uub)),Uua) ).

% ATP.lambda_427
tff(fact_10634_ATP_Olambda__428,axiom,
    ! [Uu: fun_o_fun_int_int,Uua: int,Uub: $o] : aa_o_int(aa_int_fun_o_int(aTP_Lamm_zz(Uu),Uua),(Uub)) = aa_int_int(aa_o_fun_int_int(Uu,(Uub)),Uua) ).

% ATP.lambda_428
tff(fact_10635_ATP_Olambda__429,axiom,
    ! [Uu: fun_na5492297256294505719er_nat,Uua: code_integer,Uub: nat] : aa_nat_nat(aa_Cod8670293052248272184at_nat(aTP_Lamm_aav(Uu),Uua),Uub) = aa_Code_integer_nat(aa_nat5852537043073606456er_nat(Uu,Uub),Uua) ).

% ATP.lambda_429
tff(fact_10636_ATP_Olambda__430,axiom,
    ! [Uu: fun_nat_fun_o_nat,Uua: $o,Uub: nat] : aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_aan(Uu),(Uua)),Uub) = aa_o_nat(aa_nat_fun_o_nat(Uu,Uub),(Uua)) ).

% ATP.lambda_430
tff(fact_10637_ATP_Olambda__431,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_431
tff(fact_10638_ATP_Olambda__432,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ta(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_432
tff(fact_10639_ATP_Olambda__433,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_aar(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ).

% ATP.lambda_433
tff(fact_10640_ATP_Olambda__434,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_td(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_434
tff(fact_10641_ATP_Olambda__435,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_zp(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_435
tff(fact_10642_ATP_Olambda__436,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_tg(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_436
tff(fact_10643_ATP_Olambda__437,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ot(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_437
tff(fact_10644_ATP_Olambda__438,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_vt(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_438
tff(fact_10645_ATP_Olambda__439,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_tw(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ta(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_439
tff(fact_10646_ATP_Olambda__440,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tu(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_440
tff(fact_10647_ATP_Olambda__441,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uo(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_441
tff(fact_10648_ATP_Olambda__442,axiom,
    ! [Uu: produc7822875418678951345atural,Uua: code_natural,Uub: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aTP_Lamm_ans(Uu),Uua),Uub) = produc8282080750456430313atural(aa_Cod8811915439870959971atural(aTP_Lamm_anr(Uua),Uub),produc6591912806276919810atural(aa_Pro7165575656943818355atural(next,Uu))) ).

% ATP.lambda_442
tff(fact_10649_ATP_Olambda__443,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aor(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aoq(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_443
tff(fact_10650_ATP_Olambda__444,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aop(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aoo(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_444
tff(fact_10651_ATP_Olambda__445,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aon(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aom(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_445
tff(fact_10652_ATP_Olambda__446,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aol(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_aok(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_446
tff(fact_10653_ATP_Olambda__447,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aoj(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_aoi(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_447
tff(fact_10654_ATP_Olambda__448,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aof(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aoe(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_448
tff(fact_10655_ATP_Olambda__449,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_th(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_tg(Uu),Uub),Uua) ).

% ATP.lambda_449
tff(fact_10656_ATP_Olambda__450,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_te(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_td(Uu),Uub)),Uua) ).

% ATP.lambda_450
tff(fact_10657_ATP_Olambda__451,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_tb(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ta(Uu),Uub)),Uua) ).

% ATP.lambda_451
tff(fact_10658_ATP_Olambda__452,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_tj(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_ot(Uu),Uub),Uua) ).

% ATP.lambda_452
tff(fact_10659_ATP_Olambda__453,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_sy(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Uu),Uub)),Uua) ).

% ATP.lambda_453
tff(fact_10660_ATP_Olambda__454,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_ou(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ot(Uu),Uub),Uua) ).

% ATP.lambda_454
tff(fact_10661_ATP_Olambda__455,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_or(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Uu),Uub)),Uua) ).

% ATP.lambda_455
tff(fact_10662_ATP_Olambda__456,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_wo(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(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_456
tff(fact_10663_ATP_Olambda__457,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aox(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_457
tff(fact_10664_ATP_Olambda__458,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aoy(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_458
tff(fact_10665_ATP_Olambda__459,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aow(Uu),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_459
tff(fact_10666_ATP_Olambda__460,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: heap_e7401611519738050253t_unit,Uub: nat] :
      ( aa_nat_o(aa_hea9023553427033036112_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,lim_Product_unit(Uu)),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),lim_Product_unit(Uua)) ) ) ).

% ATP.lambda_460
tff(fact_10667_ATP_Olambda__461,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_anj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_461
tff(fact_10668_ATP_Olambda__462,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_apd(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_462
tff(fact_10669_ATP_Olambda__463,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_uf(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_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_463
tff(fact_10670_ATP_Olambda__464,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ah(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),lim_Product_unit(Uu))
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_464
tff(fact_10671_ATP_Olambda__465,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_apz(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_465
tff(fact_10672_ATP_Olambda__466,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_nat2915507816091602832eger_o(aTP_Lamm_alo(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uub),Uu)
        & ( finite4902975817058060853nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_466
tff(fact_10673_ATP_Olambda__467,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_nat1727191234135941576_nat_o(aTP_Lamm_alp(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Uub),Uu)
        & ( finite711546835091564841at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_467
tff(fact_10674_ATP_Olambda__468,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: nat,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_nat4194932087953772816_int_o(aTP_Lamm_alk(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(ord_le2843351958646193337nt_int(Uub),Uu)
        & ( aa_set4753215920113914285nt_nat(finite6756421564338198497nt_int,Uub) = Uua ) ) ) ).

% ATP.lambda_468
tff(fact_10675_ATP_Olambda__469,axiom,
    ! [Uu: set_list_nat,Uua: nat,Uub: set_list_nat] :
      ( aa_set_list_nat_o(aa_nat7721107121239158477_nat_o(aTP_Lamm_all(Uu),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(Uub,Uu)
        & ( finite_card_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_469
tff(fact_10676_ATP_Olambda__470,axiom,
    ! [Uu: set_set_nat,Uua: nat,Uub: set_set_nat] :
      ( aa_set_set_nat_o(aa_nat5160323483289836711_nat_o(aTP_Lamm_alm(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Uub),Uu)
        & ( finite_card_set_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_470
tff(fact_10677_ATP_Olambda__471,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_aln(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uub),Uu)
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_471
tff(fact_10678_ATP_Olambda__472,axiom,
    ! [Uu: set_int,Uua: nat,Uub: set_int] :
      ( aa_set_int_o(aa_nat_fun_set_int_o(aTP_Lamm_alq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,Uub),Uu)
        & ( finite_card_int(Uub) = Uua ) ) ) ).

% ATP.lambda_472
tff(fact_10679_ATP_Olambda__473,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ank(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_473
tff(fact_10680_ATP_Olambda__474,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qt(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_474
tff(fact_10681_ATP_Olambda__475,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_gu(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Uu),Uub)
        | aa_Pro653082453696054616_nat_o(rep_assn(Uua),Uub) ) ) ).

% ATP.lambda_475
tff(fact_10682_ATP_Olambda__476,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: set_Pr8551490117392284871at_nat,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_set1219845518824332607_nat_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( member6689249552917799696at_nat(Uub,Uu)
        | member6689249552917799696at_nat(Uub,Uua) ) ) ).

% ATP.lambda_476
tff(fact_10683_ATP_Olambda__477,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uu)
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_477
tff(fact_10684_ATP_Olambda__478,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_478
tff(fact_10685_ATP_Olambda__479,axiom,
    ! [Uu: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_bb(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_479
tff(fact_10686_ATP_Olambda__480,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_bd(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_480
tff(fact_10687_ATP_Olambda__481,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_481
tff(fact_10688_ATP_Olambda__482,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_482
tff(fact_10689_ATP_Olambda__483,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_483
tff(fact_10690_ATP_Olambda__484,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_lk(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uua) ) ) ).

% ATP.lambda_484
tff(fact_10691_ATP_Olambda__485,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_ll(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_485
tff(fact_10692_ATP_Olambda__486,axiom,
    ! [Uu: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ln(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_486
tff(fact_10693_ATP_Olambda__487,axiom,
    ! [Uu: $o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_lm((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_487
tff(fact_10694_ATP_Olambda__488,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_lo(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_488
tff(fact_10695_ATP_Olambda__489,axiom,
    ! [Uu: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_lp(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_489
tff(fact_10696_ATP_Olambda__490,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_lq(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_490
tff(fact_10697_ATP_Olambda__491,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_lr(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_491
tff(fact_10698_ATP_Olambda__492,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_my(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_492
tff(fact_10699_ATP_Olambda__493,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ajo(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_493
tff(fact_10700_ATP_Olambda__494,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_mt(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_494
tff(fact_10701_ATP_Olambda__495,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ajq(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_495
tff(fact_10702_ATP_Olambda__496,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ajr(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_496
tff(fact_10703_ATP_Olambda__497,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ajp(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_497
tff(fact_10704_ATP_Olambda__498,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_gv(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(Uu),Uub)
        & aa_Pro653082453696054616_nat_o(rep_assn(Uua),Uub) ) ) ).

% ATP.lambda_498
tff(fact_10705_ATP_Olambda__499,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_apv(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_499
tff(fact_10706_ATP_Olambda__500,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_apt(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_500
tff(fact_10707_ATP_Olambda__501,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_gd(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_501
tff(fact_10708_ATP_Olambda__502,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_fz(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_502
tff(fact_10709_ATP_Olambda__503,axiom,
    ! [Uu: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_fy(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_503
tff(fact_10710_ATP_Olambda__504,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_ga(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        & aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_504
tff(fact_10711_ATP_Olambda__505,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_gb(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_505
tff(fact_10712_ATP_Olambda__506,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ge(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_506
tff(fact_10713_ATP_Olambda__507,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_gc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_507
tff(fact_10714_ATP_Olambda__508,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_ze(Uu),Uua),Uub)
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_508
tff(fact_10715_ATP_Olambda__509,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_zf(Uu),Uua),Uub)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,Uub),Uu) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,Uua),Uu) ) ) ).

% ATP.lambda_509
tff(fact_10716_ATP_Olambda__510,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_zb(Uu),Uua),Uub)
    <=> ( minus_1052850069191792384nt_int(Uub,Uu) = minus_1052850069191792384nt_int(Uua,Uu) ) ) ).

% ATP.lambda_510
tff(fact_10717_ATP_Olambda__511,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_zc(Uu),Uua),Uub)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uub),Uu) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,Uua),Uu) ) ) ).

% ATP.lambda_511
tff(fact_10718_ATP_Olambda__512,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_zd(Uu),Uua),Uub)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Uub),Uu) = aa_set_int_set_int(aa_set4234024024286713313et_int(minus_minus_set_int,Uua),Uu) ) ) ).

% ATP.lambda_512
tff(fact_10719_ATP_Olambda__513,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_adp(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_513
tff(fact_10720_ATP_Olambda__514,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_514
tff(fact_10721_ATP_Olambda__515,axiom,
    ! [Uu: set_o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ai(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_515
tff(fact_10722_ATP_Olambda__516,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_516
tff(fact_10723_ATP_Olambda__517,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_517
tff(fact_10724_ATP_Olambda__518,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_am(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_518
tff(fact_10725_ATP_Olambda__519,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_519
tff(fact_10726_ATP_Olambda__520,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_mi(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_520
tff(fact_10727_ATP_Olambda__521,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_mm(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_521
tff(fact_10728_ATP_Olambda__522,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_mj(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_522
tff(fact_10729_ATP_Olambda__523,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_mn((Uu)),Uua),(Uub))
    <=> ( ( (Uu)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_523
tff(fact_10730_ATP_Olambda__524,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_mk(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_524
tff(fact_10731_ATP_Olambda__525,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ml(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_525
tff(fact_10732_ATP_Olambda__526,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_mo(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_526
tff(fact_10733_ATP_Olambda__527,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_mp(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_527
tff(fact_10734_ATP_Olambda__528,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_ma(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_528
tff(fact_10735_ATP_Olambda__529,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_me(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_529
tff(fact_10736_ATP_Olambda__530,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_mb(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_530
tff(fact_10737_ATP_Olambda__531,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_mf((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_531
tff(fact_10738_ATP_Olambda__532,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_mc(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_532
tff(fact_10739_ATP_Olambda__533,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_md(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_533
tff(fact_10740_ATP_Olambda__534,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_mg(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_534
tff(fact_10741_ATP_Olambda__535,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_mh(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_535
tff(fact_10742_ATP_Olambda__536,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aTP_Lamm_ace(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod1783974652533396728nteger(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_536
tff(fact_10743_ATP_Olambda__537,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aTP_Lamm_acm(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_537
tff(fact_10744_ATP_Olambda__538,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun3864128940607509264eger_o(aTP_Lamm_ajz(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_538
tff(fact_10745_ATP_Olambda__539,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_aby(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_Code_integer(Uua,(Uub)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_539
tff(fact_10746_ATP_Olambda__540,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_acg(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_rat(Uua,(Uub)) != zero_zero_rat ) ) ) ).

% ATP.lambda_540
tff(fact_10747_ATP_Olambda__541,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_aco(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_nat(Uua,(Uub)) != zero_zero_nat ) ) ) ).

% ATP.lambda_541
tff(fact_10748_ATP_Olambda__542,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o] :
      ( aa_o_o(aa_fun_o_int_fun_o_o(aTP_Lamm_aka(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_int(Uua,(Uub)) != zero_zero_int ) ) ) ).

% ATP.lambda_542
tff(fact_10749_ATP_Olambda__543,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_aca(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_543
tff(fact_10750_ATP_Olambda__544,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_aci(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_544
tff(fact_10751_ATP_Olambda__545,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_acq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_545
tff(fact_10752_ATP_Olambda__546,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_akb(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_int(Uua,Uub) != zero_zero_int ) ) ) ).

% ATP.lambda_546
tff(fact_10753_ATP_Olambda__547,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_acc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_Code_integer(Uua,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_547
tff(fact_10754_ATP_Olambda__548,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ack(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_548
tff(fact_10755_ATP_Olambda__549,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aTP_Lamm_acy(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod1783974652533396728nteger(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_549
tff(fact_10756_ATP_Olambda__550,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun6538703224075513620eger_o(aTP_Lamm_adg(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_550
tff(fact_10757_ATP_Olambda__551,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aTP_Lamm_acs(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_Code_integer(Uua,(Uub)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_551
tff(fact_10758_ATP_Olambda__552,axiom,
    ! [Uu: set_o,Uua: fun_o_assn,Uub: $o] :
      ( aa_o_o(aa_fun8382565201914911066un_o_o(aTP_Lamm_ada(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_assn(Uua,(Uub)) != one_one_assn ) ) ) ).

% ATP.lambda_552
tff(fact_10759_ATP_Olambda__553,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aTP_Lamm_adi(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_rat(Uua,(Uub)) != one_one_rat ) ) ) ).

% ATP.lambda_553
tff(fact_10760_ATP_Olambda__554,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_integer,Uub: nat] :
      ( aa_nat_o(aa_fun3480067038916266681_nat_o(aTP_Lamm_acu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_554
tff(fact_10761_ATP_Olambda__555,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aTP_Lamm_adc(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_555
tff(fact_10762_ATP_Olambda__556,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_adk(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_556
tff(fact_10763_ATP_Olambda__557,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_integer,Uub: int] :
      ( aa_int_o(aa_fun3336537925265328129_int_o(aTP_Lamm_acw(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_Code_integer(Uua,Uub) != one_one_Code_integer ) ) ) ).

% ATP.lambda_557
tff(fact_10764_ATP_Olambda__558,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aTP_Lamm_ade(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_558
tff(fact_10765_ATP_Olambda__559,axiom,
    ! [Uu: fun_o_Code_integer,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_alr(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uua)
        & ( aa_o_Code_integer(Uu,(Uub)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_559
tff(fact_10766_ATP_Olambda__560,axiom,
    ! [Uu: fun_o_rat,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_alu(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uua)
        & ( aa_o_rat(Uu,(Uub)) != zero_zero_rat ) ) ) ).

% ATP.lambda_560
tff(fact_10767_ATP_Olambda__561,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_alx(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uua)
        & ( aa_o_nat(Uu,(Uub)) != zero_zero_nat ) ) ) ).

% ATP.lambda_561
tff(fact_10768_ATP_Olambda__562,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_ama(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uua)
        & ( aa_o_int(Uu,(Uub)) != zero_zero_int ) ) ) ).

% ATP.lambda_562
tff(fact_10769_ATP_Olambda__563,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_als(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uua)
        & ( aa_nat_Code_integer(Uu,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_563
tff(fact_10770_ATP_Olambda__564,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_alv(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uua)
        & ( aa_nat_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_564
tff(fact_10771_ATP_Olambda__565,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_aly(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uua)
        & ( aa_nat_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_565
tff(fact_10772_ATP_Olambda__566,axiom,
    ! [Uu: fun_int_Code_integer,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_alt(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_Code_integer(Uu,Uub) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_566
tff(fact_10773_ATP_Olambda__567,axiom,
    ! [Uu: fun_int_rat,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_alw(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_rat(Uu,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_567
tff(fact_10774_ATP_Olambda__568,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_alz(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uua)
        & ( aa_int_nat(Uu,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_568
tff(fact_10775_ATP_Olambda__569,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3734316457315071362atural,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun1294772355873590804eger_o(aTP_Lamm_akl(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_Cod607569804290114243atural(Uua,Uub)) ) ) ).

% ATP.lambda_569
tff(fact_10776_ATP_Olambda__570,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_ake(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(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_Code_integer_nat(Uua,Uub)) ) ) ).

% ATP.lambda_570
tff(fact_10777_ATP_Olambda__571,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun3864128940607509264eger_o(aTP_Lamm_akh(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(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_Code_integer_int(Uua,Uub)) ) ) ).

% ATP.lambda_571
tff(fact_10778_ATP_Olambda__572,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_natural,Uub: $o] :
      ( aa_o_o(aa_fun2872501519293664346un_o_o(aTP_Lamm_aki(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_o_Code_natural(Uua,(Uub))) ) ) ).

% ATP.lambda_572
tff(fact_10779_ATP_Olambda__573,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aTP_Lamm_akc(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((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_o_nat(Uua,(Uub))) ) ) ).

% ATP.lambda_573
tff(fact_10780_ATP_Olambda__574,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o] :
      ( aa_o_o(aa_fun_o_int_fun_o_o(aTP_Lamm_akf(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((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_o_int(Uua,(Uub))) ) ) ).

% ATP.lambda_574
tff(fact_10781_ATP_Olambda__575,axiom,
    ! [Uu: set_nat,Uua: fun_nat_Code_natural,Uub: nat] :
      ( aa_nat_o(aa_fun440078198722963374_nat_o(aTP_Lamm_akj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_nat_Code_natural(Uua,Uub)) ) ) ).

% ATP.lambda_575
tff(fact_10782_ATP_Olambda__576,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      ( aa_nat_o(aa_fun2180201864171524266_nat_o(aTP_Lamm_akg(Uu),Uua),Uub)
    <=> ( aa_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_576
tff(fact_10783_ATP_Olambda__577,axiom,
    ! [Uu: set_int,Uua: fun_int_Code_natural,Uub: int] :
      ( aa_int_o(aa_fun296549085072024822_int_o(aTP_Lamm_akk(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_Code_natural_o(dvd_dvd_Code_natural(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,one))),aa_int_Code_natural(Uua,Uub)) ) ) ).

% ATP.lambda_577
tff(fact_10784_ATP_Olambda__578,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_akd(Uu),Uua),Uub)
    <=> ( aa_set_int_o(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_578
tff(fact_10785_ATP_Olambda__579,axiom,
    ! [Uu: array_o,Uua: list_o,Uub: nat] : aa_nat_assn(aa_lis7996121028773434071t_assn(aTP_Lamm_hf(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_o(Uu,Uua)),aa_o_assn(pure_assn,Uub = size_size_list_o(Uua))) ).

% ATP.lambda_579
tff(fact_10786_ATP_Olambda__580,axiom,
    ! [Uu: array_nat,Uua: list_nat,Uub: nat] : aa_nat_assn(aa_lis6041139530851742867t_assn(aTP_Lamm_he(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uu,Uua)),aa_o_assn(pure_assn,Uub = aa_list_nat_nat(size_size_list_nat,Uua))) ).

% ATP.lambda_580
tff(fact_10787_ATP_Olambda__581,axiom,
    ! [Uu: array_int,Uua: list_int,Uub: nat] : aa_nat_assn(aa_lis2816288184639765743t_assn(aTP_Lamm_hg(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uu,Uua)),aa_o_assn(pure_assn,Uub = size_size_list_int(Uua))) ).

% ATP.lambda_581
tff(fact_10788_ATP_Olambda__582,axiom,
    ! [Uu: ref_Product_unit,Uua: product_unit,Uub: product_unit] : aa_Product_unit_assn(aa_Pro6018021088111102263t_assn(aTP_Lamm_fc(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Product_unit_assn(sngr_a5825115052027484668t_unit(Uu),Uua)),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_582
tff(fact_10789_ATP_Olambda__583,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_ig(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uua) ) ) ).

% ATP.lambda_583
tff(fact_10790_ATP_Olambda__584,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ib(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_584
tff(fact_10791_ATP_Olambda__585,axiom,
    ! [Uu: set_o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_ia(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ~ aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_585
tff(fact_10792_ATP_Olambda__586,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: set_se6260736226359567993nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4062574822058255039_int_o(aTP_Lamm_ic(Uu),Uua),Uub)
    <=> ( aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uu)
        & ~ aa_set6820988702716878619_int_o(member2340774599025711042nt_int(Uub),Uua) ) ) ).

% ATP.lambda_586
tff(fact_10793_ATP_Olambda__587,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_id(Uu),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_587
tff(fact_10794_ATP_Olambda__588,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ie(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_588
tff(fact_10795_ATP_Olambda__589,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_if(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_589
tff(fact_10796_ATP_Olambda__590,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_apu(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_590
tff(fact_10797_ATP_Olambda__591,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qq(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_591
tff(fact_10798_ATP_Olambda__592,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vu(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_592
tff(fact_10799_ATP_Olambda__593,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_ee(Uu),Uua),Uub)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_593
tff(fact_10800_ATP_Olambda__594,axiom,
    ! [Uu: set_Pr8551490117392284871at_nat,Uua: multis2468970476368604999at_nat,Uub: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> member6689249552917799696at_nat(produc4348348721325984599at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_594
tff(fact_10801_ATP_Olambda__595,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_cr(Uu),Uua),Uub)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uub)),Uu) ) ).

% ATP.lambda_595
tff(fact_10802_ATP_Olambda__596,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ef(Uu),Uua),Uub)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)),Uu) ) ).

% ATP.lambda_596
tff(fact_10803_ATP_Olambda__597,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)),Uu) ) ).

% ATP.lambda_597
tff(fact_10804_ATP_Olambda__598,axiom,
    ! [Uu: set_Pr1312406793078187350et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_598
tff(fact_10805_ATP_Olambda__599,axiom,
    ! [Uu: set_Pr4040394263813967220et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_599
tff(fact_10806_ATP_Olambda__600,axiom,
    ! [Uu: set_Pr7600907837789447088it_nat,Uua: b,Uub: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> member8682313912305727889it_nat(aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_600
tff(fact_10807_ATP_Olambda__601,axiom,
    ! [Uu: set_Pr7098220151150636591it_nat,Uua: a,Uub: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> member4554412811331277712it_nat(aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_601
tff(fact_10808_ATP_Olambda__602,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_aev(Uu),Uua),Uub)
    <=> ( aa_set_rat_o(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_602
tff(fact_10809_ATP_Olambda__603,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zg(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_603
tff(fact_10810_ATP_Olambda__604,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_tk(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_604
tff(fact_10811_ATP_Olambda__605,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_tl(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_605
tff(fact_10812_ATP_Olambda__606,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tm(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_606
tff(fact_10813_ATP_Olambda__607,axiom,
    ! [Uu: fun_o_Code_integer,Uua: fun_o_Code_integer,Uub: $o] : aa_o_Code_integer(aa_fun8520278569782659900nteger(aTP_Lamm_of(Uu),Uua),(Uub)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(Uua,(Uub))),aa_o_Code_integer(Uu,(Uub))) ).

% ATP.lambda_607
tff(fact_10814_ATP_Olambda__608,axiom,
    ! [Uu: fun_o_rat,Uua: fun_o_rat,Uub: $o] : aa_o_rat(aa_fun4336163974892599420_o_rat(aTP_Lamm_oi(Uu),Uua),(Uub)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(Uua,(Uub))),aa_o_rat(Uu,(Uub))) ).

% ATP.lambda_608
tff(fact_10815_ATP_Olambda__609,axiom,
    ! [Uu: fun_o_int,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_ol(Uu),Uua),(Uub)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(Uua,(Uub))),aa_o_int(Uu,(Uub))) ).

% ATP.lambda_609
tff(fact_10816_ATP_Olambda__610,axiom,
    ! [Uu: fun_se7586849273513198688nteger,Uua: fun_se7586849273513198688nteger,Uub: set_nat] : aa_set871585252387937313nteger(aa_fun4587287018503165694nteger(aTP_Lamm_oo(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_set871585252387937313nteger(Uua,Uub)),aa_set871585252387937313nteger(Uu,Uub)) ).

% ATP.lambda_610
tff(fact_10817_ATP_Olambda__611,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_og(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_611
tff(fact_10818_ATP_Olambda__612,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_oj(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_612
tff(fact_10819_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_om(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_613
tff(fact_10820_ATP_Olambda__614,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_oh(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_614
tff(fact_10821_ATP_Olambda__615,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ok(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_615
tff(fact_10822_ATP_Olambda__616,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_on(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_616
tff(fact_10823_ATP_Olambda__617,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pf(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_617
tff(fact_10824_ATP_Olambda__618,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_ph(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_o_nat(Uua,(Uub))),aa_o_nat(Uu,(Uub))) ).

% ATP.lambda_618
tff(fact_10825_ATP_Olambda__619,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se4101287982975402108nt_nat,Uub: set_Pr958786334691620121nt_int] : aa_set4753215920113914285nt_nat(aa_fun209864160807355708nt_nat(aTP_Lamm_pi(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_619
tff(fact_10826_ATP_Olambda__620,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_pj(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_620
tff(fact_10827_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pl(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_621
tff(fact_10828_ATP_Olambda__622,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_pk(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_622
tff(fact_10829_ATP_Olambda__623,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_rat,Uub: code_integer] : aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_and(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Uu,Uub)),aa_Code_integer_rat(Uua,Uub)) ).

% ATP.lambda_623
tff(fact_10830_ATP_Olambda__624,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] : aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aTP_Lamm_anf(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub)) ).

% ATP.lambda_624
tff(fact_10831_ATP_Olambda__625,axiom,
    ! [Uu: fun_Code_integer_int,Uua: fun_Code_integer_int,Uub: code_integer] : aa_Code_integer_int(aa_fun3291479149765983562er_int(aTP_Lamm_anh(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(Uu,Uub)),aa_Code_integer_int(Uua,Uub)) ).

% ATP.lambda_625
tff(fact_10832_ATP_Olambda__626,axiom,
    ! [Uu: fun_Pr8317260874932935276at_rat,Uua: fun_Pr8317260874932935276at_rat,Uub: product_prod_nat_nat] : aa_Pro3608043709957902237at_rat(aa_fun1488288443938188604at_rat(aTP_Lamm_ani(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro3608043709957902237at_rat(Uu,Uub)),aa_Pro3608043709957902237at_rat(Uua,Uub)) ).

% ATP.lambda_626
tff(fact_10833_ATP_Olambda__627,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_anb(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_627
tff(fact_10834_ATP_Olambda__628,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ov(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_628
tff(fact_10835_ATP_Olambda__629,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ang(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_629
tff(fact_10836_ATP_Olambda__630,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_anc(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uu,Uub)),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_630
tff(fact_10837_ATP_Olambda__631,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ane(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_631
tff(fact_10838_ATP_Olambda__632,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ow(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_632
tff(fact_10839_ATP_Olambda__633,axiom,
    ! [Uu: fun_Pr1099658227000185396_nat_o,Uua: fun_Pr1099658227000185396_nat_o,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_fun9125434449681309094_nat_o(aTP_Lamm_hl(Uu),Uua),Uub)
    <=> ( aa_Pro8143006071014924973_nat_o(Uu,Uub)
       => aa_Pro8143006071014924973_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_633
tff(fact_10840_ATP_Olambda__634,axiom,
    ! [Uu: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_hm(Uu),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu,Uub)
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_634
tff(fact_10841_ATP_Olambda__635,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_hh(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_635
tff(fact_10842_ATP_Olambda__636,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_hi(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
       => aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_636
tff(fact_10843_ATP_Olambda__637,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_hj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_637
tff(fact_10844_ATP_Olambda__638,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_hn(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_638
tff(fact_10845_ATP_Olambda__639,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_hk(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_639
tff(fact_10846_ATP_Olambda__640,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_ajm(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        | aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_640
tff(fact_10847_ATP_Olambda__641,axiom,
    ! [Uu: fun_Pr1099658227000185396_nat_o,Uua: fun_Pr1099658227000185396_nat_o,Uub: produc4166570645942440679at_nat] :
      ( aa_Pro8143006071014924973_nat_o(aa_fun9125434449681309094_nat_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> ( aa_Pro8143006071014924973_nat_o(Uu,Uub)
        | aa_Pro8143006071014924973_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_641
tff(fact_10848_ATP_Olambda__642,axiom,
    ! [Uu: fun_Pr3059359313727237556_nat_o,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ( aa_Pro5947980084102686253_nat_o(Uu,Uub)
        | aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_642
tff(fact_10849_ATP_Olambda__643,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_ajn(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        | aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_643
tff(fact_10850_ATP_Olambda__644,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_644
tff(fact_10851_ATP_Olambda__645,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_av(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
        | aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_645
tff(fact_10852_ATP_Olambda__646,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_aw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_646
tff(fact_10853_ATP_Olambda__647,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_647
tff(fact_10854_ATP_Olambda__648,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_648
tff(fact_10855_ATP_Olambda__649,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_ajl(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_649
tff(fact_10856_ATP_Olambda__650,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_gf(Uu),Uua),Uub)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_650
tff(fact_10857_ATP_Olambda__651,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_651
tff(fact_10858_ATP_Olambda__652,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_aq(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(Uu,Uub)
        & aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_652
tff(fact_10859_ATP_Olambda__653,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_653
tff(fact_10860_ATP_Olambda__654,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_654
tff(fact_10861_ATP_Olambda__655,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_655
tff(fact_10862_ATP_Olambda__656,axiom,
    ! [Uu: fun_Co8499238673658658103nteger,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Cod1783974652533396728nteger(aa_fun791458909507501263nteger(aTP_Lamm_yy(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_656
tff(fact_10863_ATP_Olambda__657,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Code_integer_rat(aa_fun25512833367352522er_rat(aTP_Lamm_yr(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_657
tff(fact_10864_ATP_Olambda__658,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun3533741391760749250er_nat(aTP_Lamm_yu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_integer_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_658
tff(fact_10865_ATP_Olambda__659,axiom,
    ! [Uu: fun_Code_integer_int,Uua: fun_Code_integer_o,Uub: code_integer] : aa_Code_integer_int(aa_fun8579262409106328350er_int(aTP_Lamm_yv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(Uu,Uub)),aa_o_int(zero_n2684676970156552555ol_int,aa_Code_integer_o(Uua,Uub))) ).

% ATP.lambda_659
tff(fact_10866_ATP_Olambda__660,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_o,Uub: nat] : aa_nat_Code_integer(aa_fun3969837373441324047nteger(aTP_Lamm_yz(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(Uua,Uub))) ).

% ATP.lambda_660
tff(fact_10867_ATP_Olambda__661,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] : aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_ys(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uua,Uub))) ).

% ATP.lambda_661
tff(fact_10868_ATP_Olambda__662,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_o,Uub: nat] : aa_nat_int(aa_fun27009930614080094at_int(aTP_Lamm_yw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uua,Uub))) ).

% ATP.lambda_662
tff(fact_10869_ATP_Olambda__663,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_o,Uub: int] : aa_int_Code_integer(aa_fun6019004165459918095nteger(aTP_Lamm_yx(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uu,Uub)),aa_o_Code_integer(zero_n356916108424825756nteger,aa_int_o(Uua,Uub))) ).

% ATP.lambda_663
tff(fact_10870_ATP_Olambda__664,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] : aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_yq(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,Uub)),aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uua,Uub))) ).

% ATP.lambda_664
tff(fact_10871_ATP_Olambda__665,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] : aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_yt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,Uub)),aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uua,Uub))) ).

% ATP.lambda_665
tff(fact_10872_ATP_Olambda__666,axiom,
    ! [Uu: fun_int_option_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kk(Uu),Uua),Uub)
    <=> ( aa_int_option_int(Uu,Uua) = aa_int_option_int(some_int,Uub) ) ) ).

% ATP.lambda_666
tff(fact_10873_ATP_Olambda__667,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_sx(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_667
tff(fact_10874_ATP_Olambda__668,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_int(aa_set2359242532610849469at_int(aTP_Lamm_sz(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_668
tff(fact_10875_ATP_Olambda__669,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: set_int,Uub: nat] : aa_nat_int(aa_set6310687161026558745at_int(aTP_Lamm_tc(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_669
tff(fact_10876_ATP_Olambda__670,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: set_nat,Uub: int] : aa_int_int(aa_set3358596244511904025nt_int(aTP_Lamm_tf(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(aa_int_fun_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_670
tff(fact_10877_ATP_Olambda__671,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_ti(Uu),Uua),Uub) = groups1705073143266064639nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_671
tff(fact_10878_ATP_Olambda__672,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_op(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_672
tff(fact_10879_ATP_Olambda__673,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_os(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_673
tff(fact_10880_ATP_Olambda__674,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zh(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_674
tff(fact_10881_ATP_Olambda__675,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_675
tff(fact_10882_ATP_Olambda__676,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_tq(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_676
tff(fact_10883_ATP_Olambda__677,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_oa(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_677
tff(fact_10884_ATP_Olambda__678,axiom,
    ! [Uu: fun_Co3734316457315071362atural,Uua: code_natural,Uub: code_integer] : aa_Cod607569804290114243atural(aa_Cod1476545362447692776atural(aTP_Lamm_ail(Uu),Uua),Uub) = divide5121882707175180666atural(aa_Cod607569804290114243atural(Uu,Uub),Uua) ).

% ATP.lambda_678
tff(fact_10885_ATP_Olambda__679,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_ahz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_Code_integer_nat(Uu,Uub)),Uua) ).

% ATP.lambda_679
tff(fact_10886_ATP_Olambda__680,axiom,
    ! [Uu: fun_Code_integer_int,Uua: int,Uub: code_integer] : aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_aic(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(Uu,Uub)),Uua) ).

% ATP.lambda_680
tff(fact_10887_ATP_Olambda__681,axiom,
    ! [Uu: fun_list_nat_nat,Uua: nat,Uub: list_nat] : aa_list_nat_nat(aa_nat3122681260899759163at_nat(aTP_Lamm_aix(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_list_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_681
tff(fact_10888_ATP_Olambda__682,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] : aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_aio(Uu),Uua),Uub) = divide5121882707175180666atural(aa_nat_Code_natural(Uu,Uub),Uua) ).

% ATP.lambda_682
tff(fact_10889_ATP_Olambda__683,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_air(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_683
tff(fact_10890_ATP_Olambda__684,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_aif(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_684
tff(fact_10891_ATP_Olambda__685,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] : aa_int_Code_natural(aa_Cod5692187177200718263atural(aTP_Lamm_aii(Uu),Uua),Uub) = divide5121882707175180666atural(aa_int_Code_natural(Uu,Uub),Uua) ).

% ATP.lambda_685
tff(fact_10892_ATP_Olambda__686,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_ahw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(Uu,Uub)),Uua) ).

% ATP.lambda_686
tff(fact_10893_ATP_Olambda__687,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aiu(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_687
tff(fact_10894_ATP_Olambda__688,axiom,
    ! [Uu: fun_Pr6626665806141890376t_assn,Uua: assn,Uub: product_unit] : aa_Product_unit_assn(aa_ass3177796706412234594t_assn(aTP_Lamm_fu(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Product_unit_assn(Uu,Uub)),Uua) ).

% ATP.lambda_688
tff(fact_10895_ATP_Olambda__689,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_689
tff(fact_10896_ATP_Olambda__690,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_pc(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_690
tff(fact_10897_ATP_Olambda__691,axiom,
    ! [Uu: fun_b_assn,Uua: assn,Uub: b] : aa_b_assn(aa_assn_fun_b_assn(aTP_Lamm_ft(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_b_assn(Uu,Uub)),Uua) ).

% ATP.lambda_691
tff(fact_10898_ATP_Olambda__692,axiom,
    ! [Uu: fun_a_assn,Uua: assn,Uub: a] : aa_a_assn(aa_assn_fun_a_assn(aTP_Lamm_fs(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_a_assn(Uu,Uub)),Uua) ).

% ATP.lambda_692
tff(fact_10899_ATP_Olambda__693,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_693
tff(fact_10900_ATP_Olambda__694,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_to(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_694
tff(fact_10901_ATP_Olambda__695,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_tp(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_695
tff(fact_10902_ATP_Olambda__696,axiom,
    ! [Uu: fun_Co1105491363547889651atural,Uua: fun_Co1105491363547889651atural,Uub: code_natural] : aa_Cod6352003912080677724atural(aa_fun5568814847722588540atural(aTP_Lamm_nd(Uu),Uua),Uub) = produc5538323210962509403atural(aa_Cod6352003912080677724atural(Uu,Uub),Uua) ).

% ATP.lambda_696
tff(fact_10903_ATP_Olambda__697,axiom,
    ! [Uu: produc3843707927480180839at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_ls(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_697
tff(fact_10904_ATP_Olambda__698,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_lt(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_698
tff(fact_10905_ATP_Olambda__699,axiom,
    ! [Uu: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_lv(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_699
tff(fact_10906_ATP_Olambda__700,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_lu((Uu)),Uua),(Uub))
    <=> ( ~ ( (Uub)
          <=> (Uu) )
       => aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_700
tff(fact_10907_ATP_Olambda__701,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_lw(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set6170482117830564987_int_o(Uua,Uub) ) ) ).

% ATP.lambda_701
tff(fact_10908_ATP_Olambda__702,axiom,
    ! [Uu: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_lx(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_702
tff(fact_10909_ATP_Olambda__703,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ly(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_703
tff(fact_10910_ATP_Olambda__704,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_lz(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_704
tff(fact_10911_ATP_Olambda__705,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ug(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(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_705
tff(fact_10912_ATP_Olambda__706,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Co8499238673658658103nteger,Uub: code_integer] : aa_Cod1783974652533396728nteger(aa_fun4435975922079263404nteger(aTP_Lamm_yo(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_Code_integer_o(Uu,Uub))),aa_Cod1783974652533396728nteger(Uua,Uub)) ).

% ATP.lambda_706
tff(fact_10913_ATP_Olambda__707,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_rat,Uub: code_integer] : aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aTP_Lamm_yh(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_Code_integer_o(Uu,Uub))),aa_Code_integer_rat(Uua,Uub)) ).

% ATP.lambda_707
tff(fact_10914_ATP_Olambda__708,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_nat,Uub: code_integer] : aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aTP_Lamm_yk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_Code_integer_o(Uu,Uub))),aa_Code_integer_nat(Uua,Uub)) ).

% ATP.lambda_708
tff(fact_10915_ATP_Olambda__709,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_int,Uub: code_integer] : aa_Code_integer_int(aa_fun3291479149765983562er_int(aTP_Lamm_yl(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_Code_integer_o(Uu,Uub))),aa_Code_integer_int(Uua,Uub)) ).

% ATP.lambda_709
tff(fact_10916_ATP_Olambda__710,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_yp(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_nat_o(Uu,Uub))),aa_nat_Code_integer(Uua,Uub)) ).

% ATP.lambda_710
tff(fact_10917_ATP_Olambda__711,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_yi(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_nat_o(Uu,Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_711
tff(fact_10918_ATP_Olambda__712,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ym(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Uu,Uub))),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_712
tff(fact_10919_ATP_Olambda__713,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_yn(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(zero_n356916108424825756nteger,aa_int_o(Uu,Uub))),aa_int_Code_integer(Uua,Uub)) ).

% ATP.lambda_713
tff(fact_10920_ATP_Olambda__714,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_yg(Uu),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(zero_n2052037380579107095ol_rat,aa_int_o(Uu,Uub))),aa_int_rat(Uua,Uub)) ).

% ATP.lambda_714
tff(fact_10921_ATP_Olambda__715,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_yj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_int_o(Uu,Uub))),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_715
tff(fact_10922_ATP_Olambda__716,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wc(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(Uu),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_716
tff(fact_10923_ATP_Olambda__717,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wd(Uu),Uua),Uub) = aa_set_nat_int(groups705719431365010083at_int(Uu),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_717
tff(fact_10924_ATP_Olambda__718,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wb(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(Uu),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_718
tff(fact_10925_ATP_Olambda__719,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: nat] : aa_nat_Code_natural(aa_Cod5320533616963873627atural(aTP_Lamm_sg(Uu),Uua),Uub) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uu),times_2397367101498566445atural(aa_nat_Code_natural(semiri3763490453095760265atural,Uub),Uua)) ).

% ATP.lambda_719
tff(fact_10926_ATP_Olambda__720,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_si(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_720
tff(fact_10927_ATP_Olambda__721,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_sf(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_721
tff(fact_10928_ATP_Olambda__722,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sj(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_722
tff(fact_10929_ATP_Olambda__723,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_sh(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_723
tff(fact_10930_ATP_Olambda__724,axiom,
    ! [Uu: list_o,Uua: array_o,Uub: heap_e7401611519738050253t_unit] : aa_hea6592385862050124941it_nat(aa_arr3163342345564908675it_nat(aTP_Lamm_wq(Uu),Uua),Uub) = produc2890842502232952598it_nat(Uua,produc584006145561248582it_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),size_size_list_o(Uu)))) ).

% ATP.lambda_724
tff(fact_10931_ATP_Olambda__725,axiom,
    ! [Uu: list_nat,Uua: array_nat,Uub: heap_e7401611519738050253t_unit] : aa_hea4612425101438060313it_nat(aa_arr1916294878505178839it_nat(aTP_Lamm_wp(Uu),Uua),Uub) = produc4907525368653469954it_nat(Uua,produc584006145561248582it_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_list_nat_nat(size_size_list_nat,Uu)))) ).

% ATP.lambda_725
tff(fact_10932_ATP_Olambda__726,axiom,
    ! [Uu: list_int,Uua: array_int,Uub: heap_e7401611519738050253t_unit] : aa_hea5056872858427967349it_nat(aa_arr7976763211072384471it_nat(aTP_Lamm_wr(Uu),Uua),Uub) = produc5351973125643376990it_nat(Uua,produc584006145561248582it_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),size_size_list_int(Uu)))) ).

% ATP.lambda_726
tff(fact_10933_ATP_Olambda__727,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_wl(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_727
tff(fact_10934_ATP_Olambda__728,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_wm(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_728
tff(fact_10935_ATP_Olambda__729,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wn(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_729
tff(fact_10936_ATP_Olambda__730,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oc(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_730
tff(fact_10937_ATP_Olambda__731,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_od(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_731
tff(fact_10938_ATP_Olambda__732,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sl(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_732
tff(fact_10939_ATP_Olambda__733,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_733
tff(fact_10940_ATP_Olambda__734,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pe(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_734
tff(fact_10941_ATP_Olambda__735,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_uc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_735
tff(fact_10942_ATP_Olambda__736,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_ud(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_736
tff(fact_10943_ATP_Olambda__737,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_ue(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_737
tff(fact_10944_ATP_Olambda__738,axiom,
    ! [Uu: fun_Co3734316457315071362atural,Uua: code_natural,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod403035634466734574eger_o(aTP_Lamm_aim(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_Cod607569804290114243atural(Uu,Uub)) ) ).

% ATP.lambda_738
tff(fact_10945_ATP_Olambda__739,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aTP_Lamm_aia(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_Code_integer_nat(Uu,Uub)) ) ).

% ATP.lambda_739
tff(fact_10946_ATP_Olambda__740,axiom,
    ! [Uu: fun_Code_integer_int,Uua: int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aTP_Lamm_aid(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_Code_integer_int(Uu,Uub)) ) ).

% ATP.lambda_740
tff(fact_10947_ATP_Olambda__741,axiom,
    ! [Uu: fun_list_nat_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aiy(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_list_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_741
tff(fact_10948_ATP_Olambda__742,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] :
      ( aa_nat_o(aa_Cod9120328075896114107_nat_o(aTP_Lamm_aip(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_nat_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_742
tff(fact_10949_ATP_Olambda__743,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ais(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_743
tff(fact_10950_ATP_Olambda__744,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_aig(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_744
tff(fact_10951_ATP_Olambda__745,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] :
      ( aa_int_o(aa_Cod7299285730463426143_int_o(aTP_Lamm_aij(Uu),Uua),Uub)
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_int_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_745
tff(fact_10952_ATP_Olambda__746,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_ahx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_746
tff(fact_10953_ATP_Olambda__747,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aiv(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_747
tff(fact_10954_ATP_Olambda__748,axiom,
    ! [Uu: assn,Uua: product_unit,Uub: product_unit] : aa_Product_unit_assn(aa_Pro6018021088111102263t_assn(aTP_Lamm_fx(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_748
tff(fact_10955_ATP_Olambda__749,axiom,
    ! [Uu: assn,Uua: b,Uub: b] : aa_b_assn(aa_b_fun_b_assn(aTP_Lamm_fw(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_749
tff(fact_10956_ATP_Olambda__750,axiom,
    ! [Uu: assn,Uua: a,Uub: a] : aa_a_assn(aa_a_fun_a_assn(aTP_Lamm_fv(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_750
tff(fact_10957_ATP_Olambda__751,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_uv(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_751
tff(fact_10958_ATP_Olambda__752,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_uu(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_752
tff(fact_10959_ATP_Olambda__753,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uw(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_753
tff(fact_10960_ATP_Olambda__754,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ux(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_754
tff(fact_10961_ATP_Olambda__755,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_afb(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uu,Uub))) ) ).

% ATP.lambda_755
tff(fact_10962_ATP_Olambda__756,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_afc(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(Uu,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub))) ) ).

% ATP.lambda_756
tff(fact_10963_ATP_Olambda__757,axiom,
    ! [Uu: fun_list_nat_nat,Uua: list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aTP_Lamm_aew(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_list_nat_nat(Uu,Uub)),aa_list_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_757
tff(fact_10964_ATP_Olambda__758,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: set_Pr958786334691620121nt_int,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_set4993530631398190431_int_o(aTP_Lamm_aex(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_set4753215920113914285nt_nat(Uu,Uub)),aa_set4753215920113914285nt_nat(Uu,Uub))) ) ).

% ATP.lambda_758
tff(fact_10965_ATP_Olambda__759,axiom,
    ! [Uu: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_aey(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_759
tff(fact_10966_ATP_Olambda__760,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aez(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_760
tff(fact_10967_ATP_Olambda__761,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_afa(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_761
tff(fact_10968_ATP_Olambda__762,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_aei(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_762
tff(fact_10969_ATP_Olambda__763,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_aek(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Pro6150637604445964947_nat_o(Uua,Uub),aa_Pro4243173770044397973at_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_763
tff(fact_10970_ATP_Olambda__764,axiom,
    ! [Uu: fun_list_nat_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ady(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_list_nat_o(Uua,Uub),aa_list_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_764
tff(fact_10971_ATP_Olambda__765,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se7249947736802449410_int_o,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun6889641154785167298_int_o(aTP_Lamm_aea(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_set6170482117830564987_int_o(Uua,Uub),aa_set4753215920113914285nt_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_765
tff(fact_10972_ATP_Olambda__766,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_aec(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_set_nat_o(Uua,Uub),aa_set_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_766
tff(fact_10973_ATP_Olambda__767,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aee(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_767
tff(fact_10974_ATP_Olambda__768,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aeg(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_768
tff(fact_10975_ATP_Olambda__769,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_agc(Uu),Uua),Uub)
    <=> 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,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub))) ) ).

% ATP.lambda_769
tff(fact_10976_ATP_Olambda__770,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun741873263805325634_nat_o(aTP_Lamm_agd(Uu),Uua),Uub)
    <=> 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,aa_Pro4243173770044397973at_nat(Uu,Uub)),aa_Pro4243173770044397973at_nat(Uua,Uub))) ) ).

% ATP.lambda_770
tff(fact_10977_ATP_Olambda__771,axiom,
    ! [Uu: fun_list_nat_nat,Uua: fun_list_nat_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun7915899554530022_nat_o(aTP_Lamm_afx(Uu),Uua),Uub)
    <=> 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,aa_list_nat_nat(Uu,Uub)),aa_list_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_771
tff(fact_10978_ATP_Olambda__772,axiom,
    ! [Uu: fun_se4101287982975402108nt_nat,Uua: fun_se4101287982975402108nt_nat,Uub: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aa_fun3785083049450726210_int_o(aTP_Lamm_afy(Uu),Uua),Uub)
    <=> 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,aa_set4753215920113914285nt_nat(Uu,Uub)),aa_set4753215920113914285nt_nat(Uua,Uub))) ) ).

% ATP.lambda_772
tff(fact_10979_ATP_Olambda__773,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun1332161875654541722_nat_o(aTP_Lamm_afz(Uu),Uua),Uub)
    <=> 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,aa_set_nat_nat(Uu,Uub)),aa_set_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_773
tff(fact_10980_ATP_Olambda__774,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_aga(Uu),Uua),Uub)
    <=> 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,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_774
tff(fact_10981_ATP_Olambda__775,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_agb(Uu),Uua),Uub)
    <=> 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,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub))) ) ).

% ATP.lambda_775
tff(fact_10982_ATP_Olambda__776,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vw(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_776
tff(fact_10983_ATP_Olambda__777,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vx(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_777
tff(fact_10984_ATP_Olambda__778,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aqc(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_778
tff(fact_10985_ATP_Olambda__779,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_779
tff(fact_10986_ATP_Olambda__780,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sw(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_780
tff(fact_10987_ATP_Olambda__781,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_jg(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_781
tff(fact_10988_ATP_Olambda__782,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_jh(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_782
tff(fact_10989_ATP_Olambda__783,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ji(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_783
tff(fact_10990_ATP_Olambda__784,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jj(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_784
tff(fact_10991_ATP_Olambda__785,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jk(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_785
tff(fact_10992_ATP_Olambda__786,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_num_option_num,Uub: num] : aa_num_option_num(aa_fun3636977833914307392on_num(aTP_Lamm_db(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_786
tff(fact_10993_ATP_Olambda__787,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_nat_option_num,Uub: nat] : aa_nat_option_num(aa_fun7716400287237121964on_num(aTP_Lamm_ni(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_nat_option_num(Uua,Uub)) ).

% ATP.lambda_787
tff(fact_10994_ATP_Olambda__788,axiom,
    ! [Uu: fun_option_num_o,Uua: fun_nat_option_num,Uub: nat] :
      ( aa_nat_o(aa_fun1675964276245824108_nat_o(aTP_Lamm_nk(Uu),Uua),Uub)
    <=> aa_option_num_o(Uu,aa_nat_option_num(Uua,Uub)) ) ).

% ATP.lambda_788
tff(fact_10995_ATP_Olambda__789,axiom,
    ! [Uu: fun_option_num_num,Uua: fun_num_option_num,Uub: num] : aa_num_num(aa_fun7142274290070961008um_num(aTP_Lamm_dc(Uu),Uua),Uub) = aa_option_num_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_789
tff(fact_10996_ATP_Olambda__790,axiom,
    ! [Uu: fun_option_num_nat,Uua: fun_nat_option_num,Uub: nat] : aa_nat_nat(aa_fun7387201323300553874at_nat(aTP_Lamm_nj(Uu),Uua),Uub) = aa_option_num_nat(Uu,aa_nat_option_num(Uua,Uub)) ).

% ATP.lambda_790
tff(fact_10997_ATP_Olambda__791,axiom,
    ! [Uu: fun_option_num_int,Uua: fun_num_option_num,Uub: num] : aa_num_int(aa_fun2256179333612926594um_int(aTP_Lamm_da(Uu),Uua),Uub) = aa_option_num_int(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_791
tff(fact_10998_ATP_Olambda__792,axiom,
    ! [Uu: fun_o_option_num,Uua: fun_nat_o,Uub: nat] : aa_nat_option_num(aa_fun6990899178391151900on_num(aTP_Lamm_no(Uu),Uua),Uub) = aa_o_option_num(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_792
tff(fact_10999_ATP_Olambda__793,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_nq(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_nat_o(Uua,Uub)) ) ).

% ATP.lambda_793
tff(fact_11000_ATP_Olambda__794,axiom,
    ! [Uu: fun_o_nat,Uua: fun_nat_o,Uub: nat] : aa_nat_nat(aa_fun4204860950123276802at_nat(aTP_Lamm_np(Uu),Uua),Uub) = aa_o_nat(Uu,aa_nat_o(Uua,Uub)) ).

% ATP.lambda_794
tff(fact_11001_ATP_Olambda__795,axiom,
    ! [Uu: fun_num_option_num,Uua: fun_num_num,Uub: num] : aa_num_option_num(aa_fun612943910911948912on_num(aTP_Lamm_de(Uu),Uua),Uub) = aa_num_option_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_795
tff(fact_11002_ATP_Olambda__796,axiom,
    ! [Uu: fun_num_num,Uua: fun_num_num,Uub: num] : aa_num_num(aa_fun8048995283964520608um_num(aTP_Lamm_df(Uu),Uua),Uub) = aa_num_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_796
tff(fact_11003_ATP_Olambda__797,axiom,
    ! [Uu: fun_num_int,Uua: fun_num_num,Uub: num] : aa_num_int(aa_fun3162900327506486194um_int(aTP_Lamm_dd(Uu),Uua),Uub) = aa_num_int(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_797
tff(fact_11004_ATP_Olambda__798,axiom,
    ! [Uu: fun_nat_option_num,Uua: fun_nat_nat,Uub: nat] : aa_nat_option_num(aa_fun6517009837249254290on_num(aTP_Lamm_nl(Uu),Uua),Uub) = aa_nat_option_num(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_798
tff(fact_11005_ATP_Olambda__799,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_nn(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_799
tff(fact_11006_ATP_Olambda__800,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nm(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_800
tff(fact_11007_ATP_Olambda__801,axiom,
    ! [Uu: fun_int_option_num,Uua: fun_num_int,Uub: num] : aa_num_option_num(aa_fun3858197967482829954on_num(aTP_Lamm_cy(Uu),Uua),Uub) = aa_int_option_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_801
tff(fact_11008_ATP_Olambda__802,axiom,
    ! [Uu: fun_int_num,Uua: fun_num_int,Uub: num] : aa_num_num(aa_fun5215869348392759218um_num(aTP_Lamm_cz(Uu),Uua),Uub) = aa_int_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_802
tff(fact_11009_ATP_Olambda__803,axiom,
    ! [Uu: fun_int_int,Uua: fun_num_int,Uub: num] : aa_num_int(aa_fun329774391934724804um_int(aTP_Lamm_cx(Uu),Uua),Uub) = aa_int_int(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_803
tff(fact_11010_ATP_Olambda__804,axiom,
    ! [Uu: fun_Pr1003294667014919570it_nat,Uua: produc8664842809031399944it_nat,Uub: produc8664842809031399944it_nat] : aa_Pro8996696869555642526it_nat(aa_Pro5635350972512700067it_nat(aTP_Lamm_ea(Uu),Uua),Uub) = aa_Pro8996696869555642526it_nat(some_P1914260805536162275it_nat,aa_Pro550487673543749710it_nat(aa_Pro4096753284154920275it_nat(Uu,Uua),Uub)) ).

% ATP.lambda_804
tff(fact_11011_ATP_Olambda__805,axiom,
    ! [Uu: fun_Pr774283325640823912it_nat,Uua: produc7388388658123137530it_nat,Uub: produc7388388658123137530it_nat] : aa_Pro1383761575262803586it_nat(aa_Pro8047546174521504121it_nat(aTP_Lamm_dw(Uu),Uua),Uub) = aa_Pro1383761575262803586it_nat(some_P2818173045054083285it_nat,aa_Pro88785284265212210it_nat(aa_Pro3306356554137902633it_nat(Uu,Uua),Uub)) ).

% ATP.lambda_805
tff(fact_11012_ATP_Olambda__806,axiom,
    ! [Uu: fun_Pr7750636411365408485it_nat,Uua: produc3260487557148687353it_nat,Uub: produc3260487557148687353it_nat] : aa_Pro5302094286594872192it_nat(aa_Pro7038079567744603766it_nat(aTP_Lamm_dy(Uu),Uua),Uub) = aa_Pro5302094286594872192it_nat(some_P7913643980934408916it_nat,aa_Pro4202338132779708464it_nat(aa_Pro2263720180353132454it_nat(Uu,Uua),Uub)) ).

% ATP.lambda_806
tff(fact_11013_ATP_Olambda__807,axiom,
    ! [Uu: fun_num_fun_num_num,Uua: num,Uub: num] : aa_num_option_num(aa_num6390622462783740633on_num(aTP_Lamm_ec(Uu),Uua),Uub) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Uu,Uua),Uub)) ).

% ATP.lambda_807
tff(fact_11014_ATP_Olambda__808,axiom,
    ! [Uu: fun_Co3734316457315071362atural,Uua: code_natural,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod403035634466734574eger_o(aTP_Lamm_ain(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_Cod607569804290114243atural(Uu,Uub)) ) ).

% ATP.lambda_808
tff(fact_11015_ATP_Olambda__809,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aTP_Lamm_aib(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_Code_integer_nat(Uu,Uub)) ) ).

% ATP.lambda_809
tff(fact_11016_ATP_Olambda__810,axiom,
    ! [Uu: fun_Code_integer_int,Uua: int,Uub: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aTP_Lamm_aie(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_Code_integer_int(Uu,Uub)) ) ).

% ATP.lambda_810
tff(fact_11017_ATP_Olambda__811,axiom,
    ! [Uu: fun_list_nat_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aiz(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_list_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_811
tff(fact_11018_ATP_Olambda__812,axiom,
    ! [Uu: fun_nat_Code_natural,Uua: code_natural,Uub: nat] :
      ( aa_nat_o(aa_Cod9120328075896114107_nat_o(aTP_Lamm_aiq(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_nat_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_812
tff(fact_11019_ATP_Olambda__813,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ait(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_813
tff(fact_11020_ATP_Olambda__814,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aTP_Lamm_aih(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_814
tff(fact_11021_ATP_Olambda__815,axiom,
    ! [Uu: fun_int_Code_natural,Uua: code_natural,Uub: int] :
      ( aa_int_o(aa_Cod7299285730463426143_int_o(aTP_Lamm_aik(Uu),Uua),Uub)
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(Uua),aa_int_Code_natural(Uu,Uub)) ) ).

% ATP.lambda_815
tff(fact_11022_ATP_Olambda__816,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: int] :
      ( aa_int_o(aa_nat_fun_int_o(aTP_Lamm_ahy(Uu),Uua),Uub)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_816
tff(fact_11023_ATP_Olambda__817,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aiw(Uu),Uua),Uub)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_817
tff(fact_11024_ATP_Olambda__818,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co2081214437782444048eger_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun8504338775591962366er_nat(aTP_Lamm_amu(Uu),Uua),Uub) = finite4902975817058060853nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_amt(Uu),Uua),Uub))) ).

% ATP.lambda_818
tff(fact_11025_ATP_Olambda__819,axiom,
    ! [Uu: set_o,Uua: fun_Co480733329897759773un_o_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun7670077909833392133er_nat(aTP_Lamm_amq(Uu),Uua),Uub) = finite_card_o(collect_o(aa_Cod6379977994291289310un_o_o(aa_fun8978408420733827192un_o_o(aTP_Lamm_amp(Uu),Uua),Uub))) ).

% ATP.lambda_819
tff(fact_11026_ATP_Olambda__820,axiom,
    ! [Uu: set_Code_integer,Uua: fun_in4891666191172872833eger_o,Uub: int] : aa_int_nat(aa_fun511150112467848510nt_nat(aTP_Lamm_amn(Uu),Uua),Uub) = finite4902975817058060853nteger(collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun5642168435487881468eger_o(aTP_Lamm_amm(Uu),Uua),Uub))) ).

% ATP.lambda_820
tff(fact_11027_ATP_Olambda__821,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int] : aa_int_nat(aa_fun4460091444687325999nt_nat(aTP_Lamm_aml(Uu),Uua),Uub) = finite_card_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_amk(Uu),Uua),Uub))) ).

% ATP.lambda_821
tff(fact_11028_ATP_Olambda__822,axiom,
    ! [Uu: set_o,Uua: fun_int_fun_o_o,Uub: int] : aa_int_nat(aa_fun7603103345900947235nt_nat(aTP_Lamm_amj(Uu),Uua),Uub) = finite_card_o(collect_o(aa_int_fun_o_o(aa_fun3869439686586307222un_o_o(aTP_Lamm_ami(Uu),Uua),Uub))) ).

% ATP.lambda_822
tff(fact_11029_ATP_Olambda__823,axiom,
    ! [Uu: set_Code_integer,Uua: fun_o_4021313873839786239eger_o,Uub: $o] : aa_o_nat(aa_fun5400594822745170686_o_nat(aTP_Lamm_amg(Uu),Uua),(Uub)) = finite4902975817058060853nteger(collect_Code_integer(aa_o_f8517747772616379256eger_o(aa_fun3267419118247850812eger_o(aTP_Lamm_amf(Uu),Uua),(Uub)))) ).

% ATP.lambda_823
tff(fact_11030_ATP_Olambda__824,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_o_o,Uub: $o] : aa_o_nat(aa_fun8807093888949732041_o_nat(aTP_Lamm_amd(Uu),Uua),(Uub)) = finite_card_o(collect_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_amc(Uu),Uua),(Uub)))) ).

% ATP.lambda_824
tff(fact_11031_ATP_Olambda__825,axiom,
    ! [Uu: set_int,Uua: fun_Co8353275306710069249_int_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun726042532089134319er_nat(aTP_Lamm_amr(Uu),Uua),Uub) = finite_card_int(collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_aag(Uu),Uua),Uub))) ).

% ATP.lambda_825
tff(fact_11032_ATP_Olambda__826,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_o,Uub: $o] : aa_o_nat(aa_fun2417465416094144205_o_nat(aTP_Lamm_ame(Uu),Uua),(Uub)) = finite_card_int(collect_int(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_zx(Uu),Uua),(Uub)))) ).

% ATP.lambda_826
tff(fact_11033_ATP_Olambda__827,axiom,
    ! [Uu: set_nat,Uua: fun_Co1375048237525739357_nat_o,Uub: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aTP_Lamm_anq(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_zs(Uu),Uua),Uub))) ).

% ATP.lambda_827
tff(fact_11034_ATP_Olambda__828,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aTP_Lamm_anp(Uu),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_zn(Uu),Uua),Uub))) ).

% ATP.lambda_828
tff(fact_11035_ATP_Olambda__829,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_o,Uub: $o] : aa_o_nat(aa_fun6229755271245457521_o_nat(aTP_Lamm_amv(Uu),Uua),(Uub)) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_zi(Uu),Uua),(Uub)))) ).

% ATP.lambda_829
tff(fact_11036_ATP_Olambda__830,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_iu(Uu),Uua),Uub),Uuc) = aa_Pro262675715343573244at_nat(aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uuc),Uua) ).

% ATP.lambda_830
tff(fact_11037_ATP_Olambda__831,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_iv(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_831
tff(fact_11038_ATP_Olambda__832,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_rn(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_832
tff(fact_11039_ATP_Olambda__833,axiom,
    ! [Uu: nat,Uua: fun_na2338080690716861136at_nat,Uub: fun_na2338080690716861136at_nat,Uuc: nat] :
      aa_nat2240542744625857721at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_rp(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat2240542744625857721at_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_z1048942125864253310at_nat,aa_nat2240542744625857721at_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_833
tff(fact_11040_ATP_Olambda__834,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rr(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_834
tff(fact_11041_ATP_Olambda__835,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rv(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_835
tff(fact_11042_ATP_Olambda__836,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rt(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_836
tff(fact_11043_ATP_Olambda__837,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_vn(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_837
tff(fact_11044_ATP_Olambda__838,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_vo(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_assn(Uua,Uuc),
        $ite(Uuc = Uu,one_one_assn,aa_nat_assn(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_838
tff(fact_11045_ATP_Olambda__839,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_vq(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_839
tff(fact_11046_ATP_Olambda__840,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vr(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_840
tff(fact_11047_ATP_Olambda__841,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_vs(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_841
tff(fact_11048_ATP_Olambda__842,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_ro(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_842
tff(fact_11049_ATP_Olambda__843,axiom,
    ! [Uu: nat,Uua: fun_na2338080690716861136at_nat,Uub: fun_na2338080690716861136at_nat,Uuc: nat] :
      aa_nat2240542744625857721at_nat(aa_fun4141531684813184444at_nat(aa_fun8230936523094545239at_nat(aTP_Lamm_rq(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat2240542744625857721at_nat(Uua,Uuc),aa_nat2240542744625857721at_nat(Uub,Uuc)) ).

% ATP.lambda_843
tff(fact_11050_ATP_Olambda__844,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_vp(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ).

% ATP.lambda_844
tff(fact_11051_ATP_Olambda__845,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rs(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_845
tff(fact_11052_ATP_Olambda__846,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_rw(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_846
tff(fact_11053_ATP_Olambda__847,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ru(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_847
tff(fact_11054_ATP_Olambda__848,axiom,
    ! [Uu: code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_aja(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_assn(Uua,Uuc),aa_Code_integer_assn(Uub,Uuc)) ).

% ATP.lambda_848
tff(fact_11055_ATP_Olambda__849,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ahm(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_rat(Uua,Uuc),aa_Code_integer_rat(Uub,Uuc)) ).

% ATP.lambda_849
tff(fact_11056_ATP_Olambda__850,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: code_integer] :
      aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_ahn(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_nat(Uua,Uuc),aa_Code_integer_nat(Uub,Uuc)) ).

% ATP.lambda_850
tff(fact_11057_ATP_Olambda__851,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_int,Uub: fun_Code_integer_int,Uuc: code_integer] :
      aa_Code_integer_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_aho(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_int(Uua,Uuc),aa_Code_integer_int(Uub,Uuc)) ).

% ATP.lambda_851
tff(fact_11058_ATP_Olambda__852,axiom,
    ! [Uu: $o,Uua: fun_o_assn,Uub: fun_o_assn,Uuc: $o] :
      aa_o_assn(aa_fun4536517337817438332o_assn(aa_fun3110273112593859354o_assn(aTP_Lamm_ajb((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_assn(Uua,(Uuc)),aa_o_assn(Uub,(Uuc))) ).

% ATP.lambda_852
tff(fact_11059_ATP_Olambda__853,axiom,
    ! [Uu: $o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      aa_o_rat(aa_fun4336163974892599420_o_rat(aa_fun2919298112881465802_o_rat(aTP_Lamm_ahp((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_rat(Uua,(Uuc)),aa_o_rat(Uub,(Uuc))) ).

% ATP.lambda_853
tff(fact_11060_ATP_Olambda__854,axiom,
    ! [Uu: $o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aa_fun6251207968994965954_o_nat(aTP_Lamm_ahq((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_nat(Uua,(Uuc)),aa_o_nat(Uub,(Uuc))) ).

% ATP.lambda_854
tff(fact_11061_ATP_Olambda__855,axiom,
    ! [Uu: $o,Uua: fun_o_int,Uub: fun_o_int,Uuc: $o] :
      aa_o_int(aa_fun4727766026823762812_o_int(aa_fun5648456818326528798_o_int(aTP_Lamm_ahr((Uu)),Uua),Uub),(Uuc)) = $ite((Uuc) = (Uu),aa_o_int(Uua,(Uuc)),aa_o_int(Uub,(Uuc))) ).

% ATP.lambda_855
tff(fact_11062_ATP_Olambda__856,axiom,
    ! [Uu: nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_ajc(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ).

% ATP.lambda_856
tff(fact_11063_ATP_Olambda__857,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ahs(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_857
tff(fact_11064_ATP_Olambda__858,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_aht(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_858
tff(fact_11065_ATP_Olambda__859,axiom,
    ! [Uu: int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ajd(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_assn(Uua,Uuc),aa_int_assn(Uub,Uuc)) ).

% ATP.lambda_859
tff(fact_11066_ATP_Olambda__860,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_ahu(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_860
tff(fact_11067_ATP_Olambda__861,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ahv(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_861
tff(fact_11068_ATP_Olambda__862,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      aa_Code_integer_assn(aa_fun4119713415300323650r_assn(aa_fun8067317676786998721r_assn(aTP_Lamm_ahk(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_assn(Uua,Uuc),aa_Code_integer_assn(Uub,Uuc)) ).

% ATP.lambda_862
tff(fact_11069_ATP_Olambda__863,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_aha(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_rat(Uua,Uuc),aa_Code_integer_rat(Uub,Uuc)) ).

% ATP.lambda_863
tff(fact_11070_ATP_Olambda__864,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: code_integer] :
      aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_ahc(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_nat(Uua,Uuc),aa_Code_integer_nat(Uub,Uuc)) ).

% ATP.lambda_864
tff(fact_11071_ATP_Olambda__865,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_int,Uub: fun_Code_integer_int,Uuc: code_integer] :
      aa_Code_integer_int(aa_fun3291479149765983562er_int(aa_fun4411173778969602765er_int(aTP_Lamm_ahd(Uu),Uua),Uub),Uuc) = $ite(aa_Code_integer_o(Uu,Uuc),aa_Code_integer_int(Uua,Uuc),aa_Code_integer_int(Uub,Uuc)) ).

% ATP.lambda_865
tff(fact_11072_ATP_Olambda__866,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_rat,Uub: fun_list_nat_rat,Uuc: list_nat] :
      aa_list_nat_rat(aa_fun1630595088783574696at_rat(aa_fun6629217458047550682at_rat(aTP_Lamm_ahi(Uu),Uua),Uub),Uuc) = $ite(aa_list_nat_o(Uu,Uuc),aa_list_nat_rat(Uua,Uuc),aa_list_nat_rat(Uub,Uuc)) ).

% ATP.lambda_866
tff(fact_11073_ATP_Olambda__867,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      aa_nat_assn(aa_fun785744391571715112t_assn(aa_fun6591345337174305434t_assn(aTP_Lamm_ahl(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_assn(Uua,Uuc),aa_nat_assn(Uub,Uuc)) ).

% ATP.lambda_867
tff(fact_11074_ATP_Olambda__868,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ahe(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_868
tff(fact_11075_ATP_Olambda__869,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ahg(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_869
tff(fact_11076_ATP_Olambda__870,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ahf(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_870
tff(fact_11077_ATP_Olambda__871,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      aa_int_assn(aa_fun919007281442365664t_assn(aa_fun7795160494388424622t_assn(aTP_Lamm_ahj(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_assn(Uua,Uuc),aa_int_assn(Uub,Uuc)) ).

% ATP.lambda_871
tff(fact_11078_ATP_Olambda__872,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_agz(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).

% ATP.lambda_872
tff(fact_11079_ATP_Olambda__873,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_ahb(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_873
tff(fact_11080_ATP_Olambda__874,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_ahh(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_int(Uua,Uuc),aa_int_int(Uub,Uuc)) ).

% ATP.lambda_874
tff(fact_11081_ATP_Olambda__875,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_pa(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_oz(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_875
tff(fact_11082_ATP_Olambda__876,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_oy(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ox(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_876
tff(fact_11083_ATP_Olambda__877,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co822469698833053651at_int,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_int(aa_fun3278309868433077274at_int(aa_fun1276769135904165912at_int(aTP_Lamm_abn(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_nat1674686023564409748er_int(aTP_Lamm_abm(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_zv(Uu),Uub),Uuc))) ).

% ATP.lambda_877
tff(fact_11084_ATP_Olambda__878,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_int,Uub: fun_o_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun6872859990233652739at_int(aa_fun8947132502270365866at_int(aTP_Lamm_abk(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(aa_nat_fun_o_int(aTP_Lamm_abj(Uua),Uuc),collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(Uu),Uub),Uuc))) ).

% ATP.lambda_878
tff(fact_11085_ATP_Olambda__879,axiom,
    ! [Uu: set_nat,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_abx(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_Cod8670293052248272184at_nat(aTP_Lamm_aav(Uua),Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_aaw(Uu),Uub),Uuc))) ).

% ATP.lambda_879
tff(fact_11086_ATP_Olambda__880,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_abv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_aar(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_aae(Uu),Uub),Uuc))) ).

% ATP.lambda_880
tff(fact_11087_ATP_Olambda__881,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: $o] : aa_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_abt(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_o_fun_nat_nat(aTP_Lamm_aan(Uua),(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_aao(Uu),Uub),(Uuc)))) ).

% ATP.lambda_881
tff(fact_11088_ATP_Olambda__882,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: int] : aa_int_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_abr(Uu),Uua),Uub),Uuc) = groups3188404863801439024er_int(aa_int1390634949194299888er_int(aTP_Lamm_aai(Uua),Uuc),collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_aaj(Uu),Uub),Uuc))) ).

% ATP.lambda_882
tff(fact_11089_ATP_Olambda__883,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: int] : aa_int_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_abp(Uu),Uua),Uub),Uuc) = groups3502327434004483295_o_int(aa_int_fun_o_int(aTP_Lamm_zz(Uua),Uuc),collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_aaa(Uu),Uub),Uuc))) ).

% ATP.lambda_883
tff(fact_11090_ATP_Olambda__884,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_abh(Uu),Uua),Uub),Uuc) = groups3190895334310489300er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_zu(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_zv(Uu),Uub),Uuc))) ).

% ATP.lambda_884
tff(fact_11091_ATP_Olambda__885,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_abf(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_zp(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_zq(Uu),Uub),Uuc))) ).

% ATP.lambda_885
tff(fact_11092_ATP_Olambda__886,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_abd(Uu),Uua),Uub),Uuc) = groups3504817904513533571_o_nat(aa_nat_fun_o_nat(aTP_Lamm_zk(Uua),Uuc),collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(Uu),Uub),Uuc))) ).

% ATP.lambda_886
tff(fact_11093_ATP_Olambda__887,axiom,
    ! [Uu: set_nat,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun7264493923017702603er_nat(aa_fun7529132916787797541er_nat(aTP_Lamm_aax(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(aTP_Lamm_aav(Uua),Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_aaw(Uu),Uub),Uuc))) ).

% ATP.lambda_887
tff(fact_11094_ATP_Olambda__888,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_aas(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(aTP_Lamm_aar(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_aae(Uu),Uub),Uuc))) ).

% ATP.lambda_888
tff(fact_11095_ATP_Olambda__889,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: $o] : aa_o_nat(aa_fun5720900557306247341_o_nat(aa_fun4151889263603125372_o_nat(aTP_Lamm_aap(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_o_fun_nat_nat(aTP_Lamm_aan(Uua),(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_aao(Uu),Uub),(Uuc)))) ).

% ATP.lambda_889
tff(fact_11096_ATP_Olambda__890,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: int] : aa_int_int(aa_fun4086957665101867802nt_int(aa_fun5679615576903115196nt_int(aTP_Lamm_aak(Uu),Uua),Uub),Uuc) = groups7234854612051535045er_int(aa_int1390634949194299888er_int(aTP_Lamm_aai(Uua),Uuc),collect_Code_integer(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_aaj(Uu),Uub),Uuc))) ).

% ATP.lambda_890
tff(fact_11097_ATP_Olambda__891,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: int] : aa_int_int(aa_fun7681507786902443267nt_int(aa_fun5020185059765800966nt_int(aTP_Lamm_aab(Uu),Uua),Uub),Uuc) = groups8505340233167759370_o_int(aa_int_fun_o_int(aTP_Lamm_zz(Uua),Uuc),collect_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_aaa(Uu),Uub),Uuc))) ).

% ATP.lambda_891
tff(fact_11098_ATP_Olambda__892,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_zw(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_zu(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_zv(Uu),Uub),Uuc))) ).

% ATP.lambda_892
tff(fact_11099_ATP_Olambda__893,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_zr(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_zp(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_zq(Uu),Uub),Uuc))) ).

% ATP.lambda_893
tff(fact_11100_ATP_Olambda__894,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun1827338972888073639at_nat(aa_fun390978558358138354at_nat(aTP_Lamm_zm(Uu),Uua),Uub),Uuc) = groups8507830703676809646_o_nat(aa_nat_fun_o_nat(aTP_Lamm_zk(Uua),Uuc),collect_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(Uu),Uub),Uuc))) ).

% ATP.lambda_894
tff(fact_11101_ATP_Olambda__895,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_aaf(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_td(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_aae(Uu),Uub),Uuc))) ).

% ATP.lambda_895
tff(fact_11102_ATP_Olambda__896,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_abb(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Uua),Uuc)),collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_aba(Uu),Uub),Uuc))) ).

% ATP.lambda_896
tff(fact_11103_ATP_Olambda__897,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aa_Cod8811915439870959971atural(aTP_Lamm_anr(Uu),Uua),Uub),Uuc) = produc4480994950612372183atural(produc3574140220909816553atural(inc_shift(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uu),Uuc),produc3574140220909816553atural(Uub,inc_shift(aa_num_Code_natural(numera5444537566228673987atural,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uua))) ).

% ATP.lambda_897
tff(fact_11104_ATP_Olambda__898,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ant(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_898
tff(fact_11105_ATP_Olambda__899,axiom,
    ! [Uu: fun_int_nat,Uua: set_Pr958786334691620121nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_set742265667465749333_int_o(aTP_Lamm_la(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uuc))
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uuc))
          & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc)),Uua) ) ) ) ).

% ATP.lambda_899
tff(fact_11106_ATP_Olambda__900,axiom,
    ! [Uu: fun_int_option_int,Uua: fun_Pr3238374608701176162_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun5140975492127840594_int_o(aTP_Lamm_kz(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_int_option_int(Uu,Uub) = aa_int_option_int(some_int,Uuc) )
        & aa_Pro4592068669816069595_int_o(Uua,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc)) ) ) ).

% ATP.lambda_900
tff(fact_11107_ATP_Olambda__901,axiom,
    ! [Uu: fun_fu2352890940381120596et_nat,Uua: fun_Pr2620521676128474833_nat_o,Uub: fun_Pr6460944699146269935_nat_o,Uuc: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aa_fun1045796833200626456_nat_o(aTP_Lamm_kv(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun5824947343344787589et_nat(Uu,Uub) = some_P624177172695371229et_nat(Uuc) )
        & aa_Pro5643009357358181178_nat_o(Uua,aa_Pro434340453207536480et_nat(aa_fun3926903090414666066et_nat(produc6017135503889078682et_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_901
tff(fact_11108_ATP_Olambda__902,axiom,
    ! [Uu: fun_fu1905361174868762994et_nat,Uua: fun_Pr1227177680593602867_nat_o,Uub: fun_Pr6460944699146269935_nat_o,Uuc: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aa_fun4422562838209026524_nat_o(aTP_Lamm_kw(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_fun1716447855799474851et_nat(Uu,Uub) = some_P994283844982041979et_nat(Uuc) )
        & aa_Pro2671238023309943196_nat_o(Uua,aa_Pro1979504083478021276et_nat(aa_fun615918787490101902et_nat(produc7291855603632349880et_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_902
tff(fact_11109_ATP_Olambda__903,axiom,
    ! [Uu: fun_b_7832390750199427630it_nat,Uua: fun_Pr856651894144386551_nat_o,Uub: b,Uuc: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_b_f7634180679818401570_nat_o(aa_fun5445639589913049316_nat_o(aTP_Lamm_kx(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_b_o2526299221643969375it_nat(Uu,Uub) = some_P1054239925786823975it_nat(Uuc) )
        & aa_Pro1537530206668979808_nat_o(Uua,aa_Pro2983440904383910276it_nat(aa_b_f1993928172615730054it_nat(produc4082563078715348724it_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_903
tff(fact_11110_ATP_Olambda__904,axiom,
    ! [Uu: fun_a_1313742400928293293it_nat,Uua: fun_Pr5746006151868627512_nat_o,Uub: a,Uuc: produc6653097349344004940it_nat] :
      ( aa_Pro1202363709425262478_nat_o(aa_a_f337466037417904483_nat_o(aa_fun7773039478287727974_nat_o(aTP_Lamm_ky(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_a_o5231022909227610846it_nat(Uu,Uub) = some_P1054239925786823975it_nat(Uuc) )
        & aa_Pro6426884464393220769_nat_o(Uua,aa_Pro8078911840264235907it_nat(aa_a_f2234720956197727492it_nat(produc9178034014595674355it_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_904
tff(fact_11111_ATP_Olambda__905,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_rl(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_905
tff(fact_11112_ATP_Olambda__906,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rj(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_906
tff(fact_11113_ATP_Olambda__907,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rk(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_907
tff(fact_11114_ATP_Olambda__908,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_ri(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_908
tff(fact_11115_ATP_Olambda__909,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rg(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_909
tff(fact_11116_ATP_Olambda__910,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ra(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_910
tff(fact_11117_ATP_Olambda__911,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rh(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_911
tff(fact_11118_ATP_Olambda__912,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_wh(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_912
tff(fact_11119_ATP_Olambda__913,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_wi(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_913
tff(fact_11120_ATP_Olambda__914,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co2081214437782444048eger_o,Uub: code_integer,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_amt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_914
tff(fact_11121_ATP_Olambda__915,axiom,
    ! [Uu: set_Code_integer,Uua: fun_o_4021313873839786239eger_o,Uub: $o,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(aa_fun3267419118247850812eger_o(aTP_Lamm_amf(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_915
tff(fact_11122_ATP_Olambda__916,axiom,
    ! [Uu: set_Code_integer,Uua: fun_na6318815019539281885eger_o,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun8665957533616019964eger_o(aTP_Lamm_aat(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_nat7420681472272698950eger_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_916
tff(fact_11123_ATP_Olambda__917,axiom,
    ! [Uu: set_Code_integer,Uua: fun_in4891666191172872833eger_o,Uub: int,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aa_fun5642168435487881468eger_o(aTP_Lamm_amm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_int5993532643906289898eger_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_917
tff(fact_11124_ATP_Olambda__918,axiom,
    ! [Uu: set_o,Uua: fun_Co480733329897759773un_o_o,Uub: code_integer,Uuc: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aa_fun8978408420733827192un_o_o(aTP_Lamm_amp(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_o_o(aa_Cod6379977994291289310un_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_918
tff(fact_11125_ATP_Olambda__919,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_amc(Uu),Uua),(Uub)),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_o_o(aa_o_fun_o_o(Uua,(Uub)),(Uuc)) ) ) ).

% ATP.lambda_919
tff(fact_11126_ATP_Olambda__920,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_o,Uub: nat,Uuc: $o] :
      ( aa_o_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_aal(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_o_o(aa_nat_fun_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_920
tff(fact_11127_ATP_Olambda__921,axiom,
    ! [Uu: set_o,Uua: fun_int_fun_o_o,Uub: int,Uuc: $o] :
      ( aa_o_o(aa_int_fun_o_o(aa_fun3869439686586307222un_o_o(aTP_Lamm_ami(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_o_o(aa_int_fun_o_o(Uua,Uub),(Uuc)) ) ) ).

% ATP.lambda_921
tff(fact_11128_ATP_Olambda__922,axiom,
    ! [Uu: set_nat,Uua: fun_Co1375048237525739357_nat_o,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_zs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_922
tff(fact_11129_ATP_Olambda__923,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_o,Uub: $o,Uuc: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_zi(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_o_fun_nat_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_923
tff(fact_11130_ATP_Olambda__924,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_aay(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_924
tff(fact_11131_ATP_Olambda__925,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_zn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_925
tff(fact_11132_ATP_Olambda__926,axiom,
    ! [Uu: set_int,Uua: fun_Co8353275306710069249_int_o,Uub: code_integer,Uuc: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_aag(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_Cod8477617969808203498_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_926
tff(fact_11133_ATP_Olambda__927,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_o,Uub: $o,Uuc: int] :
      ( aa_int_o(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_zx(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_o_fun_int_o(Uua,(Uub)),Uuc) ) ) ).

% ATP.lambda_927
tff(fact_11134_ATP_Olambda__928,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_aac(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_928
tff(fact_11135_ATP_Olambda__929,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_amk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_929
tff(fact_11136_ATP_Olambda__930,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co2081214437782444048eger_o,Uub: code_integer,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aa_fun7683620812777033532eger_o(aTP_Lamm_ams(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_930
tff(fact_11137_ATP_Olambda__931,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co480733329897759773un_o_o,Uub: $o,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_o_f8517747772616379256eger_o(aa_fun3295616927821077850eger_o(aTP_Lamm_amo(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_o_o(aa_Cod6379977994291289310un_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_931
tff(fact_11138_ATP_Olambda__932,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1375048237525739357_nat_o,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_zv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_932
tff(fact_11139_ATP_Olambda__933,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8353275306710069249_int_o,Uub: int,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(aa_fun7355488674016760700eger_o(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_int_o(aa_Cod8477617969808203498_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_933
tff(fact_11140_ATP_Olambda__934,axiom,
    ! [Uu: set_o,Uua: fun_o_4021313873839786239eger_o,Uub: code_integer,Uuc: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aa_fun8950210611160600154un_o_o(aTP_Lamm_ajs(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_Code_integer_o(aa_o_f8517747772616379256eger_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_934
tff(fact_11141_ATP_Olambda__935,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_amb(Uu),Uua),(Uub)),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_o_o(aa_o_fun_o_o(Uua,(Uuc)),(Uub)) ) ) ).

% ATP.lambda_935
tff(fact_11142_ATP_Olambda__936,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_nat_o,Uub: nat,Uuc: $o] :
      ( aa_o_o(aa_nat_fun_o_o(aa_fun532071374735444762un_o_o(aTP_Lamm_zl(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_nat_o(aa_o_fun_nat_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_936
tff(fact_11143_ATP_Olambda__937,axiom,
    ! [Uu: set_o,Uua: fun_o_fun_int_o,Uub: int,Uuc: $o] :
      ( aa_o_o(aa_int_fun_o_o(aa_fun250872857246597658un_o_o(aTP_Lamm_aaa(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & aa_int_o(aa_o_fun_int_o(Uua,(Uuc)),Uub) ) ) ).

% ATP.lambda_937
tff(fact_11144_ATP_Olambda__938,axiom,
    ! [Uu: set_nat,Uua: fun_na6318815019539281885eger_o,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun3722190751602477436_nat_o(aTP_Lamm_aaw(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_Code_integer_o(aa_nat7420681472272698950eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_938
tff(fact_11145_ATP_Olambda__939,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_o_o,Uub: $o,Uuc: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aa_fun2937962278134185242_nat_o(aTP_Lamm_aao(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_o_o(aa_nat_fun_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_939
tff(fact_11146_ATP_Olambda__940,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_aba(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_940
tff(fact_11147_ATP_Olambda__941,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_aae(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_941
tff(fact_11148_ATP_Olambda__942,axiom,
    ! [Uu: set_int,Uua: fun_in4891666191172872833eger_o,Uub: code_integer,Uuc: int] :
      ( aa_int_o(aa_Cod8477617969808203498_int_o(aa_fun9103777551025077884_int_o(aTP_Lamm_aju(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_Code_integer_o(aa_int5993532643906289898eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_942
tff(fact_11149_ATP_Olambda__943,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_o_o,Uub: $o,Uuc: int] :
      ( aa_int_o(aa_o_fun_int_o(aa_fun5351653887499120666_int_o(aTP_Lamm_amh(Uu),Uua),(Uub)),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_o_o(aa_int_fun_o_o(Uua,Uuc),(Uub)) ) ) ).

% ATP.lambda_943
tff(fact_11150_ATP_Olambda__944,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_zq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_944
tff(fact_11151_ATP_Olambda__945,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_ajt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_945
tff(fact_11152_ATP_Olambda__946,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_wj(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_946
tff(fact_11153_ATP_Olambda__947,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_wk(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_947
tff(fact_11154_ATP_Olambda__948,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qu(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_948
tff(fact_11155_ATP_Olambda__949,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aoi(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_949
tff(fact_11156_ATP_Olambda__950,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_any(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_950
tff(fact_11157_ATP_Olambda__951,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aok(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_951
tff(fact_11158_ATP_Olambda__952,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_anw(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_952
tff(fact_11159_ATP_Olambda__953,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aoa(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_953
tff(fact_11160_ATP_Olambda__954,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aoc(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_954
tff(fact_11161_ATP_Olambda__955,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_apq(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_955
tff(fact_11162_ATP_Olambda__956,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: fun_Co8499238673658658103nteger,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_acz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(Uua,Uuc)),aa_Cod1783974652533396728nteger(Uub,Uuc)) != one_one_Code_integer ) ) ) ).

% ATP.lambda_956
tff(fact_11163_ATP_Olambda__957,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co4490034186337852034r_assn,Uub: fun_Co4490034186337852034r_assn,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun6538703224075513620eger_o(aa_fun4422309413433313771eger_o(aTP_Lamm_adh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Code_integer_assn(Uua,Uuc)),aa_Code_integer_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_957
tff(fact_11164_ATP_Olambda__958,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: fun_o_Code_integer,Uuc: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_act(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_o_Code_integer(Uua,(Uuc))),aa_o_Code_integer(Uub,(Uuc))) != one_one_Code_integer ) ) ) ).

% ATP.lambda_958
tff(fact_11165_ATP_Olambda__959,axiom,
    ! [Uu: set_o,Uua: fun_o_assn,Uub: fun_o_assn,Uuc: $o] :
      ( aa_o_o(aa_fun8382565201914911066un_o_o(aa_fun5794833117214906962un_o_o(aTP_Lamm_adb(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(Uua,(Uuc))),aa_o_assn(Uub,(Uuc))) != one_one_assn ) ) ) ).

% ATP.lambda_959
tff(fact_11166_ATP_Olambda__960,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_adj(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(Uua,(Uuc))),aa_o_rat(Uub,(Uuc))) != one_one_rat ) ) ) ).

% ATP.lambda_960
tff(fact_11167_ATP_Olambda__961,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_acv(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_961
tff(fact_11168_ATP_Olambda__962,axiom,
    ! [Uu: set_nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_add(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uua,Uuc)),aa_nat_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_962
tff(fact_11169_ATP_Olambda__963,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_adl(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_963
tff(fact_11170_ATP_Olambda__964,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_acx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(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_964
tff(fact_11171_ATP_Olambda__965,axiom,
    ! [Uu: set_int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_adf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(Uua,Uuc)),aa_int_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_965
tff(fact_11172_ATP_Olambda__966,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8499238673658658103nteger,Uub: fun_Co8499238673658658103nteger,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun7159535342743012639eger_o(aa_fun5296778454019918337eger_o(aTP_Lamm_acf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(Uua,Uuc)),aa_Cod1783974652533396728nteger(Uub,Uuc)) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_966
tff(fact_11173_ATP_Olambda__967,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_acn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(Uua,Uuc)),aa_Code_integer_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_967
tff(fact_11174_ATP_Olambda__968,axiom,
    ! [Uu: set_o,Uua: fun_o_Code_integer,Uub: fun_o_Code_integer,Uuc: $o] :
      ( aa_o_o(aa_fun8192762245619933327un_o_o(aa_fun1865401534164435218un_o_o(aTP_Lamm_abz(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(Uua,(Uuc))),aa_o_Code_integer(Uub,(Uuc))) != zero_z3403309356797280102nteger ) ) ) ).

% ATP.lambda_968
tff(fact_11175_ATP_Olambda__969,axiom,
    ! [Uu: set_o,Uua: fun_o_rat,Uub: fun_o_rat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_rat_fun_o_o(aa_fun5510981055783399506un_o_o(aTP_Lamm_ach(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(Uua,(Uuc))),aa_o_rat(Uub,(Uuc))) != zero_zero_rat ) ) ) ).

% ATP.lambda_969
tff(fact_11176_ATP_Olambda__970,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: fun_o_nat,Uuc: $o] :
      ( aa_o_o(aa_fun_o_nat_fun_o_o(aa_fun5833554292436378194un_o_o(aTP_Lamm_acp(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(Uua,(Uuc))),aa_o_nat(Uub,(Uuc))) != zero_zero_nat ) ) ) ).

% ATP.lambda_970
tff(fact_11177_ATP_Olambda__971,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_acb(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_971
tff(fact_11178_ATP_Olambda__972,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_acj(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_972
tff(fact_11179_ATP_Olambda__973,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_acr(Uu),Uua),Uub),Uuc)
    <=> ( aa_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_973
tff(fact_11180_ATP_Olambda__974,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_acd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(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_974
tff(fact_11181_ATP_Olambda__975,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_acl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(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_975
tff(fact_11182_ATP_Olambda__976,axiom,
    ! [Uu: nat,Uua: list_Product_unit,Uub: array_Product_unit,Uuc: product_unit] : aa_Product_unit_assn(aa_arr8761957006411741156t_assn(aa_lis5628126305658872932t_assn(aTP_Lamm_il(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_a4522542871529764173t_unit(Uub,Uua)),aa_o_assn(pure_assn,Uuc = nth_Product_unit(Uua,Uu))) ).

% ATP.lambda_976
tff(fact_11183_ATP_Olambda__977,axiom,
    ! [Uu: nat,Uua: list_o,Uub: array_o,Uuc: $o] : aa_o_assn(aa_arr2544151649598861476o_assn(aa_lis966790859614740569o_assn(aTP_Lamm_ij(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_o(Uub,Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(nth_o(Uua),Uu))) ).

% ATP.lambda_977
tff(fact_11184_ATP_Olambda__978,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat] : aa_nat_assn(aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_ii(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_nat(Uub,Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_nat(nth_nat(Uua),Uu))) ).

% ATP.lambda_978
tff(fact_11185_ATP_Olambda__979,axiom,
    ! [Uu: nat,Uua: list_int,Uub: array_int,Uuc: int] : aa_int_assn(aa_arr1589534196239669886t_assn(aa_lis459371148865260371t_assn(aTP_Lamm_ik(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_int(Uub,Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_int(nth_int(Uua),Uu))) ).

% ATP.lambda_979
tff(fact_11186_ATP_Olambda__980,axiom,
    ! [Uu: fun_o_rat,Uua: rat,Uub: list_o,Uuc: nat] : aa_nat_rat(aa_lis2932070769283852167at_rat(aa_rat5396404679109611328at_rat(aTP_Lamm_vf(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(Uu,aa_nat_o(nth_o(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_980
tff(fact_11187_ATP_Olambda__981,axiom,
    ! [Uu: fun_o_nat,Uua: nat,Uub: list_o,Uuc: nat] : aa_nat_nat(aa_lis6440299327677248895at_nat(aa_nat4797045371981825344at_nat(aTP_Lamm_vk(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(Uu,aa_nat_o(nth_o(Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uuc)) ).

% ATP.lambda_981
tff(fact_11188_ATP_Olambda__982,axiom,
    ! [Uu: fun_o_int,Uua: int,Uub: list_o,Uuc: nat] : aa_nat_int(aa_lis2262448308168052187at_int(aa_int7290165588788099136at_int(aTP_Lamm_vm(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(Uu,aa_nat_o(nth_o(Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_982
tff(fact_11189_ATP_Olambda__983,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: list_nat,Uuc: nat] : aa_nat_rat(aa_lis7014846069874721091at_rat(aa_rat4396832037704353944at_rat(aTP_Lamm_ve(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_983
tff(fact_11190_ATP_Olambda__984,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: list_nat,Uuc: nat] : aa_nat_nat(aa_lis1299702591413342011at_nat(aa_nat9125522253416007816at_nat(aTP_Lamm_vj(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uuc)) ).

% ATP.lambda_984
tff(fact_11191_ATP_Olambda__985,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: list_nat,Uuc: nat] : aa_nat_int(aa_lis6345223608758921111at_int(aa_int7844622935458978624at_int(aTP_Lamm_vh(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(nth_nat(Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_985
tff(fact_11192_ATP_Olambda__986,axiom,
    ! [Uu: fun_int_rat,Uua: rat,Uub: list_int,Uuc: nat] : aa_nat_rat(aa_lis1742918661435654559at_rat(aa_rat8348276666120063220at_rat(aTP_Lamm_vg(Uu),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_986
tff(fact_11193_ATP_Olambda__987,axiom,
    ! [Uu: fun_int_nat,Uua: nat,Uub: list_int,Uuc: nat] : aa_nat_nat(aa_lis5251147219829051287at_nat(aa_nat3853594844976941284at_nat(aTP_Lamm_vl(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uuc)) ).

% ATP.lambda_987
tff(fact_11194_ATP_Olambda__988,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: list_int,Uuc: nat] : aa_nat_int(aa_lis1073296200319854579at_int(aa_int2572695527019912092at_int(aTP_Lamm_vi(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,aa_nat_int(nth_int(Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_988
tff(fact_11195_ATP_Olambda__989,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_ox(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_989
tff(fact_11196_ATP_Olambda__990,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_oz(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_990
tff(fact_11197_ATP_Olambda__991,axiom,
    ! [Uu: set_Code_integer,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: nat] : aa_nat_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_abw(Uu),Uua),Uub),Uuc) = groups3190895334310489300er_nat(aa_nat5852537043073606456er_nat(Uua,Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun8665957533616019964eger_o(aTP_Lamm_aat(Uu),Uub),Uuc))) ).

% ATP.lambda_991
tff(fact_11198_ATP_Olambda__992,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: nat] : aa_nat_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_abs(Uu),Uua),Uub),Uuc) = groups3504817904513533571_o_nat(aa_nat_fun_o_nat(Uua,Uuc),collect_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_aal(Uu),Uub),Uuc))) ).

% ATP.lambda_992
tff(fact_11199_ATP_Olambda__993,axiom,
    ! [Uu: set_int,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: code_integer] : aa_Code_integer_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_abq(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_Cod5491795744640130032nt_int(Uua,Uuc),collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_aag(Uu),Uub),Uuc))) ).

% ATP.lambda_993
tff(fact_11200_ATP_Olambda__994,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_abu(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_aac(Uu),Uub),Uuc))) ).

% ATP.lambda_994
tff(fact_11201_ATP_Olambda__995,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: $o] : aa_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_abo(Uu),Uua),Uub),(Uuc)) = groups1705073143266064639nt_int(aa_o_fun_int_int(Uua,(Uuc)),collect_int(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_zx(Uu),Uub),(Uuc)))) ).

% ATP.lambda_995
tff(fact_11202_ATP_Olambda__996,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_abg(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_zs(Uu),Uub),Uuc))) ).

% ATP.lambda_996
tff(fact_11203_ATP_Olambda__997,axiom,
    ! [Uu: set_nat,Uua: fun_Co822469698833053651at_int,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_int(aa_fun2505341505705002151er_int(aa_fun493137163814837797er_int(aTP_Lamm_abl(Uu),Uua),Uub),Uuc) = aa_set_nat_int(groups705719431365010083at_int(aa_Cod4492442032739075476at_int(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_zs(Uu),Uub),Uuc))) ).

% ATP.lambda_997
tff(fact_11204_ATP_Olambda__998,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_abe(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_zn(Uu),Uub),Uuc))) ).

% ATP.lambda_998
tff(fact_11205_ATP_Olambda__999,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: $o] : aa_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_abc(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_o_fun_nat_nat(Uua,(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_zi(Uu),Uub),(Uuc)))) ).

% ATP.lambda_999
tff(fact_11206_ATP_Olambda__1000,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_int,Uub: fun_o_fun_nat_o,Uuc: $o] : aa_o_int(aa_fun2051904251736260813_o_int(aa_fun7508458648369839740_o_int(aTP_Lamm_abi(Uu),Uua),Uub),(Uuc)) = aa_set_nat_int(groups705719431365010083at_int(aa_o_fun_nat_int(Uua,(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_zi(Uu),Uub),(Uuc)))) ).

% ATP.lambda_1000
tff(fact_11207_ATP_Olambda__1001,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_aaz(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_aay(Uu),Uub),Uuc))) ).

% ATP.lambda_1001
tff(fact_11208_ATP_Olambda__1002,axiom,
    ! [Uu: set_Code_integer,Uua: fun_na5492297256294505719er_nat,Uub: fun_na6318815019539281885eger_o,Uuc: nat] : aa_nat_nat(aa_fun2058600731543071806at_nat(aa_fun4864976072519958552at_nat(aTP_Lamm_aau(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(Uua,Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun8665957533616019964eger_o(aTP_Lamm_aat(Uu),Uub),Uuc))) ).

% ATP.lambda_1002
tff(fact_11209_ATP_Olambda__1003,axiom,
    ! [Uu: set_o,Uua: fun_nat_fun_o_nat,Uub: fun_nat_fun_o_o,Uuc: nat] : aa_nat_nat(aa_fun4178193321845537899at_nat(aa_fun3391070971708407026at_nat(aTP_Lamm_aam(Uu),Uua),Uub),Uuc) = groups8507830703676809646_o_nat(aa_nat_fun_o_nat(Uua,Uuc),collect_o(aa_nat_fun_o_o(aa_fun3555063939605388254un_o_o(aTP_Lamm_aal(Uu),Uub),Uuc))) ).

% ATP.lambda_1003
tff(fact_11210_ATP_Olambda__1004,axiom,
    ! [Uu: set_int,Uua: fun_Co3009391273526075439nt_int,Uub: fun_Co8353275306710069249_int_o,Uuc: code_integer] : aa_Code_integer_int(aa_fun5771563549434713419er_int(aa_fun4265095070934345581er_int(aTP_Lamm_aah(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_Cod5491795744640130032nt_int(Uua,Uuc),collect_int(aa_Cod8477617969808203498_int_o(aa_fun1593725752699181308_int_o(aTP_Lamm_aag(Uu),Uub),Uuc))) ).

% ATP.lambda_1004
tff(fact_11211_ATP_Olambda__1005,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_aaq(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_aac(Uu),Uub),Uuc))) ).

% ATP.lambda_1005
tff(fact_11212_ATP_Olambda__1006,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_aad(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_aac(Uu),Uub),Uuc))) ).

% ATP.lambda_1006
tff(fact_11213_ATP_Olambda__1007,axiom,
    ! [Uu: set_int,Uua: fun_o_fun_int_int,Uub: fun_o_fun_int_o,Uuc: $o] : aa_o_int(aa_fun7462986433439723305_o_int(aa_fun4201161283678742396_o_int(aTP_Lamm_zy(Uu),Uua),Uub),(Uuc)) = groups4538972089207619220nt_int(aa_o_fun_int_int(Uua,(Uuc)),collect_int(aa_o_fun_int_o(aa_fun1733087058159411102_int_o(aTP_Lamm_zx(Uu),Uub),(Uuc)))) ).

% ATP.lambda_1007
tff(fact_11214_ATP_Olambda__1008,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_zt(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_zs(Uu),Uub),Uuc))) ).

% ATP.lambda_1008
tff(fact_11215_ATP_Olambda__1009,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_zo(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_zn(Uu),Uub),Uuc))) ).

% ATP.lambda_1009
tff(fact_11216_ATP_Olambda__1010,axiom,
    ! [Uu: set_nat,Uua: fun_o_fun_nat_nat,Uub: fun_o_fun_nat_o,Uuc: $o] : aa_o_nat(aa_fun6229755271245457521_o_nat(aa_fun802073206315457404_o_nat(aTP_Lamm_zj(Uu),Uua),Uub),(Uuc)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_o_fun_nat_nat(Uua,(Uuc))),collect_nat(aa_o_fun_nat_o(aa_fun9138341750119017558_nat_o(aTP_Lamm_zi(Uu),Uub),(Uuc)))) ).

% ATP.lambda_1010
tff(fact_11217_ATP_Olambda__1011,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_lb((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1011
tff(fact_11218_ATP_Olambda__1012,axiom,
    ! [Uu: fun_b_6677255003012645673t_unit,Uua: b,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3804989708481012326it_nat(aa_hea5083147195637387702it_nat(aa_b_f7124492953218388226it_nat(aTP_Lamm_kp(Uu),Uua),Uub),Uuc) = heap_T3616969660504097270t_unit(Uuc,aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_b_H8549598898354664410t_unit(Uu,Uua)),Uub)) ).

% ATP.lambda_1012
tff(fact_11219_ATP_Olambda__1013,axiom,
    ! [Uu: fun_b_8426044978055006235Heap_b,Uua: b,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3693463769312751704it_nat(aa_hea7391015955736993704it_nat(aa_b_f3177101503181015796it_nat(aTP_Lamm_kc(Uu),Uua),Uub),Uuc) = heap_T7616092557645711336rame_b(Uuc,aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_b_H1867786022829710796Heap_b(Uu,Uua)),Uub)) ).

% ATP.lambda_1013
tff(fact_11220_ATP_Olambda__1014,axiom,
    ! [Uu: fun_b_8355010938008660250Heap_a,Uua: b,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3190776082673941207it_nat(aa_hea414212936809421479it_nat(aa_b_f8802815735423186547it_nat(aTP_Lamm_kg(Uu),Uua),Uub),Uuc) = heap_T7616092557645711335rame_a(Uuc,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_b_H1867786018526481995Heap_a(Uu,Uua)),Uub)) ).

% ATP.lambda_1014
tff(fact_11221_ATP_Olambda__1015,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_ks(Uu),Uua),Uub),Uuc) = heap_T3616969660504097270t_unit(Uuc,aa_hea684056619475940329it_nat(heap_T875086893843062177t_unit(aa_a_H4862568901524997081t_unit(Uu,Uua)),Uub)) ).

% ATP.lambda_1015
tff(fact_11222_ATP_Olambda__1016,axiom,
    ! [Uu: fun_a_6164325696594227994Heap_b,Uua: a,Uub: heap_e7401611519738050253t_unit,Uuc: nat] : aa_nat3693463769312751704it_nat(aa_hea7391015955736993704it_nat(aa_a_f5359256761190816565it_nat(aTP_Lamm_ke(Uu),Uua),Uub),Uuc) = heap_T7616092557645711336rame_b(Uuc,aa_hea5227063804513972443it_nat(heap_Time_execute_b(aa_a_H8829438778223708363Heap_b(Uu,Uua)),Uub)) ).

% ATP.lambda_1016
tff(fact_11223_ATP_Olambda__1017,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_ki(Uu),Uua),Uub),Uuc) = heap_T7616092557645711335rame_a(Uuc,aa_hea4724376117875161946it_nat(heap_Time_execute_a(aa_a_H8829438773920479562Heap_a(Uu,Uua)),Uub)) ).

% ATP.lambda_1017
tff(fact_11224_ATP_Olambda__1018,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1018
tff(fact_11225_ATP_Olambda__1019,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_ja(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_1019
tff(fact_11226_ATP_Olambda__1020,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_jb(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1020
tff(fact_11227_ATP_Olambda__1021,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_jc(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_1021
tff(fact_11228_ATP_Olambda__1022,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_iw(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_1022
tff(fact_11229_ATP_Olambda__1023,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_ix(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_1023
tff(fact_11230_ATP_Olambda__1024,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_jd(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1024
tff(fact_11231_ATP_Olambda__1025,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_iy(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_1025
tff(fact_11232_ATP_Olambda__1026,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_iz(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_1026
tff(fact_11233_ATP_Olambda__1027,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_jf(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_1027
tff(fact_11234_ATP_Olambda__1028,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_tr(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_1028
tff(fact_11235_ATP_Olambda__1029,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ts(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_1029
tff(fact_11236_ATP_Olambda__1030,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_py(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_1030
tff(fact_11237_ATP_Olambda__1031,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_tx(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_1031
tff(fact_11238_ATP_Olambda__1032,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_set1501693898512841689_nat_o(aTP_Lamm_kb(Uu),Uua),Uub),Uuc) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_set1591873763996598871_nat_o(aTP_Lamm_ka(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1032
tff(fact_11239_ATP_Olambda__1033,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: int,Uuc: int] : aa_int4542576447706434252_int_o(aa_int6955917519153030133_int_o(aa_set7124137317651323025_int_o(aTP_Lamm_jz(Uu),Uua),Uub),Uuc) = aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aa_int4499255921420784437_int_o(aa_set4999213539451412527_int_o(aTP_Lamm_jy(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1033
tff(fact_11240_ATP_Olambda__1034,axiom,
    ! [Uu: fun_nat_o,Uua: fun_na7044673412300895210at_nat,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun1888629816020924642_nat_o(aTP_Lamm_jx(Uu),Uua),Uub),Uuc) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_fun5017297687458839182_nat_o(aTP_Lamm_jw(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1034
tff(fact_11241_ATP_Olambda__1035,axiom,
    ! [Uu: fun_int_o,Uua: fun_in7209865251430457670nt_int,Uub: int,Uuc: int] : aa_int4542576447706434252_int_o(aa_int6955917519153030133_int_o(aa_fun8358116108987704894_int_o(aTP_Lamm_jv(Uu),Uua),Uub),Uuc) = aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aa_int4499255921420784437_int_o(aa_fun4962286606553988290_int_o(aTP_Lamm_ju(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1035
tff(fact_11242_ATP_Olambda__1036,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aoe(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_1036
tff(fact_11243_ATP_Olambda__1037,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aoq(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_1037
tff(fact_11244_ATP_Olambda__1038,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aoo(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_1038
tff(fact_11245_ATP_Olambda__1039,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aom(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_1039
tff(fact_11246_ATP_Olambda__1040,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_sa(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(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_1040
tff(fact_11247_ATP_Olambda__1041,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_rx(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_1041
tff(fact_11248_ATP_Olambda__1042,axiom,
    ! [Uu: multis2468970476368604999at_nat,Uua: multis2468970476368604999at_nat,Uub: set_Pr8551490117392284871at_nat,Uuc: multis2468970476368604999at_nat,Uud: multis2468970476368604999at_nat] :
      ( aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(aa_set6182894787541909731_nat_o(aa_mul6910776717834277237_nat_o(aTP_Lamm_mu(Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( member6689249552917799696at_nat(produc4348348721325984599at_nat(Uuc,Uu),transi5749863603491437800at_nat(Uub))
          | ( Uuc = Uu ) )
        & ( member6689249552917799696at_nat(produc4348348721325984599at_nat(Uua,Uud),transi5749863603491437800at_nat(Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1042
tff(fact_11249_ATP_Olambda__1043,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr4329608150637261639at_nat,Uuc: set_Pr1261947904930325089at_nat,Uud: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aa_set5605188905385065315_nat_o(aa_set3198043947577432463_nat_o(aTP_Lamm_mv(Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uuc,Uu)),transi2703068831062848130at_nat(Uub))
          | ( Uuc = Uu ) )
        & ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(produc2922128104949294807at_nat(Uua,Uud)),transi2703068831062848130at_nat(Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1043
tff(fact_11250_ATP_Olambda__1044,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_Pr1261947904930325089at_nat,Uuc: nat,Uud: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_set2634951030311766213_nat_o(aa_nat5178545565753023578_nat_o(aTP_Lamm_mw(Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uuc),Uu)),transi6264000038957366511cl_nat(Uub))
          | ( Uuc = Uu ) )
        & ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uud)),transi6264000038957366511cl_nat(Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1044
tff(fact_11251_ATP_Olambda__1045,axiom,
    ! [Uu: int,Uua: int,Uub: set_Pr958786334691620121nt_int,Uuc: int,Uud: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_set742265667465749333_int_o(aa_int2891257317550437190_int_o(aTP_Lamm_mx(Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uuc),Uu)),transi6261509568448316235cl_int(Uub))
          | ( Uuc = Uu ) )
        & ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uud)),transi6261509568448316235cl_int(Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1045
tff(fact_11252_ATP_Olambda__1046,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_ry(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_1046
tff(fact_11253_ATP_Olambda__1047,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: nat,Uuc: nat,Uud: nat,Uue: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_set1591873763996598871_nat_o(aTP_Lamm_ka(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uub),Uud)),Uu)
        | ( ( Uub = Uud )
          & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uuc),Uue)),Uua) ) ) ) ).

% ATP.lambda_1047
tff(fact_11254_ATP_Olambda__1048,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: int,Uuc: int,Uud: int,Uue: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aa_int4499255921420784437_int_o(aa_set4999213539451412527_int_o(aTP_Lamm_jy(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uud)),Uu)
        | ( ( Uub = Uud )
          & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uuc),Uue)),Uua) ) ) ) ).

% ATP.lambda_1048
tff(fact_11255_ATP_Olambda__1049,axiom,
    ! [Uu: fun_nat_o,Uua: fun_na7044673412300895210at_nat,Uub: nat,Uuc: nat,Uud: nat,Uue: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_fun5017297687458839182_nat_o(aTP_Lamm_jw(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa_nat_o(Uu,Uud)
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uuc),Uue)),aa_nat7062706319057040595at_nat(Uua,Uud)) ) ) ).

% ATP.lambda_1049
tff(fact_11256_ATP_Olambda__1050,axiom,
    ! [Uu: fun_int_o,Uua: fun_in7209865251430457670nt_int,Uub: int,Uuc: int,Uud: int,Uue: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aa_int4499255921420784437_int_o(aa_fun4962286606553988290_int_o(aTP_Lamm_ju(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa_int_o(Uu,Uud)
        & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uuc),Uue)),aa_int8519581412571446831nt_int(Uua,Uud)) ) ) ).

% ATP.lambda_1050
tff(fact_11257_ATP_Olambda__1051,axiom,
    ! [Uu: heap_T5738788834812785303t_unit,Uua: product_unit] : aa_Pro1108561329362217960t_unit(aTP_Lamm_ei(Uu),Uua) = Uu ).

% ATP.lambda_1051
tff(fact_11258_ATP_Olambda__1052,axiom,
    ! [Uu: heap_Time_Heap_b,Uua: product_unit] : aa_Pro7886978639879396314Heap_b(aTP_Lamm_eh(Uu),Uua) = Uu ).

% ATP.lambda_1052
tff(fact_11259_ATP_Olambda__1053,axiom,
    ! [Uu: heap_Time_Heap_a,Uua: product_unit] : aa_Pro7886978635576167513Heap_a(aTP_Lamm_eg(Uu),Uua) = Uu ).

% ATP.lambda_1053
tff(fact_11260_ATP_Olambda__1054,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_akv(Uu),Uua) = Uu ).

% ATP.lambda_1054
tff(fact_11261_ATP_Olambda__1055,axiom,
    ! [Uu: code_integer,Uua: int] : aa_int_Code_integer(aTP_Lamm_akw(Uu),Uua) = Uu ).

% ATP.lambda_1055
tff(fact_11262_ATP_Olambda__1056,axiom,
    ! [Uu: rat,Uua: list_nat] : aa_list_nat_rat(aTP_Lamm_akx(Uu),Uua) = Uu ).

% ATP.lambda_1056
tff(fact_11263_ATP_Olambda__1057,axiom,
    ! [Uu: rat,Uua: set_nat] : aa_set_nat_rat(aTP_Lamm_aky(Uu),Uua) = Uu ).

% ATP.lambda_1057
tff(fact_11264_ATP_Olambda__1058,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_akt(Uu),Uua) = Uu ).

% ATP.lambda_1058
tff(fact_11265_ATP_Olambda__1059,axiom,
    ! [Uu: rat,Uua: int] : aa_int_rat(aTP_Lamm_aku(Uu),Uua) = Uu ).

% ATP.lambda_1059
tff(fact_11266_ATP_Olambda__1060,axiom,
    ! [Uu: nat,Uua: list_nat] : aa_list_nat_nat(aTP_Lamm_akm(Uu),Uua) = Uu ).

% ATP.lambda_1060
tff(fact_11267_ATP_Olambda__1061,axiom,
    ! [Uu: nat,Uua: set_nat] : aa_set_nat_nat(aTP_Lamm_akn(Uu),Uua) = Uu ).

% ATP.lambda_1061
tff(fact_11268_ATP_Olambda__1062,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_akq,Uu),Uua) = Uu ).

% ATP.lambda_1062
tff(fact_11269_ATP_Olambda__1063,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_ff,Uu),Uua) = Uu ).

% ATP.lambda_1063
tff(fact_11270_ATP_Olambda__1064,axiom,
    ! [Uu: int,Uua: list_nat] : aa_list_nat_int(aTP_Lamm_ako(Uu),Uua) = Uu ).

% ATP.lambda_1064
tff(fact_11271_ATP_Olambda__1065,axiom,
    ! [Uu: int,Uua: set_nat] : aa_set_nat_int(aTP_Lamm_akp(Uu),Uua) = Uu ).

% ATP.lambda_1065
tff(fact_11272_ATP_Olambda__1066,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_akr,Uu),Uua) = Uu ).

% ATP.lambda_1066
tff(fact_11273_ATP_Olambda__1067,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aks,Uu),Uua) = Uu ).

% ATP.lambda_1067
tff(fact_11274_ATP_Olambda__1068,axiom,
    ! [Uu: fun_Pr5876161306168176259nteger,Uua: code_integer] : aa_Cod1978467581589928094nteger(aTP_Lamm_fd(Uu),Uua) = Uu ).

% ATP.lambda_1068
tff(fact_11275_ATP_Olambda__1069,axiom,
    ! [Uu: fun_Pr5876161306168176259nteger,Uua: fun_Co8499238673658658103nteger] : aa_fun2509385170507305080nteger(aTP_Lamm_fe(Uu),Uua) = Uu ).

% ATP.lambda_1069
tff(fact_11276_ATP_Olambda__1070,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_im,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1070
tff(fact_11277_ATP_Olambda__1071,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_in,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1071
tff(fact_11278_ATP_Olambda__1072,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_fn,Uu) = Uu ).

% ATP.lambda_1072
tff(fact_11279_ATP_Olambda__1073,axiom,
    ! [Uu: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_aqg,Uu) = Uu ).

% ATP.lambda_1073
tff(fact_11280_ATP_Olambda__1074,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_fo,Uu) = Uu ).

% ATP.lambda_1074
tff(fact_11281_ATP_Olambda__1075,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_fp,Uu) = Uu ).

% ATP.lambda_1075
tff(fact_11282_ATP_Olambda__1076,axiom,
    ! [Uu: num] : aa_num_num(aTP_Lamm_alj,Uu) = Uu ).

% ATP.lambda_1076
tff(fact_11283_ATP_Olambda__1077,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fq,Uu) = Uu ).

% ATP.lambda_1077
tff(fact_11284_ATP_Olambda__1078,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_fr,Uu) = Uu ).

% ATP.lambda_1078
tff(fact_11285_ATP_Olambda__1079,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_fj,Uu) = zero_z3403309356797280102nteger ).

% ATP.lambda_1079
tff(fact_11286_ATP_Olambda__1080,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_fk,Uu) = zero_zero_rat ).

% ATP.lambda_1080
tff(fact_11287_ATP_Olambda__1081,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_fl,Uu) = zero_zero_nat ).

% ATP.lambda_1081
tff(fact_11288_ATP_Olambda__1082,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_fm,Uu) = zero_zero_int ).

% ATP.lambda_1082
tff(fact_11289_ATP_Olambda__1083,axiom,
    ! [Uu: product_unit] : aa_Product_unit_assn(aTP_Lamm_ey,Uu) = one_one_assn ).

% ATP.lambda_1083
tff(fact_11290_ATP_Olambda__1084,axiom,
    ! [Uu: list_nat] : aa_list_nat_nat(aTP_Lamm_amx,Uu) = one_one_nat ).

% ATP.lambda_1084
tff(fact_11291_ATP_Olambda__1085,axiom,
    ! [Uu: set_nat] : aa_set_nat_nat(aTP_Lamm_amy,Uu) = one_one_nat ).

% ATP.lambda_1085
tff(fact_11292_ATP_Olambda__1086,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_sr,Uu) = one_one_nat ).

% ATP.lambda_1086
tff(fact_11293_ATP_Olambda__1087,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ss,Uu) = one_one_int ).

% ATP.lambda_1087
tff(fact_11294_ATP_Olambda__1088,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_amw,Uu) = one_one_nat ).

% ATP.lambda_1088
tff(fact_11295_ATP_Olambda__1089,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_st,Uu) = one_one_int ).

% ATP.lambda_1089
tff(fact_11296_ATP_Olambda__1090,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_dv,Uu) = none_P9117596204409417319it_nat ).

% ATP.lambda_1090
tff(fact_11297_ATP_Olambda__1091,axiom,
    ! [Uu: nat] : aa_nat_option_nat(aTP_Lamm_eu,Uu) = none_nat ).

% ATP.lambda_1091
tff(fact_11298_ATP_Olambda__1092,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_gt,Uu)
    <=> $false ) ).

% ATP.lambda_1092
tff(fact_11299_ATP_Olambda__1093,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_em,Uu)
    <=> $false ) ).

% ATP.lambda_1093
tff(fact_11300_ATP_Olambda__1094,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ej,Uu)
    <=> $false ) ).

% ATP.lambda_1094
tff(fact_11301_ATP_Olambda__1095,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_en,(Uu))
    <=> $false ) ).

% ATP.lambda_1095
tff(fact_11302_ATP_Olambda__1096,axiom,
    ! [Uu: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(aTP_Lamm_ek,Uu)
    <=> $false ) ).

% ATP.lambda_1096
tff(fact_11303_ATP_Olambda__1097,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_el,Uu)
    <=> $false ) ).

% ATP.lambda_1097
tff(fact_11304_ATP_Olambda__1098,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_cw,Uu)
    <=> $false ) ).

% ATP.lambda_1098
tff(fact_11305_ATP_Olambda__1099,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_eo,Uu)
    <=> $false ) ).

% ATP.lambda_1099
tff(fact_11306_ATP_Olambda__1100,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_ep,Uu)
    <=> $false ) ).

% ATP.lambda_1100
tff(fact_11307_ATP_Olambda__1101,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_dj,Uu)
    <=> $true ) ).

% ATP.lambda_1101
tff(fact_11308_ATP_Olambda__1102,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ob,Uu)
    <=> $true ) ).

% ATP.lambda_1102

% Helper facts (28)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ( (X) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y))
      | ( (X) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X != Y )
      | aa_set_nat_o(fequal_set_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(fequal_set_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | aa_list_nat_o(fequal_list_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X != Y )
      | aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( ( X != Y )
      | aa_set6170482117830564987_int_o(fequal1127456680179375854nt_int(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( ~ aa_set6170482117830564987_int_o(fequal1127456680179375854nt_int(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
      | aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(fequal1430618250418080822at_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(fequal1430618250418080822at_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( X != Y )
      | aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(fequal2154631582430647452at_nat,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ~ aa_mul869302825175964237_nat_o(aa_mul3916465897682844511_nat_o(fequal2154631582430647452at_nat,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_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_T,axiom,
    ! [X: produc3260487557148687353it_nat,Y: produc3260487557148687353it_nat] :
      ( ( X != Y )
      | aa_Pro6426884464393220769_nat_o(fequal7405809337025681252it_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_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_T,axiom,
    ! [X: produc3260487557148687353it_nat,Y: produc3260487557148687353it_nat] :
      ( ~ aa_Pro6426884464393220769_nat_o(fequal7405809337025681252it_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc7388388658123137530it_nat,Y: produc7388388658123137530it_nat] :
      ( ( X != Y )
      | aa_Pro1537530206668979808_nat_o(fequal2310338401145355621it_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: produc7388388658123137530it_nat,Y: produc7388388658123137530it_nat] :
      ( ~ aa_Pro1537530206668979808_nat_o(fequal2310338401145355621it_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_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_T,axiom,
    ! [X: produc8664842809031399944it_nat,Y: produc8664842809031399944it_nat] :
      ( ( X != Y )
      | aa_Pro4744234731172327378_nat_o(fequal1407654770548522611it_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_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_T,axiom,
    ! [X: produc8664842809031399944it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ aa_Pro4744234731172327378_nat_o(fequal1407654770548522611it_nat(X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( X != Y )
      | aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ aa_Pro5947980084102686253_nat_o(fequal7949547063978738364at_nat(X),Y)
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_b_assn(q,rg)),produc7507926704131184380et_nat(h,hoare_new_addrs(h3,as,h))) ).

%------------------------------------------------------------------------------