%------------------------------------------------------------------------------
% File     : ITP211_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Assertions 00607_017823
% 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    : 0024_Assertions_00607_017823 [Des22]

% Status   : Theorem
% Rating   : 0.67 v8.1.0
% Syntax   : Number of formulae    : 10427 (1943 unt;2565 typ;   0 def)
%            Number of atoms       : 23264 (6977 equ)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 18030 (2962   ~; 296   |;1271   &)
%                                         (1994 <=>;11507  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   6 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 2361 ( 418 fml;1943 var)
%            Number of X terms     :  240 (   0  []; 236 ite;   4 let)
%            Number of types       :  520 ( 519 usr)
%            Number of type conns  : 2970 (1669   >;1301   *;   0   +;   0  <<)
%            Number of predicates  :  222 ( 219 usr;   4 prp; 0-4 aty)
%            Number of functors    : 1831 (1831 usr; 379 con; 0-8 aty)
%            Number of variables   : 22135 (21523   !; 612   ?;22135   :)
% 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 14:51:37.575
%------------------------------------------------------------------------------
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    set_Code_natural: $tType ).

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

tff(ty_n_t__fun_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
    fun_set_rat_o: $tType ).

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

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

tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_Mt__Nat__Onat_J,type,
    fun_set_o_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Set__Oset_I_Eo_J_J,type,
    fun_nat_set_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Set__Oset_I_Eo_J_J,type,
    fun_int_set_o: $tType ).

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

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

tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    fun_o_set_nat: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Int__Oint_J_J,type,
    fun_o_set_int: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Onatural_J,type,
    fun_o_Code_natural: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Ointeger_J,type,
    fun_o_Code_integer: $tType ).

tff(ty_n_t__Set__Oset_It__fun_It__Nat__Onat_M_Eo_J_J,type,
    set_fun_nat_o: $tType ).

tff(ty_n_t__Set__Oset_It__fun_It__Int__Oint_M_Eo_J_J,type,
    set_fun_int_o: $tType ).

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

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

tff(ty_n_t__fun_I_Eo_Mt__Product____Type__Ounit_J,type,
    fun_o_Product_unit: $tType ).

tff(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

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

tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
    fun_rat_nat: $tType ).

tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
    fun_rat_int: $tType ).

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

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

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

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

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

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

tff(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
    set_assn: $tType ).

tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
    fun_assn_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    fun_o_assn: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
    fun_set_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_I_Eo_J_J,type,
    fun_o_set_o: $tType ).

tff(ty_n_t__Set__Oset_It__fun_I_Eo_M_Eo_J_J,type,
    set_fun_o_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__fun_I_Eo_Mt__Rat__Orat_J,type,
    fun_o_rat: $tType ).

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

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

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

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

tff(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

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

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

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

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

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

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

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

% Explicit typings (2046)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_fu3555266933746557007_nat_o > fun_fu5135969500543783852at_o_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_Product_unit > fun_Product_unit_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: assn > fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: assn > fun_as3747249873083764768_nat_o ).

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

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_as3747249873083764768_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_int_int ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_Co1981900195452921617nteger ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: fun_fu3790787410453625514o_assn ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: fun_as3747249873083764768_nat_o ).

tff(sy_c_Assertions_Oin__range,type,
    in_range: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: fun_Pr7920380315054183110_nat_o ).

tff(sy_c_Assertions_Ois__pure__assn,type,
    is_pure_assn: fun_assn_o ).

tff(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: fun_Pr7920380315054183110_nat_o ).

tff(sy_c_Assertions_Oproper,type,
    proper: fun_fu5135969500543783852at_o_o ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_o_assn ).

tff(sy_c_Assertions_Opure__assn__raw_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a825153325127701367it_nat: $o > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Opure__assn__raw__rel_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a6022498039421069578it_nat: fun_Pr8346683996641107384_nat_o ).

tff(sy_c_Assertions_OrelH,type,
    relH: set_nat > fun_he3009834522632631726unit_o ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun_Pr3358449090565925198_nat_o ).

tff(sy_c_Assertions_Owand__assn,type,
    wand_assn: assn > fun_assn_assn ).

tff(sy_c_Assertions_Owand__raw,type,
    wand_raw: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: fun_Pr3358449090565925198_nat_o ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
    bNF_Cardinal_cone: set_Pr5094982260447487303t_unit ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Product____Type__Ounit,type,
    bNF_Ca7083678892733797993t_unit: set_Product_unit > set_Pr5094982260447487303t_unit ).

tff(sy_c_BNF__Def_OfstOp_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_fs3871268833404974121et_nat: ( fun_he5746958392356810273_nat_o * fun_se2170385732985038782_nat_o ) > fun_Pr7091840695141096449et_nat ).

tff(sy_c_BNF__Def_OfstOp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_fs8167890477030535480nt_int: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_Pr2478283820680522243nt_int ).

tff(sy_c_BNF__Def_Opick__middlep_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_pi3450177011487049898et_nat: ( fun_he3009834522632631726unit_o * fun_he5746958392356810273_nat_o * heap_e7401611519738050253t_unit * set_nat ) > heap_e7401611519738050253t_unit ).

tff(sy_c_BNF__Def_Opick__middlep_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_pi4042512755101277815et_nat: ( fun_he5746958392356810273_nat_o * fun_se2170385732985038782_nat_o * heap_e7401611519738050253t_unit ) > fun_set_nat_set_nat ).

tff(sy_c_BNF__Def_Opick__middlep_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_pi8853482479757052266nt_int: ( fun_int_fun_int_o * fun_int_fun_int_o * int ) > fun_int_int ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re3379532845092657523nteger: ( fun_in4891666191172872833eger_o * fun_in4891666191172872833eger_o ) > fun_fu726873822431485118eger_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re9130850084932126070nteger: ( fun_in4891666191172872833eger_o * fun_fu726873822431485118eger_o * fun_int_fun_int_int * fun_Co1981900195452921617nteger ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_fu4208408993564413984_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re5717476210804479843nt_int: ( fun_int_fun_int_o * fun_fu4208408993564413984_int_o * fun_int_fun_int_int * fun_int_fun_int_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    bNF_re3704215830270325841atural: ( fun_na2240068485812228818ural_o * fun_na2240068485812228818ural_o ) > fun_fu2034268335391280352ural_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re1400205106991740995atural: ( fun_na2240068485812228818ural_o * fun_fu2034268335391280352ural_o * fun_nat_fun_nat_nat * fun_Co6813818151677833714atural ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_fu9201817095025044880_nat_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re5162689864697376059at_nat: ( fun_nat_fun_nat_o * fun_fu9201817095025044880_nat_o * fun_nat_fun_nat_nat * fun_nat_fun_nat_nat ) > $o ).

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    comp_P4540793059153670726nteger: ( fun_Pr6127221218041104501nteger * fun_Pr5458885100655985245nteger ) > fun_Pr3001699213897242575nteger ).

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tff(sy_c_Fun_Ocomp_001t__fun_It__Set__Oset_I_Eo_J_Mt__Assertions__Oassn_J_001t__fun_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Assertions__Oassn_J_001t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
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tff(sy_c_Rat_OAbs__Rat,type,
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    member1641360573044772336_nat_o: ( set_Pr2890355626087285447_nat_o * set_se7341084505648470951_nat_o ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    member5877623283571906838t_unit: ( set_Product_unit * set_set_Product_unit ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: ( set_set_nat * set_set_set_nat ) > $o ).

tff(sy_c_member_001t__Set__Oset_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_J,type,
    member361801372707117048_nat_o: ( set_fu3555266933746557007_nat_o * set_se6690607124073039791_nat_o ) > $o ).

tff(sy_c_member_001t__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,type,
    member7544390268018397336_nat_o: fun_Pr6460944699146269935_nat_o > fun_se7459991829846211660at_o_o ).

tff(sy_v_h,type,
    h: produc3658429121746597890et_nat ).

% Relevant facts (7851)
tff(fact_0_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_ass4243947775541419017_nat_o(rep_assn,X) = aa_ass4243947775541419017_nat_o(rep_assn,Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_1_Rep__assn__inverse,axiom,
    ! [X: assn] : aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(rep_assn,X)) = X ).

% Rep_assn_inverse
tff(fact_2_Abs__assn__eqI_I2_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(P,H)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H) )
     => ( Pr = aa_fun5526454687433275355o_assn(abs_assn,P) ) ) ).

% Abs_assn_eqI(2)
tff(fact_3_Abs__assn__eqI_I1_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(P,H)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H) )
     => ( aa_fun5526454687433275355o_assn(abs_assn,P) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_4_empty__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),bot_bo8632255151191677115_nat_o) ).

% empty_iff
tff(fact_5_empty__iff,axiom,
    ! [C: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(C),bot_bo3957492148770167129t_unit) ).

% empty_iff
tff(fact_6_empty__iff,axiom,
    ! [C: $o] : ~ aa_set_o_o(member_o((C)),bot_bot_set_o) ).

% empty_iff
tff(fact_7_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_8_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_9_all__not__in__conv,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
    <=> ( A = bot_bo8632255151191677115_nat_o ) ) ).

% all_not_in_conv
tff(fact_10_all__not__in__conv,axiom,
    ! [A: set_Product_unit] :
      ( ! [X2: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% all_not_in_conv
tff(fact_11_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),A)
    <=> ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_12_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ! [X2: nat] : ~ aa_set_nat_o(member_nat(X2),A)
    <=> ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_13_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ! [X2: int] : ~ aa_set_int_o(member_int(X2),A)
    <=> ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_14_Collect__empty__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ( aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_15_Collect__empty__eq,axiom,
    ! [P: fun_Product_unit_o] :
      ( ( aa_fun8337552879598298401t_unit(collect_Product_unit,P) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: product_unit] : ~ aa_Product_unit_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_16_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( aa_fun_o_o_set_o(collect_o,P) = bot_bot_set_o )
    <=> ! [X2: $o] : ~ aa_o_o(P,(X2)) ) ).

% Collect_empty_eq
tff(fact_17_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( aa_fun_nat_o_set_nat(collect_nat,P) = bot_bot_set_nat )
    <=> ! [X2: nat] : ~ aa_nat_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_18_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( aa_fun_int_o_set_int(collect_int,P) = bot_bot_set_int )
    <=> ! [X2: int] : ~ aa_int_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_19_empty__Collect__eq,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ( bot_bo8632255151191677115_nat_o = aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P) )
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_20_empty__Collect__eq,axiom,
    ! [P: fun_Product_unit_o] :
      ( ( bot_bo3957492148770167129t_unit = aa_fun8337552879598298401t_unit(collect_Product_unit,P) )
    <=> ! [X2: product_unit] : ~ aa_Product_unit_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_21_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = aa_fun_o_o_set_o(collect_o,P) )
    <=> ! [X2: $o] : ~ aa_o_o(P,(X2)) ) ).

% empty_Collect_eq
tff(fact_22_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = aa_fun_nat_o_set_nat(collect_nat,P) )
    <=> ! [X2: nat] : ~ aa_nat_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_23_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = aa_fun_int_o_set_int(collect_int,P) )
    <=> ! [X2: int] : ~ aa_int_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_24_emptyE,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),bot_bo8632255151191677115_nat_o) ).

% emptyE
tff(fact_25_emptyE,axiom,
    ! [A2: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),bot_bo3957492148770167129t_unit) ).

% emptyE
tff(fact_26_emptyE,axiom,
    ! [A2: $o] : ~ aa_set_o_o(member_o((A2)),bot_bot_set_o) ).

% emptyE
tff(fact_27_emptyE,axiom,
    ! [A2: nat] : ~ aa_set_nat_o(member_nat(A2),bot_bot_set_nat) ).

% emptyE
tff(fact_28_emptyE,axiom,
    ! [A2: int] : ~ aa_set_int_o(member_int(A2),bot_bot_set_int) ).

% emptyE
tff(fact_29_equals0D,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( ( A = bot_bo3957492148770167129t_unit )
     => ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),A) ) ).

% equals0D
tff(fact_30_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A2)),A) ) ).

% equals0D
tff(fact_31_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A2),A) ) ).

% equals0D
tff(fact_32_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A2),A) ) ).

% equals0D
tff(fact_33_equals0D,axiom,
    ! [A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( A = bot_bo8632255151191677115_nat_o )
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A) ) ).

% equals0D
tff(fact_34_equals0I,axiom,
    ! [A: set_Product_unit] :
      ( ! [Y2: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(Y2),A)
     => ( A = bot_bo3957492148770167129t_unit ) ) ).

% equals0I
tff(fact_35_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y2: $o] : ~ aa_set_o_o(member_o((Y2)),A)
     => ( A = bot_bot_set_o ) ) ).

% equals0I
tff(fact_36_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y2: nat] : ~ aa_set_nat_o(member_nat(Y2),A)
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_37_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y2: int] : ~ aa_set_int_o(member_int(Y2),A)
     => ( A = bot_bot_set_int ) ) ).

% equals0I
tff(fact_38_equals0I,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
     => ( A = bot_bo8632255151191677115_nat_o ) ) ).

% equals0I
tff(fact_39_ex__in__conv,axiom,
    ! [A: set_Product_unit] :
      ( ? [X2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
    <=> ( A != bot_bo3957492148770167129t_unit ) ) ).

% ex_in_conv
tff(fact_40_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ? [X2: $o] : aa_set_o_o(member_o((X2)),A)
    <=> ( A != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_41_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ? [X2: nat] : aa_set_nat_o(member_nat(X2),A)
    <=> ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_42_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ? [X2: int] : aa_set_int_o(member_int(X2),A)
    <=> ( A != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_43_ex__in__conv,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ? [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
    <=> ( A != bot_bo8632255151191677115_nat_o ) ) ).

% ex_in_conv
tff(fact_44_set__notEmptyE,axiom,
    ! [S: set_Product_unit] :
      ( ( S != bot_bo3957492148770167129t_unit )
     => ~ ! [X3: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),S) ) ).

% set_notEmptyE
tff(fact_45_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_46_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_47_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_48_set__notEmptyE,axiom,
    ! [S: set_fu3555266933746557007_nat_o] :
      ( ( S != bot_bo8632255151191677115_nat_o )
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),S) ) ).

% set_notEmptyE
tff(fact_49_memb__imp__not__empty,axiom,
    ! [X: product_unit,S: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),S)
     => ( S != bot_bo3957492148770167129t_unit ) ) ).

% memb_imp_not_empty
tff(fact_50_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_51_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_52_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_53_memb__imp__not__empty,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
     => ( S != bot_bo8632255151191677115_nat_o ) ) ).

% memb_imp_not_empty
tff(fact_54_bot__set__def,axiom,
    bot_bo3957492148770167129t_unit = aa_fun8337552879598298401t_unit(collect_Product_unit,bot_bo6263423904857894434unit_o) ).

% bot_set_def
tff(fact_55_bot__set__def,axiom,
    bot_bot_set_o = aa_fun_o_o_set_o(collect_o,bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_56_bot__set__def,axiom,
    bot_bot_set_nat = aa_fun_nat_o_set_nat(collect_nat,bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_57_bot__set__def,axiom,
    bot_bot_set_int = aa_fun_int_o_set_int(collect_int,bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_58_bot__set__def,axiom,
    bot_bo8632255151191677115_nat_o = aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,bot_bo8793957029302321216at_o_o) ).

% bot_set_def
tff(fact_59_mod__false,axiom,
    ! [Ha: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,bot_bot_assn),Ha) ).

% mod_false
tff(fact_60_Set_Ois__empty__def,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(is_emp5529367068699929823t_unit,A)
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% Set.is_empty_def
tff(fact_61_Set_Ois__empty__def,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(is_empty_o,A)
    <=> ( A = bot_bot_set_o ) ) ).

% Set.is_empty_def
tff(fact_62_Set_Ois__empty__def,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(is_empty_nat,A)
    <=> ( A = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_63_Set_Ois__empty__def,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(is_empty_int,A)
    <=> ( A = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_64_Set_Ois__empty__def,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(is_emp7728009199165831873_nat_o,A)
    <=> ( A = bot_bo8632255151191677115_nat_o ) ) ).

% Set.is_empty_def
tff(fact_65_Collect__empty__eq__bot,axiom,
    ! [P: fun_Product_unit_o] :
      ( ( aa_fun8337552879598298401t_unit(collect_Product_unit,P) = bot_bo3957492148770167129t_unit )
    <=> ( P = bot_bo6263423904857894434unit_o ) ) ).

% Collect_empty_eq_bot
tff(fact_66_Collect__empty__eq__bot,axiom,
    ! [P: fun_o_o] :
      ( ( aa_fun_o_o_set_o(collect_o,P) = bot_bot_set_o )
    <=> ( P = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_67_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_o] :
      ( ( aa_fun_nat_o_set_nat(collect_nat,P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_68_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_o] :
      ( ( aa_fun_int_o_set_int(collect_int,P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_69_Collect__empty__eq__bot,axiom,
    ! [P: fun_fu5135969500543783852at_o_o] :
      ( ( aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P) = bot_bo8632255151191677115_nat_o )
    <=> ( P = bot_bo8793957029302321216at_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_70_bot__empty__eq,axiom,
    ! [X4: product_unit] :
      ( aa_Product_unit_o(bot_bo6263423904857894434unit_o,X4)
    <=> aa_set8780210009671629607unit_o(member_Product_unit(X4),bot_bo3957492148770167129t_unit) ) ).

% bot_empty_eq
tff(fact_71_bot__empty__eq,axiom,
    ! [X4: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X4))
    <=> aa_set_o_o(member_o((X4)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_72_bot__empty__eq,axiom,
    ! [X4: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X4)
    <=> aa_set_nat_o(member_nat(X4),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_73_bot__empty__eq,axiom,
    ! [X4: int] :
      ( aa_int_o(bot_bot_fun_int_o,X4)
    <=> aa_set_int_o(member_int(X4),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_74_bot__empty__eq,axiom,
    ! [X4: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(bot_bo8793957029302321216at_o_o,X4)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),bot_bo8632255151191677115_nat_o) ) ).

% bot_empty_eq
tff(fact_75_wand__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(wand_assn(P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(wand_raw(aa_ass4243947775541419017_nat_o(rep_assn,P)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ).

% wand_assn_def
tff(fact_76_one__assn__def,axiom,
    one_one_assn = aa_fun5526454687433275355o_assn(abs_assn,one_assn_raw) ).

% one_assn_def
tff(fact_77_Abs__assn__inverse,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper))
     => ( aa_ass4243947775541419017_nat_o(rep_assn,aa_fun5526454687433275355o_assn(abs_assn,Y)) = Y ) ) ).

% Abs_assn_inverse
tff(fact_78_one__assn__raw_Osimps,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
    <=> ( As = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_79_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,As2: set_nat] :
            ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( (Y)
            <=> ( As2 != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_80_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( As2 != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_81_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( As2 = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_82_is__singletonI_H,axiom,
    ! [A: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit,Y2: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),A)
             => ( X3 = Y2 ) ) )
       => aa_set8780210009671629607unit_o(is_sin2160648248035936513t_unit,A) ) ) ).

% is_singletonI'
tff(fact_83_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o,Y2: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ( aa_set_o_o(member_o((Y2)),A)
             => ( (X3)
              <=> (Y2) ) ) )
       => aa_set_o_o(is_singleton_o,A) ) ) ).

% is_singletonI'
tff(fact_84_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( X3 = Y2 ) ) )
       => aa_set_nat_o(is_singleton_nat,A) ) ) ).

% is_singletonI'
tff(fact_85_is__singletonI_H,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int,Y2: int] :
            ( aa_set_int_o(member_int(X3),A)
           => ( aa_set_int_o(member_int(Y2),A)
             => ( X3 = Y2 ) ) )
       => aa_set_int_o(is_singleton_int,A) ) ) ).

% is_singletonI'
tff(fact_86_is__singletonI_H,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
           => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
             => ( X3 = Y2 ) ) )
       => aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,A) ) ) ).

% is_singletonI'
tff(fact_87_wand__proper,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(wand_raw(P),Q)) ).

% wand_proper
tff(fact_88_one__assn__proper,axiom,
    aa_fun1675921005893230629at_o_o(proper,one_assn_raw) ).

% one_assn_proper
tff(fact_89_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_90_bex2I,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat,P: fun_he5746958392356810273_nat_o] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),S)
     => ( ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),S)
         => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A2),B) )
       => ? [A3: heap_e7401611519738050253t_unit,B2: set_nat] :
            ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2),S)
            & aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_91_bex2I,axiom,
    ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),S)
     => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),S)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B) )
       => ? [A3: nat,B2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),S)
            & aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_92_bex2I,axiom,
    ! [A2: int,B: int,S: set_Pr958786334691620121nt_int,P: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),S)
     => ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),S)
         => aa_int_o(aa_int_fun_int_o(P,A2),B) )
       => ? [A3: int,B2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),S)
            & aa_int_o(aa_int_fun_int_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_93_bex2I,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat,P: fun_fu8419763957888848259_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),S)
     => ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),S)
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A2),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2),S)
            & aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_94_bex2I,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat,P: fun_fu2611741329380808165_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),S)
     => ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),S)
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A2),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2),S)
            & aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B2) ) ) ) ).

% bex2I
tff(fact_95_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] : X != aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) ).

% one_assn_raw.cases
tff(fact_96_sndE,axiom,
    ! [X: product_prod_nat_nat,A2: nat,B: nat,P: fun_nat_o] :
      ( ( X = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B) )
     => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X))
       => aa_nat_o(P,B) ) ) ).

% sndE
tff(fact_97_sndE,axiom,
    ! [X: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,P: fun_Pr6460944699146269935_nat_o] :
      ( ( X = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B) )
     => ( aa_Pro653082453696054616_nat_o(P,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,X))
       => aa_Pro653082453696054616_nat_o(P,B) ) ) ).

% sndE
tff(fact_98_sndE,axiom,
    ! [X: produc3112576180687729086et_nat,A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,P: fun_Pr2620521676128474833_nat_o] :
      ( ( X = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B) )
     => ( aa_Pro5643009357358181178_nat_o(P,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,X))
       => aa_Pro5643009357358181178_nat_o(P,B) ) ) ).

% sndE
tff(fact_99_sndE,axiom,
    ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: fun_set_nat_o] :
      ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B) )
     => ( aa_set_nat_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))
       => aa_set_nat_o(P,B) ) ) ).

% sndE
tff(fact_100_sndE,axiom,
    ! [X: product_prod_int_int,A2: int,B: int,P: fun_int_o] :
      ( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B) )
     => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))
       => aa_int_o(P,B) ) ) ).

% sndE
tff(fact_101_Rep__assn,axiom,
    ! [X: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,X)),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper)) ).

% Rep_assn
tff(fact_102_mem__Collect__eq,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P))
    <=> aa_fun1675921005893230629at_o_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_103_mem__Collect__eq,axiom,
    ! [A2: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A2),aa_fun_nat_o_set_nat(collect_nat,P))
    <=> aa_nat_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_104_mem__Collect__eq,axiom,
    ! [A2: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A2),aa_fun_int_o_set_int(collect_int,P))
    <=> aa_int_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_105_mem__Collect__eq,axiom,
    ! [A2: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o((A2)),aa_fun_o_o_set_o(collect_o,P))
    <=> aa_o_o(P,(A2)) ) ).

% mem_Collect_eq
tff(fact_106_mem__Collect__eq,axiom,
    ! [A2: product_unit,P: fun_Product_unit_o] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_fun8337552879598298401t_unit(collect_Product_unit,P))
    <=> aa_Product_unit_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_107_Collect__mem__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,aTP_Lamm_a(A)) = A ).

% Collect_mem_eq
tff(fact_108_Collect__mem__eq,axiom,
    ! [A: set_nat] : aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_aa(A)) = A ).

% Collect_mem_eq
tff(fact_109_Collect__mem__eq,axiom,
    ! [A: set_int] : aa_fun_int_o_set_int(collect_int,aTP_Lamm_ab(A)) = A ).

% Collect_mem_eq
tff(fact_110_Collect__mem__eq,axiom,
    ! [A: set_o] : aa_fun_o_o_set_o(collect_o,aTP_Lamm_ac(A)) = A ).

% Collect_mem_eq
tff(fact_111_Collect__mem__eq,axiom,
    ! [A: set_Product_unit] : aa_fun8337552879598298401t_unit(collect_Product_unit,aTP_Lamm_ad(A)) = A ).

% Collect_mem_eq
tff(fact_112_Collect__cong,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(P,X3)
        <=> aa_fun1675921005893230629at_o_o(Q,X3) )
     => ( aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P) = aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,Q) ) ) ).

% Collect_cong
tff(fact_113_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
        <=> aa_nat_o(Q,X3) )
     => ( aa_fun_nat_o_set_nat(collect_nat,P) = aa_fun_nat_o_set_nat(collect_nat,Q) ) ) ).

% Collect_cong
tff(fact_114_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(Q,X3) )
     => ( aa_fun_int_o_set_int(collect_int,P) = aa_fun_int_o_set_int(collect_int,Q) ) ) ).

% Collect_cong
tff(fact_115_Collect__cong,axiom,
    ! [P: fun_o_o,Q: fun_o_o] :
      ( ! [X3: $o] :
          ( aa_o_o(P,(X3))
        <=> aa_o_o(Q,(X3)) )
     => ( aa_fun_o_o_set_o(collect_o,P) = aa_fun_o_o_set_o(collect_o,Q) ) ) ).

% Collect_cong
tff(fact_116_Collect__cong,axiom,
    ! [P: fun_Product_unit_o,Q: fun_Product_unit_o] :
      ( ! [X3: product_unit] :
          ( aa_Product_unit_o(P,X3)
        <=> aa_Product_unit_o(Q,X3) )
     => ( aa_fun8337552879598298401t_unit(collect_Product_unit,P) = aa_fun8337552879598298401t_unit(collect_Product_unit,Q) ) ) ).

% Collect_cong
tff(fact_117_Rep__assn__cases,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper))
     => ~ ! [X3: assn] : Y != aa_ass4243947775541419017_nat_o(rep_assn,X3) ) ).

% Rep_assn_cases
tff(fact_118_Rep__assn__induct,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper))
     => ( ! [X3: assn] : aa_fun1675921005893230629at_o_o(P,aa_ass4243947775541419017_nat_o(rep_assn,X3))
       => aa_fun1675921005893230629at_o_o(P,Y) ) ) ).

% Rep_assn_induct
tff(fact_119_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( ( X = aa_fun5526454687433275355o_assn(abs_assn,Y2) )
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper)) ) ).

% Abs_assn_cases
tff(fact_120_Abs__assn__induct,axiom,
    ! [P: fun_assn_o,X: assn] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper))
         => aa_assn_o(P,aa_fun5526454687433275355o_assn(abs_assn,Y2)) )
     => aa_assn_o(P,X) ) ).

% Abs_assn_induct
tff(fact_121_Abs__assn__inject,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper))
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper))
       => ( ( aa_fun5526454687433275355o_assn(abs_assn,X) = aa_fun5526454687433275355o_assn(abs_assn,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_122_mod__h__bot__indep,axiom,
    ! [P: assn,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_123_old_Oprod_Oinject,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A4: heap_e7401611519738050253t_unit,B3: set_nat] :
      ( ( aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_124_old_Oprod_Oinject,axiom,
    ! [A2: nat,B: nat,A4: nat,B3: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_125_old_Oprod_Oinject,axiom,
    ! [A2: int,B: int,A4: int,B3: int] :
      ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B) = aa_int7462565939943104719nt_int(product_Pair_int_int(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_126_old_Oprod_Oinject,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A4: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_127_old_Oprod_Oinject,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A4: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
tff(fact_128_prod_Oinject,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X22: set_nat,Y1: heap_e7401611519738050253t_unit,Y22: set_nat] :
      ( ( aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X1),X22) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_129_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_130_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(X1),X22) = aa_int7462565939943104719nt_int(product_Pair_int_int(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_131_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y22: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X1),X22) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_132_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y22: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X1),X22) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_133_sndI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
      ( ( X = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z) )
     => ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X) = Z ) ) ).

% sndI
tff(fact_134_sndI,axiom,
    ! [X: produc8799533983703829792et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc3658429121746597890et_nat] :
      ( ( X = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y),Z) )
     => ( aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,X) = Z ) ) ).

% sndI
tff(fact_135_sndI,axiom,
    ! [X: produc3112576180687729086et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc8799533983703829792et_nat] :
      ( ( X = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y),Z) )
     => ( aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,X) = Z ) ) ).

% sndI
tff(fact_136_sndI,axiom,
    ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
      ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y),Z) )
     => ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X) = Z ) ) ).

% sndI
tff(fact_137_sndI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z) )
     => ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X) = Z ) ) ).

% sndI
tff(fact_138_snd__eqD,axiom,
    ! [X: nat,Y: nat,A2: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_139_snd__eqD,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat] :
      ( ( aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_140_snd__eqD,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A2: produc8799533983703829792et_nat] :
      ( ( aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_141_snd__eqD,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: set_nat] :
      ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_142_snd__eqD,axiom,
    ! [X: int,Y: int,A2: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_143_snd__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = X22 ).

% snd_conv
tff(fact_144_snd__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat] : aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X1),X22)) = X22 ).

% snd_conv
tff(fact_145_snd__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat] : aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X1),X22)) = X22 ).

% snd_conv
tff(fact_146_snd__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X1),X22)) = X22 ).

% snd_conv
tff(fact_147_snd__conv,axiom,
    ! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X1),X22)) = X22 ).

% snd_conv
tff(fact_148_eq__snd__iff,axiom,
    ! [B: nat,P2: product_prod_nat_nat] :
      ( ( B = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) )
    <=> ? [A5: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B) ) ).

% eq_snd_iff
tff(fact_149_eq__snd__iff,axiom,
    ! [B: produc3658429121746597890et_nat,P2: produc8799533983703829792et_nat] :
      ( ( B = aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,P2) )
    <=> ? [A5: fun_Pr6460944699146269935_nat_o] : P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B) ) ).

% eq_snd_iff
tff(fact_150_eq__snd__iff,axiom,
    ! [B: produc8799533983703829792et_nat,P2: produc3112576180687729086et_nat] :
      ( ( B = aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,P2) )
    <=> ? [A5: fun_Pr6460944699146269935_nat_o] : P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B) ) ).

% eq_snd_iff
tff(fact_151_eq__snd__iff,axiom,
    ! [B: set_nat,P2: produc3658429121746597890et_nat] :
      ( ( B = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2) )
    <=> ? [A5: heap_e7401611519738050253t_unit] : P2 = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A5),B) ) ).

% eq_snd_iff
tff(fact_152_eq__snd__iff,axiom,
    ! [B: int,P2: product_prod_int_int] :
      ( ( B = aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) )
    <=> ? [A5: int] : P2 = aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B) ) ).

% eq_snd_iff
tff(fact_153_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,As2: set_nat] :
            ( ( Xa = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( ( As2 = bot_bot_set_nat )
              & (X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_154_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,As2: set_nat] :
            ( ( Xa = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ~ ( ( As2 = bot_bot_set_nat )
                & (X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_155_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,As2: set_nat] :
            ( ( Xa = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( (Y)
            <=> ~ ( ( As2 = bot_bot_set_nat )
                  & (X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_156_pure__assn__raw_Osimps,axiom,
    ! [B: $o,Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((B)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
    <=> ( ( As = bot_bot_set_nat )
        & (B) ) ) ).

% pure_assn_raw.simps
tff(fact_157_pure__assn__proper,axiom,
    ! [B: $o] : aa_fun1675921005893230629at_o_o(proper,pure_a825153325127701367it_nat((B))) ).

% pure_assn_proper
tff(fact_158_times__assn__raw_Ocases,axiom,
    ! [X: produc3112576180687729086et_nat] :
      ~ ! [P3: fun_Pr6460944699146269935_nat_o,Q2: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As2: set_nat] : X != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(P3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Q2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))) ).

% times_assn_raw.cases
tff(fact_159_pure__assn__raw_Ocases,axiom,
    ! [X: produc6197004810343482825et_nat] :
      ~ ! [B2: $o,H: heap_e7401611519738050253t_unit,As2: set_nat] : X != produc3762353314782720579et_nat((B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)) ).

% pure_assn_raw.cases
tff(fact_160_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3658429121746597890et_nat] :
      ~ ! [A3: heap_e7401611519738050253t_unit,B2: set_nat] : Y != aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_161_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B2: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_162_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B2: int] : Y != aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2) ).

% old.prod.exhaust
tff(fact_163_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] : Y != aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_164_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] : Y != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_165_surj__pair,axiom,
    ! [P2: produc3658429121746597890et_nat] :
    ? [X3: heap_e7401611519738050253t_unit,Y2: set_nat] : P2 = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2) ).

% surj_pair
tff(fact_166_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y2: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) ).

% surj_pair
tff(fact_167_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y2: int] : P2 = aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2) ).

% surj_pair
tff(fact_168_surj__pair,axiom,
    ! [P2: produc8799533983703829792et_nat] :
    ? [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] : P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2) ).

% surj_pair
tff(fact_169_surj__pair,axiom,
    ! [P2: produc3112576180687729086et_nat] :
    ? [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] : P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2) ).

% surj_pair
tff(fact_170_prod__cases,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,P2: produc3658429121746597890et_nat] :
      ( ! [A3: heap_e7401611519738050253t_unit,B2: set_nat] : aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2))
     => aa_Pro653082453696054616_nat_o(P,P2) ) ).

% prod_cases
tff(fact_171_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B2: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_172_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A3: int,B2: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_173_prod__cases,axiom,
    ! [P: fun_Pr2620521676128474833_nat_o,P2: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(P,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2))
     => aa_Pro5643009357358181178_nat_o(P,P2) ) ).

% prod_cases
tff(fact_174_prod__cases,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,P2: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2))
     => aa_Pro2671238023309943196_nat_o(P,P2) ) ).

% prod_cases
tff(fact_175_Pair__inject,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A4: heap_e7401611519738050253t_unit,B3: set_nat] :
      ( ( aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_176_Pair__inject,axiom,
    ! [A2: nat,B: nat,A4: nat,B3: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_177_Pair__inject,axiom,
    ! [A2: int,B: int,A4: int,B3: int] :
      ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B) = aa_int7462565939943104719nt_int(product_Pair_int_int(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_178_Pair__inject,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A4: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
      ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_179_Pair__inject,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A4: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
      ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
tff(fact_180_prod__cases3,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : Y != aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B2),C2)) ).

% prod_cases3
tff(fact_181_prod__cases3,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: produc3658429121746597890et_nat] : Y != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),C2)) ).

% prod_cases3
tff(fact_182_prod__cases4,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: heap_e7401611519738050253t_unit,D: set_nat] : Y != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(C2),D))) ).

% prod_cases4
tff(fact_183_prod__induct3,axiom,
    ! [P: fun_Pr2620521676128474833_nat_o,X: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : aa_Pro5643009357358181178_nat_o(P,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B2),C2)))
     => aa_Pro5643009357358181178_nat_o(P,X) ) ).

% prod_induct3
tff(fact_184_prod__induct3,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),C2)))
     => aa_Pro2671238023309943196_nat_o(P,X) ) ).

% prod_induct3
tff(fact_185_prod__induct4,axiom,
    ! [P: fun_Pr1227177680593602867_nat_o,X: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: fun_Pr6460944699146269935_nat_o,C2: heap_e7401611519738050253t_unit,D: set_nat] : aa_Pro2671238023309943196_nat_o(P,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(C2),D))))
     => aa_Pro2671238023309943196_nat_o(P,X) ) ).

% prod_induct4
tff(fact_186_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
    <=> ( X = one_one_assn ) ) ).

% one_reorient
tff(fact_187_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_188_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_189_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
    <=> ( X = one_one_Code_integer ) ) ).

% one_reorient
tff(fact_190_one__reorient,axiom,
    ! [X: code_natural] :
      ( ( one_one_Code_natural = X )
    <=> ( X = one_one_Code_natural ) ) ).

% one_reorient
tff(fact_191_type__definition__assn,axiom,
    aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(rep_assn,abs_assn),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,proper)) ).

% type_definition_assn
tff(fact_192_times__assn__proper,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(times_assn_raw(P),Q)) ) ) ).

% times_assn_proper
tff(fact_193_is__pure__assn__basic__simps_I2_J,axiom,
    aa_assn_o(is_pure_assn,one_one_assn) ).

% is_pure_assn_basic_simps(2)
tff(fact_194_bijective__def,axiom,
    ! [R: set_Pr3948176798113811640et_nat] :
      ( biject2219996528949639174et_nat(R)
    <=> ( ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat,Z2: set_nat] :
            ( ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X2),Y3),R)
              & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X2),Z2),R) )
           => ( Y3 = Z2 ) )
        & ! [X2: heap_e7401611519738050253t_unit,Y3: heap_e7401611519738050253t_unit,Z2: set_nat] :
            ( ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X2),Z2),R)
              & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y3),Z2),R) )
           => ( X2 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_195_bijective__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( bijective_nat_nat(R)
    <=> ( ! [X2: nat,Y3: nat,Z2: nat] :
            ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y3),R)
              & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Z2),R) )
           => ( Y3 = Z2 ) )
        & ! [X2: nat,Y3: nat,Z2: nat] :
            ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Z2),R)
              & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z2),R) )
           => ( X2 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_196_bijective__def,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( bijective_int_int(R)
    <=> ( ! [X2: int,Y3: int,Z2: int] :
            ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y3),R)
              & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Z2),R) )
           => ( Y3 = Z2 ) )
        & ! [X2: int,Y3: int,Z2: int] :
            ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Z2),R)
              & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y3),Z2),R) )
           => ( X2 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_197_bijective__def,axiom,
    ! [R: set_Pr1312406793078187350et_nat] :
      ( biject6574446902887564708et_nat(R)
    <=> ( ! [X2: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y3),R)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Z2),R) )
           => ( Y3 = Z2 ) )
        & ! [X2: fun_Pr6460944699146269935_nat_o,Y3: fun_Pr6460944699146269935_nat_o,Z2: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Z2),R)
              & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y3),Z2),R) )
           => ( X2 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_198_bijective__def,axiom,
    ! [R: set_Pr4040394263813967220et_nat] :
      ( biject4066639966817638850et_nat(R)
    <=> ( ! [X2: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat,Z2: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Y3),R)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Z2),R) )
           => ( Y3 = Z2 ) )
        & ! [X2: fun_Pr6460944699146269935_nat_o,Y3: fun_Pr6460944699146269935_nat_o,Z2: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Z2),R)
              & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y3),Z2),R) )
           => ( X2 = Y3 ) ) ) ) ).

% bijective_def
tff(fact_199_ssubst__Pair__rhs,axiom,
    ! [R2: heap_e7401611519738050253t_unit,S2: set_nat,R: set_Pr3948176798113811640et_nat,S3: set_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(R2),S2),R)
     => ( ( S3 = S2 )
       => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_200_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S2),R)
     => ( ( S3 = S2 )
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_201_ssubst__Pair__rhs,axiom,
    ! [R2: int,S2: int,R: set_Pr958786334691620121nt_int,S3: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S2),R)
     => ( ( S3 = S2 )
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_202_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr6460944699146269935_nat_o,S2: produc3658429121746597890et_nat,R: set_Pr1312406793078187350et_nat,S3: produc3658429121746597890et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(R2),S2),R)
     => ( ( S3 = S2 )
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_203_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr6460944699146269935_nat_o,S2: produc8799533983703829792et_nat,R: set_Pr4040394263813967220et_nat,S3: produc8799533983703829792et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(R2),S2),R)
     => ( ( S3 = S2 )
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(R2),S3),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_204_is__singletonI,axiom,
    ! [X: product_unit] : aa_set8780210009671629607unit_o(is_sin2160648248035936513t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) ).

% is_singletonI
tff(fact_205_is__singletonI,axiom,
    ! [X: $o] : aa_set_o_o(is_singleton_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) ).

% is_singletonI
tff(fact_206_is__singletonI,axiom,
    ! [X: nat] : aa_set_nat_o(is_singleton_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) ).

% is_singletonI
tff(fact_207_is__singletonI,axiom,
    ! [X: int] : aa_set_int_o(is_singleton_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) ).

% is_singletonI
tff(fact_208_is__singletonI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) ).

% is_singletonI
tff(fact_209_curryI,axiom,
    ! [F: fun_Pr6460944699146269935_nat_o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
      ( aa_Pro653082453696054616_nat_o(F,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B))
     => aa_set_nat_o(aa_hea8622544166608216090_nat_o(produc1807123492089482958_nat_o(F),A2),B) ) ).

% curryI
tff(fact_210_curryI,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A2: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(F,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B))
     => aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(F),A2),B) ) ).

% curryI
tff(fact_211_curryI,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,A2: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(F,aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B))
     => aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(F),A2),B) ) ).

% curryI
tff(fact_212_curryI,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B))
     => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A2),B) ) ).

% curryI
tff(fact_213_curryI,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B))
     => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A2),B) ) ).

% curryI
tff(fact_214_insertCI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
       => ( A2 = B ) )
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),B4)) ) ).

% insertCI
tff(fact_215_insertCI,axiom,
    ! [A2: nat,B4: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A2),B4)
       => ( A2 = B ) )
     => aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),B4)) ) ).

% insertCI
tff(fact_216_insertCI,axiom,
    ! [A2: int,B4: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A2),B4)
       => ( A2 = B ) )
     => aa_set_int_o(member_int(A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),B4)) ) ).

% insertCI
tff(fact_217_insertCI,axiom,
    ! [A2: $o,B4: set_o,B: $o] :
      ( ( ~ aa_set_o_o(member_o((A2)),B4)
       => ( (A2)
        <=> (B) ) )
     => aa_set_o_o(member_o((A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),B4)) ) ).

% insertCI
tff(fact_218_insertCI,axiom,
    ! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
      ( ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
       => ( A2 = B ) )
     => aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),B4)) ) ).

% insertCI
tff(fact_219_insert__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),A))
    <=> ( ( A2 = B )
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A) ) ) ).

% insert_iff
tff(fact_220_insert__iff,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),A))
    <=> ( ( A2 = B )
        | aa_set_nat_o(member_nat(A2),A) ) ) ).

% insert_iff
tff(fact_221_insert__iff,axiom,
    ! [A2: int,B: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),A))
    <=> ( ( A2 = B )
        | aa_set_int_o(member_int(A2),A) ) ) ).

% insert_iff
tff(fact_222_insert__iff,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),A))
    <=> ( ( (A2)
        <=> (B) )
        | aa_set_o_o(member_o((A2)),A) ) ) ).

% insert_iff
tff(fact_223_insert__iff,axiom,
    ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),A))
    <=> ( ( A2 = B )
        | aa_set8780210009671629607unit_o(member_Product_unit(A2),A) ) ) ).

% insert_iff
tff(fact_224_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A) ).

% insert_absorb2
tff(fact_225_insert__absorb2,axiom,
    ! [X: int,A: set_int] : aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A) ).

% insert_absorb2
tff(fact_226_insert__absorb2,axiom,
    ! [X: $o,A: set_o] : aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A) ).

% insert_absorb2
tff(fact_227_insert__absorb2,axiom,
    ! [X: product_unit,A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A) ).

% insert_absorb2
tff(fact_228_singletonI,axiom,
    ! [A2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) ).

% singletonI
tff(fact_229_singletonI,axiom,
    ! [A2: $o] : aa_set_o_o(member_o((A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) ).

% singletonI
tff(fact_230_singletonI,axiom,
    ! [A2: nat] : aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) ).

% singletonI
tff(fact_231_singletonI,axiom,
    ! [A2: int] : aa_set_int_o(member_int(A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) ).

% singletonI
tff(fact_232_singletonI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) ).

% singletonI
tff(fact_233_insertE,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),A))
     => ( ( A2 != B )
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A) ) ) ).

% insertE
tff(fact_234_insertE,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),A))
     => ( ( A2 != B )
       => aa_set_nat_o(member_nat(A2),A) ) ) ).

% insertE
tff(fact_235_insertE,axiom,
    ! [A2: int,B: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),A))
     => ( ( A2 != B )
       => aa_set_int_o(member_int(A2),A) ) ) ).

% insertE
tff(fact_236_insertE,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),A))
     => ( ( (A2)
        <=> ~ (B) )
       => aa_set_o_o(member_o((A2)),A) ) ) ).

% insertE
tff(fact_237_insertE,axiom,
    ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),A))
     => ( ( A2 != B )
       => aa_set8780210009671629607unit_o(member_Product_unit(A2),A) ) ) ).

% insertE
tff(fact_238_insertI1,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) ).

% insertI1
tff(fact_239_insertI1,axiom,
    ! [A2: nat,B4: set_nat] : aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) ).

% insertI1
tff(fact_240_insertI1,axiom,
    ! [A2: int,B4: set_int] : aa_set_int_o(member_int(A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) ).

% insertI1
tff(fact_241_insertI1,axiom,
    ! [A2: $o,B4: set_o] : aa_set_o_o(member_o((A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) ).

% insertI1
tff(fact_242_insertI1,axiom,
    ! [A2: product_unit,B4: set_Product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) ).

% insertI1
tff(fact_243_insertI2,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),B4)) ) ).

% insertI2
tff(fact_244_insertI2,axiom,
    ! [A2: nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A2),B4)
     => aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),B4)) ) ).

% insertI2
tff(fact_245_insertI2,axiom,
    ! [A2: int,B4: set_int,B: int] :
      ( aa_set_int_o(member_int(A2),B4)
     => aa_set_int_o(member_int(A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),B4)) ) ).

% insertI2
tff(fact_246_insertI2,axiom,
    ! [A2: $o,B4: set_o,B: $o] :
      ( aa_set_o_o(member_o((A2)),B4)
     => aa_set_o_o(member_o((A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),B4)) ) ).

% insertI2
tff(fact_247_insertI2,axiom,
    ! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
     => aa_set8780210009671629607unit_o(member_Product_unit(A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),B4)) ) ).

% insertI2
tff(fact_248_Set_Oset__insert,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ~ ! [B5: set_fu3555266933746557007_nat_o] :
            ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B5) )
           => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B5) ) ) ).

% Set.set_insert
tff(fact_249_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => ~ ! [B5: set_nat] :
            ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B5) )
           => aa_set_nat_o(member_nat(X),B5) ) ) ).

% Set.set_insert
tff(fact_250_Set_Oset__insert,axiom,
    ! [X: int,A: set_int] :
      ( aa_set_int_o(member_int(X),A)
     => ~ ! [B5: set_int] :
            ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B5) )
           => aa_set_int_o(member_int(X),B5) ) ) ).

% Set.set_insert
tff(fact_251_Set_Oset__insert,axiom,
    ! [X: $o,A: set_o] :
      ( aa_set_o_o(member_o((X)),A)
     => ~ ! [B5: set_o] :
            ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B5) )
           => aa_set_o_o(member_o((X)),B5) ) ) ).

% Set.set_insert
tff(fact_252_Set_Oset__insert,axiom,
    ! [X: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ~ ! [B5: set_Product_unit] :
            ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B5) )
           => aa_set8780210009671629607unit_o(member_Product_unit(X),B5) ) ) ).

% Set.set_insert
tff(fact_253_insert__ident,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4)
       => ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B4) )
        <=> ( A = B4 ) ) ) ) ).

% insert_ident
tff(fact_254_insert__ident,axiom,
    ! [X: nat,A: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( ~ aa_set_nat_o(member_nat(X),B4)
       => ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4) )
        <=> ( A = B4 ) ) ) ) ).

% insert_ident
tff(fact_255_insert__ident,axiom,
    ! [X: int,A: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A)
     => ( ~ aa_set_int_o(member_int(X),B4)
       => ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4) )
        <=> ( A = B4 ) ) ) ) ).

% insert_ident
tff(fact_256_insert__ident,axiom,
    ! [X: $o,A: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A)
     => ( ~ aa_set_o_o(member_o((X)),B4)
       => ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4) )
        <=> ( A = B4 ) ) ) ) ).

% insert_ident
tff(fact_257_insert__ident,axiom,
    ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),B4)
       => ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4) )
        <=> ( A = B4 ) ) ) ) ).

% insert_ident
tff(fact_258_insert__absorb,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) = A ) ) ).

% insert_absorb
tff(fact_259_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) = A ) ) ).

% insert_absorb
tff(fact_260_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) = A ) ) ).

% insert_absorb
tff(fact_261_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) = A ) ) ).

% insert_absorb
tff(fact_262_insert__absorb,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) = A ) ) ).

% insert_absorb
tff(fact_263_insert__eq__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),B4)
       => ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),B4) )
        <=> $ite(
              A2 = B,
              A = B4,
              ? [C3: set_fu3555266933746557007_nat_o] :
                ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),C3) )
                & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),C3)
                & ( B4 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),C3) )
                & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_264_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A2),A)
     => ( ~ aa_set_nat_o(member_nat(B),B4)
       => ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),B4) )
        <=> $ite(
              A2 = B,
              A = B4,
              ? [C3: set_nat] :
                ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),C3) )
                & ~ aa_set_nat_o(member_nat(B),C3)
                & ( B4 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),C3) )
                & ~ aa_set_nat_o(member_nat(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_265_insert__eq__iff,axiom,
    ! [A2: int,A: set_int,B: int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A2),A)
     => ( ~ aa_set_int_o(member_int(B),B4)
       => ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),B4) )
        <=> $ite(
              A2 = B,
              A = B4,
              ? [C3: set_int] :
                ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),C3) )
                & ~ aa_set_int_o(member_int(B),C3)
                & ( B4 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),C3) )
                & ~ aa_set_int_o(member_int(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_266_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B: $o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A2)),A)
     => ( ~ aa_set_o_o(member_o((B)),B4)
       => ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),B4) )
        <=> $ite(
              (A2) = (B),
              A = B4,
              ? [C3: set_o] :
                ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),C3) )
                & ~ aa_set_o_o(member_o((B)),C3)
                & ( B4 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),C3) )
                & ~ aa_set_o_o(member_o((A2)),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_267_insert__eq__iff,axiom,
    ! [A2: product_unit,A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(B),B4)
       => ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),B4) )
        <=> $ite(
              A2 = B,
              A = B4,
              ? [C3: set_Product_unit] :
                ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),C3) )
                & ~ aa_set8780210009671629607unit_o(member_Product_unit(B),C3)
                & ( B4 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),C3) )
                & ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_268_insert__commute,axiom,
    ! [X: nat,Y: nat,A: set_nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),A)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) ).

% insert_commute
tff(fact_269_insert__commute,axiom,
    ! [X: int,Y: int,A: set_int] : aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),A)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) ).

% insert_commute
tff(fact_270_insert__commute,axiom,
    ! [X: $o,Y: $o,A: set_o] : aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),A)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) ).

% insert_commute
tff(fact_271_insert__commute,axiom,
    ! [X: product_unit,Y: product_unit,A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),A)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) ).

% insert_commute
tff(fact_272_mk__disjoint__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ? [B5: set_fu3555266933746557007_nat_o] :
          ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B5) )
          & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_273_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ? [B5: set_nat] :
          ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B5) )
          & ~ aa_set_nat_o(member_nat(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_274_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ? [B5: set_int] :
          ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B5) )
          & ~ aa_set_int_o(member_int(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_275_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ? [B5: set_o] :
          ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B5) )
          & ~ aa_set_o_o(member_o((A2)),B5) ) ) ).

% mk_disjoint_insert
tff(fact_276_mk__disjoint__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ? [B5: set_Product_unit] :
          ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B5) )
          & ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_277_singletonD,axiom,
    ! [B: product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(B),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))
     => ( B = A2 ) ) ).

% singletonD
tff(fact_278_singletonD,axiom,
    ! [B: $o,A2: $o] :
      ( aa_set_o_o(member_o((B)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))
     => ( (B)
      <=> (A2) ) ) ).

% singletonD
tff(fact_279_singletonD,axiom,
    ! [B: nat,A2: nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))
     => ( B = A2 ) ) ).

% singletonD
tff(fact_280_singletonD,axiom,
    ! [B: int,A2: int] :
      ( aa_set_int_o(member_int(B),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))
     => ( B = A2 ) ) ).

% singletonD
tff(fact_281_singletonD,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))
     => ( B = A2 ) ) ).

% singletonD
tff(fact_282_singleton__iff,axiom,
    ! [B: product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(B),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))
    <=> ( B = A2 ) ) ).

% singleton_iff
tff(fact_283_singleton__iff,axiom,
    ! [B: $o,A2: $o] :
      ( aa_set_o_o(member_o((B)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))
    <=> ( (B)
      <=> (A2) ) ) ).

% singleton_iff
tff(fact_284_singleton__iff,axiom,
    ! [B: nat,A2: nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))
    <=> ( B = A2 ) ) ).

% singleton_iff
tff(fact_285_singleton__iff,axiom,
    ! [B: int,A2: int] :
      ( aa_set_int_o(member_int(B),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))
    <=> ( B = A2 ) ) ).

% singleton_iff
tff(fact_286_singleton__iff,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))
    <=> ( B = A2 ) ) ).

% singleton_iff
tff(fact_287_doubleton__eq__iff,axiom,
    ! [A2: product_unit,B: product_unit,C: product_unit,D2: product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,C),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,D2),bot_bo3957492148770167129t_unit)) )
    <=> ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_288_doubleton__eq__iff,axiom,
    ! [A2: $o,B: $o,C: $o,D2: $o] :
      ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(C)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(D2)),bot_bot_set_o)) )
    <=> ( ( ( (A2)
          <=> (C) )
          & ( (B)
          <=> (D2) ) )
        | ( ( (A2)
          <=> (D2) )
          & ( (B)
          <=> (C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_289_doubleton__eq__iff,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,C),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,D2),bot_bot_set_nat)) )
    <=> ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_290_doubleton__eq__iff,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,C),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,D2),bot_bot_set_int)) )
    <=> ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_291_doubleton__eq__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,C: fun_Pr6460944699146269935_nat_o,D2: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(C),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(D2),bot_bo8632255151191677115_nat_o)) )
    <=> ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_292_insert__not__empty,axiom,
    ! [A2: product_unit,A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) != bot_bo3957492148770167129t_unit ).

% insert_not_empty
tff(fact_293_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] : aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) != bot_bot_set_o ).

% insert_not_empty
tff(fact_294_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_295_insert__not__empty,axiom,
    ! [A2: int,A: set_int] : aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) != bot_bot_set_int ).

% insert_not_empty
tff(fact_296_insert__not__empty,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) != bot_bo8632255151191677115_nat_o ).

% insert_not_empty
tff(fact_297_singleton__inject,axiom,
    ! [A2: product_unit,B: product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit) )
     => ( A2 = B ) ) ).

% singleton_inject
tff(fact_298_singleton__inject,axiom,
    ! [A2: $o,B: $o] :
      ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o) )
     => ( (A2)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_299_singleton__inject,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat) )
     => ( A2 = B ) ) ).

% singleton_inject
tff(fact_300_singleton__inject,axiom,
    ! [A2: int,B: int] :
      ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int) )
     => ( A2 = B ) ) ).

% singleton_inject
tff(fact_301_singleton__inject,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o) )
     => ( A2 = B ) ) ).

% singleton_inject
tff(fact_302_curryE,axiom,
    ! [F: fun_Pr6460944699146269935_nat_o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
      ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(produc1807123492089482958_nat_o(F),A2),B)
     => aa_Pro653082453696054616_nat_o(F,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B)) ) ).

% curryE
tff(fact_303_curryE,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(F),A2),B)
     => aa_Pro6150637604445964947_nat_o(F,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) ) ).

% curryE
tff(fact_304_curryE,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(F),A2),B)
     => aa_Pro4592068669816069595_int_o(F,aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) ) ).

% curryE
tff(fact_305_curryE,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A2),B)
     => aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B)) ) ).

% curryE
tff(fact_306_curryE,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A2),B)
     => aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B)) ) ).

% curryE
tff(fact_307_curryD,axiom,
    ! [F: fun_Pr6460944699146269935_nat_o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
      ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(produc1807123492089482958_nat_o(F),A2),B)
     => aa_Pro653082453696054616_nat_o(F,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B)) ) ).

% curryD
tff(fact_308_curryD,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(F),A2),B)
     => aa_Pro6150637604445964947_nat_o(F,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) ) ).

% curryD
tff(fact_309_curryD,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(F),A2),B)
     => aa_Pro4592068669816069595_int_o(F,aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) ) ).

% curryD
tff(fact_310_curryD,axiom,
    ! [F: fun_Pr2620521676128474833_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(produc4701769847872139056_nat_o(F),A2),B)
     => aa_Pro5643009357358181178_nat_o(F,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B)) ) ).

% curryD
tff(fact_311_curryD,axiom,
    ! [F: fun_Pr1227177680593602867_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(produc3333590387107866898_nat_o(F),A2),B)
     => aa_Pro2671238023309943196_nat_o(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B)) ) ).

% curryD
tff(fact_312_is__singleton__def,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(is_sin2160648248035936513t_unit,A)
    <=> ? [X2: product_unit] : A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X2),bot_bo3957492148770167129t_unit) ) ).

% is_singleton_def
tff(fact_313_is__singleton__def,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(is_singleton_o,A)
    <=> ? [X2: $o] : A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X2)),bot_bot_set_o) ) ).

% is_singleton_def
tff(fact_314_is__singleton__def,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A)
    <=> ? [X2: nat] : A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X2),bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_315_is__singleton__def,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(is_singleton_int,A)
    <=> ? [X2: int] : A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X2),bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_316_is__singleton__def,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,A)
    <=> ? [X2: fun_Pr6460944699146269935_nat_o] : A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),bot_bo8632255151191677115_nat_o) ) ).

% is_singleton_def
tff(fact_317_is__singletonE,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(is_sin2160648248035936513t_unit,A)
     => ~ ! [X3: product_unit] : A != aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit) ) ).

% is_singletonE
tff(fact_318_is__singletonE,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(is_singleton_o,A)
     => ~ ! [X3: $o] : A != aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o) ) ).

% is_singletonE
tff(fact_319_is__singletonE,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A)
     => ~ ! [X3: nat] : A != aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_320_is__singletonE,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(is_singleton_int,A)
     => ~ ! [X3: int] : A != aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int) ) ).

% is_singletonE
tff(fact_321_is__singletonE,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,A)
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] : A != aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),bot_bo8632255151191677115_nat_o) ) ).

% is_singletonE
tff(fact_322_is__pure__assn__basic__simps_I1_J,axiom,
    aa_assn_o(is_pure_assn,bot_bot_assn) ).

% is_pure_assn_basic_simps(1)
tff(fact_323_is__singleton__the__elem,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(is_sin2160648248035936513t_unit,A)
    <=> ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,the_el608902216710161154t_unit(A)),bot_bo3957492148770167129t_unit) ) ) ).

% is_singleton_the_elem
tff(fact_324_is__singleton__the__elem,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(is_singleton_o,A)
    <=> ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,aa_set_o_o(the_elem_o,A)),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_325_is__singleton__the__elem,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A)
    <=> ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_set_nat_nat(the_elem_nat,A)),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_326_is__singleton__the__elem,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(is_singleton_int,A)
    <=> ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,the_elem_int(A)),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_327_is__singleton__the__elem,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(is_sin624665226112233955_nat_o,A)
    <=> ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(the_el1896834595806683492_nat_o(A)),bot_bo8632255151191677115_nat_o) ) ) ).

% is_singleton_the_elem
tff(fact_328_the__elem__eq,axiom,
    ! [X: product_unit] : the_el608902216710161154t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% the_elem_eq
tff(fact_329_the__elem__eq,axiom,
    ! [X: $o] :
      ( aa_set_o_o(the_elem_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% the_elem_eq
tff(fact_330_the__elem__eq,axiom,
    ! [X: nat] : aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_331_the__elem__eq,axiom,
    ! [X: int] : the_elem_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% the_elem_eq
tff(fact_332_the__elem__eq,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : the_el1896834595806683492_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X ).

% the_elem_eq
tff(fact_333_pure__assn__def,axiom,
    ! [B: $o] : aa_o_assn(pure_assn,(B)) = aa_fun5526454687433275355o_assn(abs_assn,pure_a825153325127701367it_nat((B))) ).

% pure_assn_def
tff(fact_334_times__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(times_times_assn(P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_fun2638059832795814268_nat_o(times_assn_raw(aa_ass4243947775541419017_nat_o(rep_assn,P)),aa_ass4243947775541419017_nat_o(rep_assn,Q))) ).

% times_assn_def
tff(fact_335_type__definition__def,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
    <=> ( ! [X2: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(Rep,X2)),A)
        & ! [X2: assn] : aa_fun5526454687433275355o_assn(Abs,aa_ass4243947775541419017_nat_o(Rep,X2)) = X2
        & ! [Y3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y3),A)
           => ( aa_ass4243947775541419017_nat_o(Rep,aa_fun5526454687433275355o_assn(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_336_type__definition__def,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
    <=> ( ! [X2: code_integer] : aa_set_int_o(member_int(aa_Code_integer_int(Rep,X2)),A)
        & ! [X2: code_integer] : aa_int_Code_integer(Abs,aa_Code_integer_int(Rep,X2)) = X2
        & ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),A)
           => ( aa_Code_integer_int(Rep,aa_int_Code_integer(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_337_type__definition__def,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
    <=> ( ! [X2: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(Rep,X2)),A)
        & ! [X2: product_unit] : aa_o_Product_unit(Abs,aa_Product_unit_o(Rep,X2)) = X2
        & ! [Y3: $o] :
            ( aa_set_o_o(member_o((Y3)),A)
           => ( aa_Product_unit_o(Rep,aa_o_Product_unit(Abs,(Y3)))
            <=> (Y3) ) ) ) ) ).

% type_definition_def
tff(fact_338_type__definition__def,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
    <=> ( ! [X2: code_natural] : aa_set_nat_o(member_nat(aa_Code_natural_nat(Rep,X2)),A)
        & ! [X2: code_natural] : aa_nat_Code_natural(Abs,aa_Code_natural_nat(Rep,X2)) = X2
        & ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),A)
           => ( aa_Code_natural_nat(Rep,aa_nat_Code_natural(Abs,Y3)) = Y3 ) ) ) ) ).

% type_definition_def
tff(fact_339_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( aa_fun5526454687433275355o_assn(Abs,aa_ass4243947775541419017_nat_o(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_340_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,X: code_integer] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( aa_int_Code_integer(Abs,aa_Code_integer_int(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_341_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( aa_o_Product_unit(Abs,aa_Product_unit_o(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_342_type__definition_ORep__inverse,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,X: code_natural] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( aa_nat_Code_natural(Abs,aa_Code_natural_nat(Rep,X)) = X ) ) ).

% type_definition.Rep_inverse
tff(fact_343_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
       => ( aa_ass4243947775541419017_nat_o(Rep,aa_fun5526454687433275355o_assn(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_344_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( aa_set_int_o(member_int(Y),A)
       => ( aa_Code_integer_int(Rep,aa_int_Code_integer(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_345_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( aa_set_o_o(member_o((Y)),A)
       => ( aa_Product_unit_o(Rep,aa_o_Product_unit(Abs,(Y)))
        <=> (Y) ) ) ) ).

% type_definition.Abs_inverse
tff(fact_346_type__definition_OAbs__inverse,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,Y: nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( aa_set_nat_o(member_nat(Y),A)
       => ( aa_Code_natural_nat(Rep,aa_nat_Code_natural(Abs,Y)) = Y ) ) ) ).

% type_definition.Abs_inverse
tff(fact_347_type__definition_ORep__inject,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,X: assn,Y: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( ( aa_ass4243947775541419017_nat_o(Rep,X) = aa_ass4243947775541419017_nat_o(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_348_type__definition_ORep__inject,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,X: code_integer,Y: code_integer] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( ( aa_Code_integer_int(Rep,X) = aa_Code_integer_int(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_349_type__definition_ORep__inject,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,X: product_unit,Y: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( ( aa_Product_unit_o(Rep,X)
        <=> aa_Product_unit_o(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_350_type__definition_ORep__inject,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,X: code_natural,Y: code_natural] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( ( aa_Code_natural_nat(Rep,X) = aa_Code_natural_nat(Rep,Y) )
      <=> ( X = Y ) ) ) ).

% type_definition.Rep_inject
tff(fact_351_type__definition_ORep__induct,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
       => ( ! [X3: assn] : aa_fun1675921005893230629at_o_o(P,aa_ass4243947775541419017_nat_o(Rep,X3))
         => aa_fun1675921005893230629at_o_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_352_type__definition_ORep__induct,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,Y: int,P: fun_int_o] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( aa_set_int_o(member_int(Y),A)
       => ( ! [X3: code_integer] : aa_int_o(P,aa_Code_integer_int(Rep,X3))
         => aa_int_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_353_type__definition_ORep__induct,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,Y: $o,P: fun_o_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( aa_set_o_o(member_o((Y)),A)
       => ( ! [X3: product_unit] : aa_o_o(P,aa_Product_unit_o(Rep,X3))
         => aa_o_o(P,(Y)) ) ) ) ).

% type_definition.Rep_induct
tff(fact_354_type__definition_ORep__induct,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,Y: nat,P: fun_nat_o] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( aa_set_nat_o(member_nat(Y),A)
       => ( ! [X3: code_natural] : aa_nat_o(P,aa_Code_natural_nat(Rep,X3))
         => aa_nat_o(P,Y) ) ) ) ).

% type_definition.Rep_induct
tff(fact_355_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
         => ( ( aa_fun5526454687433275355o_assn(Abs,X) = aa_fun5526454687433275355o_assn(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_356_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,X: int,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( aa_set_int_o(member_int(X),A)
       => ( aa_set_int_o(member_int(Y),A)
         => ( ( aa_int_Code_integer(Abs,X) = aa_int_Code_integer(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_357_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,X: $o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(member_o((Y)),A)
         => ( ( aa_o_Product_unit(Abs,(X)) = aa_o_Product_unit(Abs,(Y)) )
          <=> ( (X)
            <=> (Y) ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_358_type__definition_OAbs__inject,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,X: nat,Y: nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_o(member_nat(Y),A)
         => ( ( aa_nat_Code_natural(Abs,X) = aa_nat_Code_natural(Abs,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% type_definition.Abs_inject
tff(fact_359_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = aa_o_assn(pure_assn,(Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_360_mult_Oright__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),one_one_Code_integer) = A2 ).

% mult.right_neutral
tff(fact_361_mult_Oright__neutral,axiom,
    ! [A2: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),one_one_Code_natural) = A2 ).

% mult.right_neutral
tff(fact_362_mult_Oright__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% mult.right_neutral
tff(fact_363_mult_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),one_one_rat) = A2 ).

% mult.right_neutral
tff(fact_364_mult_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.right_neutral
tff(fact_365_mult_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.right_neutral
tff(fact_366_mult__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A2) = A2 ).

% mult_1
tff(fact_367_mult__1,axiom,
    ! [A2: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural),A2) = A2 ).

% mult_1
tff(fact_368_mult__1,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% mult_1
tff(fact_369_mult__1,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A2) = A2 ).

% mult_1
tff(fact_370_mult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% mult_1
tff(fact_371_mult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% mult_1
tff(fact_372_merge__pure__star,axiom,
    ! [A2: $o,B: $o] :
      aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,(A2))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A2)
        & (B) )) ).

% merge_pure_star
tff(fact_373_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_374_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_375_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_376_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = bot_bot_assn )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_377_is__pure__assn__starI,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(is_pure_assn,A2)
     => ( aa_assn_o(is_pure_assn,B)
       => aa_assn_o(is_pure_assn,aa_assn_assn(times_times_assn(A2),B)) ) ) ).

% is_pure_assn_starI
tff(fact_378_is__pure__assn__pure,axiom,
    ! [P: $o] : aa_assn_o(is_pure_assn,aa_o_assn(pure_assn,(P))) ).

% is_pure_assn_pure
tff(fact_379_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_380_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = one_one_assn )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_381_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(times_times_assn(P),Q) = aa_assn_assn(times_times_assn(Q),P) ).

% assn_times_comm
tff(fact_382_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),R) = aa_assn_assn(times_times_assn(P),aa_assn_assn(times_times_assn(Q),R)) ).

% assn_times_assoc
tff(fact_383_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A2: assn,C: assn] : aa_assn_assn(times_times_assn(B),aa_assn_assn(times_times_assn(A2),C)) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_384_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_385_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_386_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_387_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: assn,B: assn] : aa_assn_assn(times_times_assn(A2),B) = aa_assn_assn(times_times_assn(B),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_388_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_389_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_390_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_391_mult_Oassoc,axiom,
    ! [A2: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B)),C) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B),C)) ).

% mult.assoc
tff(fact_392_mult_Oassoc,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_393_mult_Oassoc,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_394_mult_Oassoc,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_395_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_396_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_397_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_398_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_399_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_400_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_401_mult_Ocomm__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),one_one_Code_integer) = A2 ).

% mult.comm_neutral
tff(fact_402_mult_Ocomm__neutral,axiom,
    ! [A2: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),one_one_Code_natural) = A2 ).

% mult.comm_neutral
tff(fact_403_mult_Ocomm__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% mult.comm_neutral
tff(fact_404_mult_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),one_one_rat) = A2 ).

% mult.comm_neutral
tff(fact_405_mult_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.comm_neutral
tff(fact_406_mult_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.comm_neutral
tff(fact_407_mod__starE,axiom,
    ! [A2: assn,B: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(times_times_assn(A2),B)),Ha)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B),H_2) ) ) ).

% mod_starE
tff(fact_408_mod__starD,axiom,
    ! [A: assn,B4: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(times_times_assn(A),B4)),Ha)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A),H1)
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B4),H22) ) ) ).

% mod_starD
tff(fact_409_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(one_one_assn),P) = P ).

% assn_one_left
tff(fact_410_is__pure__assnE,axiom,
    ! [A2: assn] :
      ( aa_assn_o(is_pure_assn,A2)
     => ~ ! [P3: $o] : A2 != aa_o_assn(pure_assn,(P3)) ) ).

% is_pure_assnE
tff(fact_411_is__pure__assn__def,axiom,
    ! [A2: assn] :
      ( aa_assn_o(is_pure_assn,A2)
    <=> ? [P4: $o] : A2 = aa_o_assn(pure_assn,(P4)) ) ).

% is_pure_assn_def
tff(fact_412_type__definition_ORep,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(Rep,X)),A) ) ).

% type_definition.Rep
tff(fact_413_type__definition_ORep,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,X: code_integer] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => aa_set_int_o(member_int(aa_Code_integer_int(Rep,X)),A) ) ).

% type_definition.Rep
tff(fact_414_type__definition_ORep,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => aa_set_o_o(member_o(aa_Product_unit_o(Rep,X)),A) ) ).

% type_definition.Rep
tff(fact_415_type__definition_ORep,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,X: code_natural] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => aa_set_nat_o(member_nat(aa_Code_natural_nat(Rep,X)),A) ) ).

% type_definition.Rep
tff(fact_416_type__definition_Ointro,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,A: set_fu3555266933746557007_nat_o,Abs: fun_fu3790787410453625514o_assn] :
      ( ! [X3: assn] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(Rep,X3)),A)
     => ( ! [X3: assn] : aa_fun5526454687433275355o_assn(Abs,aa_ass4243947775541419017_nat_o(Rep,X3)) = X3
       => ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
             => ( aa_ass4243947775541419017_nat_o(Rep,aa_fun5526454687433275355o_assn(Abs,Y2)) = Y2 ) )
         => aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A) ) ) ) ).

% type_definition.intro
tff(fact_417_type__definition_Ointro,axiom,
    ! [Rep: fun_Code_integer_int,A: set_int,Abs: fun_int_Code_integer] :
      ( ! [X3: code_integer] : aa_set_int_o(member_int(aa_Code_integer_int(Rep,X3)),A)
     => ( ! [X3: code_integer] : aa_int_Code_integer(Abs,aa_Code_integer_int(Rep,X3)) = X3
       => ( ! [Y2: int] :
              ( aa_set_int_o(member_int(Y2),A)
             => ( aa_Code_integer_int(Rep,aa_int_Code_integer(Abs,Y2)) = Y2 ) )
         => aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A) ) ) ) ).

% type_definition.intro
tff(fact_418_type__definition_Ointro,axiom,
    ! [Rep: fun_Product_unit_o,A: set_o,Abs: fun_o_Product_unit] :
      ( ! [X3: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(Rep,X3)),A)
     => ( ! [X3: product_unit] : aa_o_Product_unit(Abs,aa_Product_unit_o(Rep,X3)) = X3
       => ( ! [Y2: $o] :
              ( aa_set_o_o(member_o((Y2)),A)
             => ( aa_Product_unit_o(Rep,aa_o_Product_unit(Abs,(Y2)))
              <=> (Y2) ) )
         => aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A) ) ) ) ).

% type_definition.intro
tff(fact_419_type__definition_Ointro,axiom,
    ! [Rep: fun_Code_natural_nat,A: set_nat,Abs: fun_nat_Code_natural] :
      ( ! [X3: code_natural] : aa_set_nat_o(member_nat(aa_Code_natural_nat(Rep,X3)),A)
     => ( ! [X3: code_natural] : aa_nat_Code_natural(Abs,aa_Code_natural_nat(Rep,X3)) = X3
       => ( ! [Y2: nat] :
              ( aa_set_nat_o(member_nat(Y2),A)
             => ( aa_Code_natural_nat(Rep,aa_nat_Code_natural(Abs,Y2)) = Y2 ) )
         => aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A) ) ) ) ).

% type_definition.intro
tff(fact_420_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
            ( ( X = aa_fun5526454687433275355o_assn(Abs,Y2) )
           => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A) ) ) ).

% type_definition.Abs_cases
tff(fact_421_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,X: code_integer] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ~ ! [Y2: int] :
            ( ( X = aa_int_Code_integer(Abs,Y2) )
           => ~ aa_set_int_o(member_int(Y2),A) ) ) ).

% type_definition.Abs_cases
tff(fact_422_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ~ ! [Y2: $o] :
            ( ( X = aa_o_Product_unit(Abs,(Y2)) )
           => ~ aa_set_o_o(member_o((Y2)),A) ) ) ).

% type_definition.Abs_cases
tff(fact_423_type__definition_OAbs__cases,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,X: code_natural] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ~ ! [Y2: nat] :
            ( ( X = aa_nat_Code_natural(Abs,Y2) )
           => ~ aa_set_nat_o(member_nat(Y2),A) ) ) ).

% type_definition.Abs_cases
tff(fact_424_type__definition_ORep__cases,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
       => ~ ! [X3: assn] : Y != aa_ass4243947775541419017_nat_o(Rep,X3) ) ) ).

% type_definition.Rep_cases
tff(fact_425_type__definition_ORep__cases,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( aa_set_int_o(member_int(Y),A)
       => ~ ! [X3: code_integer] : Y != aa_Code_integer_int(Rep,X3) ) ) ).

% type_definition.Rep_cases
tff(fact_426_type__definition_ORep__cases,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( aa_set_o_o(member_o((Y)),A)
       => ~ ! [X3: product_unit] :
              ( (Y)
            <=> ~ aa_Product_unit_o(Rep,X3) ) ) ) ).

% type_definition.Rep_cases
tff(fact_427_type__definition_ORep__cases,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,Y: nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( aa_set_nat_o(member_nat(Y),A)
       => ~ ! [X3: code_natural] : Y != aa_Code_natural_nat(Rep,X3) ) ) ).

% type_definition.Rep_cases
tff(fact_428_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o,P: fun_assn_o,X: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
           => aa_assn_o(P,aa_fun5526454687433275355o_assn(Abs,Y2)) )
       => aa_assn_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_429_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int,P: fun_Code_integer_o,X: code_integer] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),A)
           => aa_Code_integer_o(P,aa_int_Code_integer(Abs,Y2)) )
       => aa_Code_integer_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_430_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o,P: fun_Product_unit_o,X: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( ! [Y2: $o] :
            ( aa_set_o_o(member_o((Y2)),A)
           => aa_Product_unit_o(P,aa_o_Product_unit(Abs,(Y2))) )
       => aa_Product_unit_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_431_type__definition_OAbs__induct,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat,P: fun_Code_natural_o,X: code_natural] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),A)
           => aa_Code_natural_o(P,aa_nat_Code_natural(Abs,Y2)) )
       => aa_Code_natural_o(P,X) ) ) ).

% type_definition.Abs_induct
tff(fact_432_total__on__singleton,axiom,
    ! [X: product_unit] : total_5665700600833547665t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit),insert4616891024742550839t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),X),bot_bo1654022762305692595t_unit)) ).

% total_on_singleton
tff(fact_433_total__on__singleton,axiom,
    ! [X: $o] : total_on_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o),insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(X)),bot_bo7073875226086086771od_o_o)) ).

% total_on_singleton
tff(fact_434_total__on__singleton,axiom,
    ! [X: nat] : total_on_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat),insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),bot_bo2099793752762293965at_nat)) ).

% total_on_singleton
tff(fact_435_total__on__singleton,axiom,
    ! [X: int] : total_on_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),X),bot_bo1796632182523588997nt_int)) ).

% total_on_singleton
tff(fact_436_total__on__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : total_3645986146778848883_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o),insert4451378821651018679_nat_o(produc4714251849521319255_nat_o(X,X),bot_bo6763312114214544691_nat_o)) ).

% total_on_singleton
tff(fact_437_refl__on__singleton,axiom,
    ! [X: product_unit] : refl_on_Product_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit),insert4616891024742550839t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),X),bot_bo1654022762305692595t_unit)) ).

% refl_on_singleton
tff(fact_438_refl__on__singleton,axiom,
    ! [X: $o] : refl_on_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o),insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(X)),bot_bo7073875226086086771od_o_o)) ).

% refl_on_singleton
tff(fact_439_refl__on__singleton,axiom,
    ! [X: nat] : refl_on_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat),insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),bot_bo2099793752762293965at_nat)) ).

% refl_on_singleton
tff(fact_440_refl__on__singleton,axiom,
    ! [X: int] : refl_on_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),X),bot_bo1796632182523588997nt_int)) ).

% refl_on_singleton
tff(fact_441_refl__on__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : refl_o5094713388964525678_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o),insert4451378821651018679_nat_o(produc4714251849521319255_nat_o(X,X),bot_bo6763312114214544691_nat_o)) ).

% refl_on_singleton
tff(fact_442_Id__on__empty,axiom,
    id_on_Product_unit(bot_bo3957492148770167129t_unit) = bot_bo1654022762305692595t_unit ).

% Id_on_empty
tff(fact_443_Id__on__empty,axiom,
    id_on_o(bot_bot_set_o) = bot_bo7073875226086086771od_o_o ).

% Id_on_empty
tff(fact_444_Id__on__empty,axiom,
    id_on_nat(bot_bot_set_nat) = bot_bo2099793752762293965at_nat ).

% Id_on_empty
tff(fact_445_Id__on__empty,axiom,
    id_on_int(bot_bot_set_int) = bot_bo1796632182523588997nt_int ).

% Id_on_empty
tff(fact_446_Id__on__empty,axiom,
    id_on_7912660205148244492_nat_o(bot_bo8632255151191677115_nat_o) = bot_bo6763312114214544691_nat_o ).

% Id_on_empty
tff(fact_447_linear__order__on__singleton,axiom,
    ! [X: product_unit] : order_888282469287196226t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit),insert4616891024742550839t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),X),bot_bo1654022762305692595t_unit)) ).

% linear_order_on_singleton
tff(fact_448_linear__order__on__singleton,axiom,
    ! [X: $o] : order_6238756238976269133r_on_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o),insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(X)),bot_bo7073875226086086771od_o_o)) ).

% linear_order_on_singleton
tff(fact_449_linear__order__on__singleton,axiom,
    ! [X: nat] : order_4473980167227706203on_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat),insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),bot_bo2099793752762293965at_nat)) ).

% linear_order_on_singleton
tff(fact_450_linear__order__on__singleton,axiom,
    ! [X: int] : order_4471489696718655927on_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),X),bot_bo1796632182523588997nt_int)) ).

% linear_order_on_singleton
tff(fact_451_linear__order__on__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : order_53248997263766692_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o),insert4451378821651018679_nat_o(produc4714251849521319255_nat_o(X,X),bot_bo6763312114214544691_nat_o)) ).

% linear_order_on_singleton
tff(fact_452_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,As2: set_nat] :
              ( ( Xa = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( ( (Y)
                <=> ( ( As2 = bot_bot_set_nat )
                    & (X) ) )
               => ~ accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))) ) ) ) ) ).

% pure_assn_raw.pelims(1)
tff(fact_453_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,As2: set_nat] :
              ( ( Xa = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)))
               => ~ ( ( As2 = bot_bot_set_nat )
                    & (X) ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
tff(fact_454_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,As2: set_nat] :
              ( ( Xa = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( accp_P8458817951426537472et_nat(pure_a6022498039421069578it_nat,produc3762353314782720579et_nat((X),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)))
               => ( ( As2 = bot_bot_set_nat )
                  & (X) ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
tff(fact_455_Image__singleton__iff,axiom,
    ! [B: nat,R2: set_Pr1763845938948868674it_nat,A2: product_unit] :
      ( aa_set_nat_o(member_nat(B),image_5657742817519485770it_nat(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))
    <=> member423003724879500579it_nat(produc4803288329813530246it_nat(A2,B),R2) ) ).

% Image_singleton_iff
tff(fact_456_Image__singleton__iff,axiom,
    ! [B: int,R2: set_Pr8497134830871917726it_int,A2: product_unit] :
      ( aa_set_int_o(member_int(B),image_5655252347010435494it_int(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))
    <=> member5468524742225079679it_int(produc4800797859304479970it_int(A2,B),R2) ) ).

% Image_singleton_iff
tff(fact_457_Image__singleton__iff,axiom,
    ! [B: $o,R2: set_Pr5464989305201766866unit_o,A2: product_unit] :
      ( aa_set_o_o(member_o((B)),image_Product_unit_o(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))
    <=> member3689400097658526619unit_o(aa_o_P5708869953080425366unit_o(produc8487917018914037602unit_o(A2),(B)),R2) ) ).

% Image_singleton_iff
tff(fact_458_Image__singleton__iff,axiom,
    ! [B: product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(B),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))
    <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ).

% Image_singleton_iff
tff(fact_459_Image__singleton__iff,axiom,
    ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A2: $o] :
      ( aa_set_nat_o(member_nat(B),image_o_nat(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))
    <=> member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A2)),B),R2) ) ).

% Image_singleton_iff
tff(fact_460_Image__singleton__iff,axiom,
    ! [B: int,R2: set_Pr8834758594704517033_o_int,A2: $o] :
      ( aa_set_int_o(member_int(B),image_o_int(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))
    <=> member7847949116333733898_o_int(product_Pair_o_int((A2),B),R2) ) ).

% Image_singleton_iff
tff(fact_461_Image__singleton__iff,axiom,
    ! [B: $o,R2: set_Product_prod_o_o,A2: $o] :
      ( aa_set_o_o(member_o((B)),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))
    <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ).

% Image_singleton_iff
tff(fact_462_Image__singleton__iff,axiom,
    ! [B: product_unit,R2: set_Pr6671815714118170108t_unit,A2: $o] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(B),image_o_Product_unit(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))
    <=> member9215336023991391429t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A2)),B),R2) ) ).

% Image_singleton_iff
tff(fact_463_Image__singleton__iff,axiom,
    ! [B: int,R2: set_Pr7995236796853374141at_int,A2: nat] :
      ( aa_set_int_o(member_int(B),image_nat_int(R2,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)))
    <=> member4262671552274231302at_int(product_Pair_nat_int(A2,B),R2) ) ).

% Image_singleton_iff
tff(fact_464_Image__singleton__iff,axiom,
    ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A2: nat] :
      ( aa_set_o_o(member_o((B)),image_nat_o(R2,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)))
    <=> member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A2),(B)),R2) ) ).

% Image_singleton_iff
tff(fact_465_mult_Oright__assoc,axiom,
    ! [A2: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B)),C) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B),C)) ).

% mult.right_assoc
tff(fact_466_mult_Oright__assoc,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.right_assoc
tff(fact_467_mult_Oright__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.right_assoc
tff(fact_468_mult_Oright__assoc,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.right_assoc
tff(fact_469_mult_Oright__commute,axiom,
    ! [A2: assn,B: assn,C: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B)),C) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),C)),B) ).

% mult.right_commute
tff(fact_470_mult_Oright__commute,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),B) ).

% mult.right_commute
tff(fact_471_mult_Oright__commute,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),B) ).

% mult.right_commute
tff(fact_472_mult_Oright__commute,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),B) ).

% mult.right_commute
tff(fact_473_in__range__empty,axiom,
    ! [Ha: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),bot_bot_set_nat)) ).

% in_range_empty
tff(fact_474_ImageI,axiom,
    ! [A2: nat,B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
      ( member4262671552274231302at_int(product_Pair_nat_int(A2,B),R2)
     => ( aa_set_nat_o(member_nat(A2),A)
       => aa_set_int_o(member_int(B),image_nat_int(R2,A)) ) ) ).

% ImageI
tff(fact_475_ImageI,axiom,
    ! [A2: nat,B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
      ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A2),(B)),R2)
     => ( aa_set_nat_o(member_nat(A2),A)
       => aa_set_o_o(member_o((B)),image_nat_o(R2,A)) ) ) ).

% ImageI
tff(fact_476_ImageI,axiom,
    ! [A2: nat,B: product_unit,R2: set_Pr4334478416066269672t_unit,A: set_nat] :
      ( member348563419383025609t_unit(aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(A2),B),R2)
     => ( aa_set_nat_o(member_nat(A2),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_4288904962331477416t_unit(R2,A)) ) ) ).

% ImageI
tff(fact_477_ImageI,axiom,
    ! [A2: int,B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
      ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A2),B),R2)
     => ( aa_set_int_o(member_int(A2),A)
       => aa_set_nat_o(member_nat(B),image_int_nat(R2,A)) ) ) ).

% ImageI
tff(fact_478_ImageI,axiom,
    ! [A2: int,B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
      ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A2),(B)),R2)
     => ( aa_set_int_o(member_int(A2),A)
       => aa_set_o_o(member_o((B)),image_int_o(R2,A)) ) ) ).

% ImageI
tff(fact_479_ImageI,axiom,
    ! [A2: int,B: product_unit,R2: set_Pr5889264611543340172t_unit,A: set_int] :
      ( member6688357403439981677t_unit(aa_Pro399436582157352231t_unit(produc550872421827702152t_unit(A2),B),R2)
     => ( aa_set_int_o(member_int(A2),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_1405326909533657676t_unit(R2,A)) ) ) ).

% ImageI
tff(fact_480_ImageI,axiom,
    ! [A2: $o,B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
      ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A2)),B),R2)
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set_nat_o(member_nat(B),image_o_nat(R2,A)) ) ) ).

% ImageI
tff(fact_481_ImageI,axiom,
    ! [A2: $o,B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
      ( member7847949116333733898_o_int(product_Pair_o_int((A2),B),R2)
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set_int_o(member_int(B),image_o_int(R2,A)) ) ) ).

% ImageI
tff(fact_482_ImageI,axiom,
    ! [A2: $o,B: $o,R2: set_Product_prod_o_o,A: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set_o_o(member_o((B)),image_o_o(R2,A)) ) ) ).

% ImageI
tff(fact_483_ImageI,axiom,
    ! [A2: $o,B: product_unit,R2: set_Pr6671815714118170108t_unit,A: set_o] :
      ( member9215336023991391429t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A2)),B),R2)
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_o_Product_unit(R2,A)) ) ) ).

% ImageI
tff(fact_484_Image__empty2,axiom,
    ! [R: set_Pr5094982260447487303t_unit] : image_8431784415008483091t_unit(R,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% Image_empty2
tff(fact_485_Image__empty2,axiom,
    ! [R: set_Pr5464989305201766866unit_o] : image_Product_unit_o(R,bot_bo3957492148770167129t_unit) = bot_bot_set_o ).

% Image_empty2
tff(fact_486_Image__empty2,axiom,
    ! [R: set_Pr1763845938948868674it_nat] : image_5657742817519485770it_nat(R,bot_bo3957492148770167129t_unit) = bot_bot_set_nat ).

% Image_empty2
tff(fact_487_Image__empty2,axiom,
    ! [R: set_Pr8497134830871917726it_int] : image_5655252347010435494it_int(R,bot_bo3957492148770167129t_unit) = bot_bot_set_int ).

% Image_empty2
tff(fact_488_Image__empty2,axiom,
    ! [R: set_Pr6671815714118170108t_unit] : image_o_Product_unit(R,bot_bot_set_o) = bot_bo3957492148770167129t_unit ).

% Image_empty2
tff(fact_489_Image__empty2,axiom,
    ! [R: set_Product_prod_o_o] : image_o_o(R,bot_bot_set_o) = bot_bot_set_o ).

% Image_empty2
tff(fact_490_Image__empty2,axiom,
    ! [R: set_Pr2101469702781467981_o_nat] : image_o_nat(R,bot_bot_set_o) = bot_bot_set_nat ).

% Image_empty2
tff(fact_491_Image__empty2,axiom,
    ! [R: set_Pr8834758594704517033_o_int] : image_o_int(R,bot_bot_set_o) = bot_bot_set_int ).

% Image_empty2
tff(fact_492_Image__empty2,axiom,
    ! [R: set_Pr4334478416066269672t_unit] : image_4288904962331477416t_unit(R,bot_bot_set_nat) = bot_bo3957492148770167129t_unit ).

% Image_empty2
tff(fact_493_Image__empty2,axiom,
    ! [R: set_Pr3149072824959771635_nat_o] : image_nat_o(R,bot_bot_set_nat) = bot_bot_set_o ).

% Image_empty2
tff(fact_494_Id__onI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,A2),id_on_7912660205148244492_nat_o(A)) ) ).

% Id_onI
tff(fact_495_Id__onI,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(A2)),id_on_o(A)) ) ).

% Id_onI
tff(fact_496_Id__onI,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),A2),id_on_Product_unit(A)) ) ).

% Id_onI
tff(fact_497_Id__onI,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),id_on_nat(A)) ) ).

% Id_onI
tff(fact_498_Id__onI,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),id_on_int(A)) ) ).

% Id_onI
tff(fact_499_bool__assn__proper_I1_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,in_range) ).

% bool_assn_proper(1)
tff(fact_500_refl__on__domain,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( refl_o5094713388964525678_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
          & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A) ) ) ) ).

% refl_on_domain
tff(fact_501_refl__on__domain,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( refl_on_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
       => ( aa_set_o_o(member_o((A2)),A)
          & aa_set_o_o(member_o((B)),A) ) ) ) ).

% refl_on_domain
tff(fact_502_refl__on__domain,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( refl_on_Product_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
          & aa_set8780210009671629607unit_o(member_Product_unit(B),A) ) ) ) ).

% refl_on_domain
tff(fact_503_refl__on__domain,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( refl_on_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => ( aa_set_nat_o(member_nat(A2),A)
          & aa_set_nat_o(member_nat(B),A) ) ) ) ).

% refl_on_domain
tff(fact_504_refl__on__domain,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( refl_on_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => ( aa_set_int_o(member_int(A2),A)
          & aa_set_int_o(member_int(B),A) ) ) ) ).

% refl_on_domain
tff(fact_505_rev__ImageI,axiom,
    ! [A2: nat,A: set_nat,B: int,R2: set_Pr7995236796853374141at_int] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( member4262671552274231302at_int(product_Pair_nat_int(A2,B),R2)
       => aa_set_int_o(member_int(B),image_nat_int(R2,A)) ) ) ).

% rev_ImageI
tff(fact_506_rev__ImageI,axiom,
    ! [A2: nat,A: set_nat,B: $o,R2: set_Pr3149072824959771635_nat_o] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A2),(B)),R2)
       => aa_set_o_o(member_o((B)),image_nat_o(R2,A)) ) ) ).

% rev_ImageI
tff(fact_507_rev__ImageI,axiom,
    ! [A2: nat,A: set_nat,B: product_unit,R2: set_Pr4334478416066269672t_unit] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( member348563419383025609t_unit(aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(A2),B),R2)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_4288904962331477416t_unit(R2,A)) ) ) ).

% rev_ImageI
tff(fact_508_rev__ImageI,axiom,
    ! [A2: int,A: set_int,B: nat,R2: set_Pr3448869479623346877nt_nat] :
      ( aa_set_int_o(member_int(A2),A)
     => ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A2),B),R2)
       => aa_set_nat_o(member_nat(B),image_int_nat(R2,A)) ) ) ).

% rev_ImageI
tff(fact_509_rev__ImageI,axiom,
    ! [A2: int,A: set_int,B: $o,R2: set_Pr903927857289325719_int_o] :
      ( aa_set_int_o(member_int(A2),A)
     => ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A2),(B)),R2)
       => aa_set_o_o(member_o((B)),image_int_o(R2,A)) ) ) ).

% rev_ImageI
tff(fact_510_rev__ImageI,axiom,
    ! [A2: int,A: set_int,B: product_unit,R2: set_Pr5889264611543340172t_unit] :
      ( aa_set_int_o(member_int(A2),A)
     => ( member6688357403439981677t_unit(aa_Pro399436582157352231t_unit(produc550872421827702152t_unit(A2),B),R2)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_1405326909533657676t_unit(R2,A)) ) ) ).

% rev_ImageI
tff(fact_511_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: nat,R2: set_Pr2101469702781467981_o_nat] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A2)),B),R2)
       => aa_set_nat_o(member_nat(B),image_o_nat(R2,A)) ) ) ).

% rev_ImageI
tff(fact_512_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: int,R2: set_Pr8834758594704517033_o_int] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( member7847949116333733898_o_int(product_Pair_o_int((A2),B),R2)
       => aa_set_int_o(member_int(B),image_o_int(R2,A)) ) ) ).

% rev_ImageI
tff(fact_513_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: $o,R2: set_Product_prod_o_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
       => aa_set_o_o(member_o((B)),image_o_o(R2,A)) ) ) ).

% rev_ImageI
tff(fact_514_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: product_unit,R2: set_Pr6671815714118170108t_unit] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( member9215336023991391429t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A2)),B),R2)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_o_Product_unit(R2,A)) ) ) ).

% rev_ImageI
tff(fact_515_Image__iff,axiom,
    ! [B: set_nat,R2: set_Pr3948176798113811640et_nat,A: set_he8644174446132254637t_unit] :
      ( member_set_nat(B,image_3942343239045813184et_nat(R2,A))
    <=> ? [X2: heap_e7401611519738050253t_unit] :
          ( member242389114213480694t_unit(X2,A)
          & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X2),B),R2) ) ) ).

% Image_iff
tff(fact_516_Image__iff,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(R2),A))
    <=> ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),B),R2) ) ) ).

% Image_iff
tff(fact_517_Image__iff,axiom,
    ! [B: int,R2: set_Pr958786334691620121nt_int,A: set_int] :
      ( aa_set_int_o(member_int(B),image_int_int(R2,A))
    <=> ? [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),B),R2) ) ) ).

% Image_iff
tff(fact_518_Image__iff,axiom,
    ! [B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,A: set_fu3555266933746557007_nat_o] :
      ( member6260224972018164377et_nat(B,image_552160923151422558et_nat(R2,A))
    <=> ? [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),B),R2) ) ) ).

% Image_iff
tff(fact_519_Image__iff,axiom,
    ! [B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,A: set_fu3555266933746557007_nat_o] :
      ( member4513763875858039863et_nat(B,image_2192608224838851452et_nat(R2,A))
    <=> ? [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),B),R2) ) ) ).

% Image_iff
tff(fact_520_ImageE,axiom,
    ! [B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
      ( aa_set_nat_o(member_nat(B),image_int_nat(R2,A))
     => ~ ! [X3: int] :
            ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(X3),B),R2)
           => ~ aa_set_int_o(member_int(X3),A) ) ) ).

% ImageE
tff(fact_521_ImageE,axiom,
    ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
      ( aa_set_nat_o(member_nat(B),image_o_nat(R2,A))
     => ~ ! [X3: $o] :
            ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((X3)),B),R2)
           => ~ aa_set_o_o(member_o((X3)),A) ) ) ).

% ImageE
tff(fact_522_ImageE,axiom,
    ! [B: nat,R2: set_Pr1763845938948868674it_nat,A: set_Product_unit] :
      ( aa_set_nat_o(member_nat(B),image_5657742817519485770it_nat(R2,A))
     => ~ ! [X3: product_unit] :
            ( member423003724879500579it_nat(produc4803288329813530246it_nat(X3,B),R2)
           => ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),A) ) ) ).

% ImageE
tff(fact_523_ImageE,axiom,
    ! [B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
      ( aa_set_int_o(member_int(B),image_nat_int(R2,A))
     => ~ ! [X3: nat] :
            ( member4262671552274231302at_int(product_Pair_nat_int(X3,B),R2)
           => ~ aa_set_nat_o(member_nat(X3),A) ) ) ).

% ImageE
tff(fact_524_ImageE,axiom,
    ! [B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
      ( aa_set_int_o(member_int(B),image_o_int(R2,A))
     => ~ ! [X3: $o] :
            ( member7847949116333733898_o_int(product_Pair_o_int((X3),B),R2)
           => ~ aa_set_o_o(member_o((X3)),A) ) ) ).

% ImageE
tff(fact_525_ImageE,axiom,
    ! [B: int,R2: set_Pr8497134830871917726it_int,A: set_Product_unit] :
      ( aa_set_int_o(member_int(B),image_5655252347010435494it_int(R2,A))
     => ~ ! [X3: product_unit] :
            ( member5468524742225079679it_int(produc4800797859304479970it_int(X3,B),R2)
           => ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),A) ) ) ).

% ImageE
tff(fact_526_ImageE,axiom,
    ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
      ( aa_set_o_o(member_o((B)),image_nat_o(R2,A))
     => ~ ! [X3: nat] :
            ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(X3),(B)),R2)
           => ~ aa_set_nat_o(member_nat(X3),A) ) ) ).

% ImageE
tff(fact_527_ImageE,axiom,
    ! [B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
      ( aa_set_o_o(member_o((B)),image_int_o(R2,A))
     => ~ ! [X3: int] :
            ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(X3),(B)),R2)
           => ~ aa_set_int_o(member_int(X3),A) ) ) ).

% ImageE
tff(fact_528_ImageE,axiom,
    ! [B: $o,R2: set_Product_prod_o_o,A: set_o] :
      ( aa_set_o_o(member_o((B)),image_o_o(R2,A))
     => ~ ! [X3: $o] :
            ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X3)),(B)),R2)
           => ~ aa_set_o_o(member_o((X3)),A) ) ) ).

% ImageE
tff(fact_529_ImageE,axiom,
    ! [B: $o,R2: set_Pr5464989305201766866unit_o,A: set_Product_unit] :
      ( aa_set_o_o(member_o((B)),image_Product_unit_o(R2,A))
     => ~ ! [X3: product_unit] :
            ( member3689400097658526619unit_o(aa_o_P5708869953080425366unit_o(produc8487917018914037602unit_o(X3),(B)),R2)
           => ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),A) ) ) ).

% ImageE
tff(fact_530_refl__onD2,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( refl_o5094713388964525678_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A) ) ) ).

% refl_onD2
tff(fact_531_refl__onD2,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
      ( refl_on_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2)
       => aa_set_o_o(member_o((Y)),A) ) ) ).

% refl_onD2
tff(fact_532_refl__onD2,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
      ( refl_on_Product_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2)
       => aa_set8780210009671629607unit_o(member_Product_unit(Y),A) ) ) ).

% refl_onD2
tff(fact_533_refl__onD2,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( refl_on_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
       => aa_set_nat_o(member_nat(Y),A) ) ) ).

% refl_onD2
tff(fact_534_refl__onD2,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( refl_on_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
       => aa_set_int_o(member_int(Y),A) ) ) ).

% refl_onD2
tff(fact_535_refl__onD1,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( refl_o5094713388964525678_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A) ) ) ).

% refl_onD1
tff(fact_536_refl__onD1,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
      ( refl_on_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2)
       => aa_set_o_o(member_o((X)),A) ) ) ).

% refl_onD1
tff(fact_537_refl__onD1,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
      ( refl_on_Product_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2)
       => aa_set8780210009671629607unit_o(member_Product_unit(X),A) ) ) ).

% refl_onD1
tff(fact_538_refl__onD1,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( refl_on_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
       => aa_set_nat_o(member_nat(X),A) ) ) ).

% refl_onD1
tff(fact_539_refl__onD1,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( refl_on_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
       => aa_set_int_o(member_int(X),A) ) ) ).

% refl_onD1
tff(fact_540_refl__onD,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( refl_o5094713388964525678_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
       => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,A2),R2) ) ) ).

% refl_onD
tff(fact_541_refl__onD,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
      ( refl_on_o(A,R2)
     => ( aa_set_o_o(member_o((A2)),A)
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(A2)),R2) ) ) ).

% refl_onD
tff(fact_542_refl__onD,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
      ( refl_on_Product_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
       => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),A2),R2) ) ) ).

% refl_onD
tff(fact_543_refl__onD,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( refl_on_nat(A,R2)
     => ( aa_set_nat_o(member_nat(A2),A)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),R2) ) ) ).

% refl_onD
tff(fact_544_refl__onD,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
      ( refl_on_int(A,R2)
     => ( aa_set_int_o(member_int(A2),A)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),R2) ) ) ).

% refl_onD
tff(fact_545_total__on__def,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( total_on_nat(A,R2)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => ! [Xa2: nat] :
              ( aa_set_nat_o(member_nat(Xa2),A)
             => ( ( X2 != Xa2 )
               => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa2),R2)
                  | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),X2),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_546_total__on__def,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( total_on_int(A,R2)
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),A)
             => ( ( X2 != Xa2 )
               => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Xa2),R2)
                  | member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa2),X2),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_547_total__onI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
         => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
           => ( ( X3 != Y2 )
             => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X3,Y2),R2)
                | member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Y2,X3),R2) ) ) ) )
     => total_3645986146778848883_nat_o(A,R2) ) ).

% total_onI
tff(fact_548_total__onI,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o] :
      ( ! [X3: $o,Y2: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => ( aa_set_o_o(member_o((Y2)),A)
           => ( ~ ( (X3)
                <=> (Y2) )
             => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X3)),(Y2)),R2)
                | member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y2)),(X3)),R2) ) ) ) )
     => total_on_o(A,R2) ) ).

% total_onI
tff(fact_549_total__onI,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( ! [X3: product_unit,Y2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),A)
           => ( ( X3 != Y2 )
             => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X3),Y2),R2)
                | member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Y2),X3),R2) ) ) ) )
     => total_5665700600833547665t_unit(A,R2) ) ).

% total_onI
tff(fact_550_total__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_set_nat_o(member_nat(Y2),A)
           => ( ( X3 != Y2 )
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
                | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),X3),R2) ) ) ) )
     => total_on_nat(A,R2) ) ).

% total_onI
tff(fact_551_total__onI,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y2: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ( aa_set_int_o(member_int(Y2),A)
           => ( ( X3 != Y2 )
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
                | member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),X3),R2) ) ) ) )
     => total_on_int(A,R2) ) ).

% total_onI
tff(fact_552_total__on__empty,axiom,
    ! [R2: set_Pr5094982260447487303t_unit] : total_5665700600833547665t_unit(bot_bo3957492148770167129t_unit,R2) ).

% total_on_empty
tff(fact_553_total__on__empty,axiom,
    ! [R2: set_Product_prod_o_o] : total_on_o(bot_bot_set_o,R2) ).

% total_on_empty
tff(fact_554_total__on__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : total_on_nat(bot_bot_set_nat,R2) ).

% total_on_empty
tff(fact_555_total__on__empty,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : total_on_int(bot_bot_set_int,R2) ).

% total_on_empty
tff(fact_556_total__on__empty,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o] : total_3645986146778848883_nat_o(bot_bo8632255151191677115_nat_o,R2) ).

% total_on_empty
tff(fact_557_models__in__range,axiom,
    ! [P: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),Ha)
     => aa_Pro653082453696054616_nat_o(in_range,Ha) ) ).

% models_in_range
tff(fact_558_Id__on__iff,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),id_on_7912660205148244492_nat_o(A))
    <=> ( ( X = Y )
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A) ) ) ).

% Id_on_iff
tff(fact_559_Id__on__iff,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),id_on_o(A))
    <=> ( ( (X)
        <=> (Y) )
        & aa_set_o_o(member_o((X)),A) ) ) ).

% Id_on_iff
tff(fact_560_Id__on__iff,axiom,
    ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),id_on_Product_unit(A))
    <=> ( ( X = Y )
        & aa_set8780210009671629607unit_o(member_Product_unit(X),A) ) ) ).

% Id_on_iff
tff(fact_561_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),id_on_nat(A))
    <=> ( ( X = Y )
        & aa_set_nat_o(member_nat(X),A) ) ) ).

% Id_on_iff
tff(fact_562_Id__on__iff,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),id_on_int(A))
    <=> ( ( X = Y )
        & aa_set_int_o(member_int(X),A) ) ) ).

% Id_on_iff
tff(fact_563_Id__on__eqI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( A2 = B )
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
       => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),id_on_7912660205148244492_nat_o(A)) ) ) ).

% Id_on_eqI
tff(fact_564_Id__on__eqI,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( ( (A2)
      <=> (B) )
     => ( aa_set_o_o(member_o((A2)),A)
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),id_on_o(A)) ) ) ).

% Id_on_eqI
tff(fact_565_Id__on__eqI,axiom,
    ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
      ( ( A2 = B )
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
       => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),id_on_Product_unit(A)) ) ) ).

% Id_on_eqI
tff(fact_566_Id__on__eqI,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( A2 = B )
     => ( aa_set_nat_o(member_nat(A2),A)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),id_on_nat(A)) ) ) ).

% Id_on_eqI
tff(fact_567_Id__on__eqI,axiom,
    ! [A2: int,B: int,A: set_int] :
      ( ( A2 = B )
     => ( aa_set_int_o(member_int(A2),A)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),id_on_int(A)) ) ) ).

% Id_on_eqI
tff(fact_568_Id__onE,axiom,
    ! [C: produc6671579610312156647_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(C,id_on_7912660205148244492_nat_o(A))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
           => ( C != produc4714251849521319255_nat_o(X3,X3) ) ) ) ).

% Id_onE
tff(fact_569_Id__onE,axiom,
    ! [C: product_prod_o_o,A: set_o] :
      ( member7466972457876170832od_o_o(C,id_on_o(A))
     => ~ ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ( C != aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X3)),(X3)) ) ) ) ).

% Id_onE
tff(fact_570_Id__onE,axiom,
    ! [C: produc8459935480633519975t_unit,A: set_Product_unit] :
      ( member7821138191923116944t_unit(C,id_on_Product_unit(A))
     => ~ ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => ( C != aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X3),X3) ) ) ) ).

% Id_onE
tff(fact_571_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A: set_nat] :
      ( member8440522571783428010at_nat(C,id_on_nat(A))
     => ~ ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( C != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3) ) ) ) ).

% Id_onE
tff(fact_572_Id__onE,axiom,
    ! [C: product_prod_int_int,A: set_int] :
      ( member5262025264175285858nt_int(C,id_on_int(A))
     => ~ ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => ( C != aa_int7462565939943104719nt_int(product_Pair_int_int(X3),X3) ) ) ) ).

% Id_onE
tff(fact_573_lnear__order__on__empty,axiom,
    order_888282469287196226t_unit(bot_bo3957492148770167129t_unit,bot_bo1654022762305692595t_unit) ).

% lnear_order_on_empty
tff(fact_574_lnear__order__on__empty,axiom,
    order_6238756238976269133r_on_o(bot_bot_set_o,bot_bo7073875226086086771od_o_o) ).

% lnear_order_on_empty
tff(fact_575_lnear__order__on__empty,axiom,
    order_4473980167227706203on_nat(bot_bot_set_nat,bot_bo2099793752762293965at_nat) ).

% lnear_order_on_empty
tff(fact_576_lnear__order__on__empty,axiom,
    order_4471489696718655927on_int(bot_bot_set_int,bot_bo1796632182523588997nt_int) ).

% lnear_order_on_empty
tff(fact_577_lnear__order__on__empty,axiom,
    order_53248997263766692_nat_o(bot_bo8632255151191677115_nat_o,bot_bo6763312114214544691_nat_o) ).

% lnear_order_on_empty
tff(fact_578_refl__on__empty,axiom,
    refl_on_Product_unit(bot_bo3957492148770167129t_unit,bot_bo1654022762305692595t_unit) ).

% refl_on_empty
tff(fact_579_refl__on__empty,axiom,
    refl_on_o(bot_bot_set_o,bot_bo7073875226086086771od_o_o) ).

% refl_on_empty
tff(fact_580_refl__on__empty,axiom,
    refl_on_nat(bot_bot_set_nat,bot_bo2099793752762293965at_nat) ).

% refl_on_empty
tff(fact_581_refl__on__empty,axiom,
    refl_on_int(bot_bot_set_int,bot_bo1796632182523588997nt_int) ).

% refl_on_empty
tff(fact_582_refl__on__empty,axiom,
    refl_o5094713388964525678_nat_o(bot_bo8632255151191677115_nat_o,bot_bo6763312114214544691_nat_o) ).

% refl_on_empty
tff(fact_583_properD1,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
       => aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As)) ) ) ).

% properD1
tff(fact_584_proper__iff,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,As: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
       => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
         => ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
          <=> aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As)) ) ) ) ) ).

% proper_iff
tff(fact_585_proper__def,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
    <=> ! [H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H3),As3))
           => aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H3),As3)) )
          & ( ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H3),As3))
              & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As3),H3),H4)
              & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As3)) )
           => aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As3)) ) ) ) ).

% proper_def
tff(fact_586_properD2,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,As: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
       => ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
         => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
           => aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As)) ) ) ) ) ).

% properD2
tff(fact_587_properI,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o] :
      ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit] :
          ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
         => aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)) )
     => ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
            ( aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
           => ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H5)
             => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As2))
               => aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As2)) ) ) )
       => aa_fun1675921005893230629at_o_o(proper,P) ) ) ).

% properI
tff(fact_588_pair__vimage__is__Image,axiom,
    ! [U: heap_e7401611519738050253t_unit,E: set_Pr3948176798113811640et_nat] : vimage5412618565253405765et_nat(produc7507926704131184380et_nat(U),E) = image_3942343239045813184et_nat(E,insert8563697753282147485t_unit(U,bot_bo3686245532977126937t_unit)) ).

% pair_vimage_is_Image
tff(fact_589_pair__vimage__is__Image,axiom,
    ! [U: nat,E: set_Pr1261947904930325089at_nat] : vimage8013328719654469172at_nat(product_Pair_nat_nat(U),E) = aa_set_nat_set_nat(image_nat_nat(E),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,U),bot_bot_set_nat)) ).

% pair_vimage_is_Image
tff(fact_590_pair__vimage__is__Image,axiom,
    ! [U: int,E: set_Pr958786334691620121nt_int] : vimage7872673496510460816nt_int(product_Pair_int_int(U),E) = image_int_int(E,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,U),bot_bot_set_int)) ).

% pair_vimage_is_Image
tff(fact_591_pair__vimage__is__Image,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o,E: set_Pr1312406793078187350et_nat] : vimage1008269277602427487et_nat(produc6017135503889078682et_nat(U),E) = image_552160923151422558et_nat(E,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(U),bot_bo8632255151191677115_nat_o)) ).

% pair_vimage_is_Image
tff(fact_592_pair__vimage__is__Image,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o,E: set_Pr4040394263813967220et_nat] : vimage6782145143455462555et_nat(produc7291855603632349880et_nat(U),E) = image_2192608224838851452et_nat(E,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(U),bot_bo8632255151191677115_nat_o)) ).

% pair_vimage_is_Image
tff(fact_593_Range__insert,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] : range_9132311542094983838et_nat(insert1271470778893645362et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R2)) = insert_set_nat(B,range_9132311542094983838et_nat(R2)) ).

% Range_insert
tff(fact_594_Range__insert,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] : range_nat_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),range_nat_nat(R2)) ).

% Range_insert
tff(fact_595_Range__insert,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] : range_int_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),range_int_int(R2)) ).

% Range_insert
tff(fact_596_Range__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] : range_3211142380271236156et_nat(insert663245765287195472et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)) = insert1271470778893645362et_nat(B,range_3211142380271236156et_nat(R2)) ).

% Range_insert
tff(fact_597_Range__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] : range_1185527156928805978et_nat(insert727258735945777134et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)) = insert663245765287195472et_nat(B,range_1185527156928805978et_nat(R2)) ).

% Range_insert
tff(fact_598_Domain__insert,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] : domain2007548326387822151et_nat(insert1271470778893645362et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R2)) = insert8563697753282147485t_unit(A2,domain2007548326387822151et_nat(R2)) ).

% Domain_insert
tff(fact_599_Domain__insert,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] : domain_nat_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),domain_nat_nat(R2)) ).

% Domain_insert
tff(fact_600_Domain__insert,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] : domain_int_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),domain_int_int(R2)) ).

% Domain_insert
tff(fact_601_Domain__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] : domain2953816653896713061et_nat(insert663245765287195472et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),domain2953816653896713061et_nat(R2)) ).

% Domain_insert
tff(fact_602_Domain__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] : domain5137017734253780739et_nat(insert727258735945777134et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),domain5137017734253780739et_nat(R2)) ).

% Domain_insert
tff(fact_603_preorder__on__empty,axiom,
    order_7534304569044981540t_unit(bot_bo3957492148770167129t_unit,bot_bo1654022762305692595t_unit) ).

% preorder_on_empty
tff(fact_604_preorder__on__empty,axiom,
    order_preorder_on_o(bot_bot_set_o,bot_bo7073875226086086771od_o_o) ).

% preorder_on_empty
tff(fact_605_preorder__on__empty,axiom,
    order_4861654808422542329on_nat(bot_bot_set_nat,bot_bo2099793752762293965at_nat) ).

% preorder_on_empty
tff(fact_606_preorder__on__empty,axiom,
    order_4859164337913492053on_int(bot_bot_set_int,bot_bo1796632182523588997nt_int) ).

% preorder_on_empty
tff(fact_607_preorder__on__empty,axiom,
    order_4728425272145416326_nat_o(bot_bo8632255151191677115_nat_o,bot_bo6763312114214544691_nat_o) ).

% preorder_on_empty
tff(fact_608_trancl__single,axiom,
    ! [A2: nat,B: nat] : transi6264000038957366511cl_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),bot_bo2099793752762293965at_nat) ).

% trancl_single
tff(fact_609_trancl__single,axiom,
    ! [A2: int,B: int] : transi6261509568448316235cl_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),bot_bo1796632182523588997nt_int)) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),bot_bo1796632182523588997nt_int) ).

% trancl_single
tff(fact_610_trans__singleton,axiom,
    ! [A2: nat] : trans_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),bot_bo2099793752762293965at_nat)) ).

% trans_singleton
tff(fact_611_trans__singleton,axiom,
    ! [A2: int] : trans_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),bot_bo1796632182523588997nt_int)) ).

% trans_singleton
tff(fact_612_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(syntax7398250324933576852n_assn,aa_assn_assn(times_times_assn(X),Y)),A2)
     => ( aa_assn_assn(times_times_assn(A2),B) = aa_assn_assn(times_times_assn(B),A2) ) ) ).

% mult.safe_commute
tff(fact_613_mult_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(syntax3730441303064801268at_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),A2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A2) ) ) ).

% mult.safe_commute
tff(fact_614_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A2) ) ) ).

% mult.safe_commute
tff(fact_615_mult_Osafe__commute,axiom,
    ! [X: int,Y: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),A2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A2) ) ) ).

% mult.safe_commute
tff(fact_616_vimage__eq,axiom,
    ! [A2: nat,F: fun_nat_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(vimage_nat_nat(F),B4))
    <=> aa_set_nat_o(member_nat(aa_nat_nat(F,A2)),B4) ) ).

% vimage_eq
tff(fact_617_vimage__eq,axiom,
    ! [A2: nat,F: fun_nat_int,B4: set_int] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_int(F,B4))
    <=> aa_set_int_o(member_int(aa_nat_int(F,A2)),B4) ) ).

% vimage_eq
tff(fact_618_vimage__eq,axiom,
    ! [A2: nat,F: fun_nat_o,B4: set_o] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_o(F,B4))
    <=> aa_set_o_o(member_o(aa_nat_o(F,A2)),B4) ) ).

% vimage_eq
tff(fact_619_vimage__eq,axiom,
    ! [A2: nat,F: fun_nat_Product_unit,B4: set_Product_unit] :
      ( aa_set_nat_o(member_nat(A2),vimage4884490618288580032t_unit(F,B4))
    <=> aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(F,A2)),B4) ) ).

% vimage_eq
tff(fact_620_vimage__eq,axiom,
    ! [A2: int,F: fun_int_nat,B4: set_nat] :
      ( aa_set_int_o(member_int(A2),vimage_int_nat(F,B4))
    <=> aa_set_nat_o(member_nat(aa_int_nat(F,A2)),B4) ) ).

% vimage_eq
tff(fact_621_vimage__eq,axiom,
    ! [A2: int,F: fun_int_int,B4: set_int] :
      ( aa_set_int_o(member_int(A2),vimage_int_int(F,B4))
    <=> aa_set_int_o(member_int(aa_int_int(F,A2)),B4) ) ).

% vimage_eq
tff(fact_622_vimage__eq,axiom,
    ! [A2: int,F: fun_int_o,B4: set_o] :
      ( aa_set_int_o(member_int(A2),vimage_int_o(F,B4))
    <=> aa_set_o_o(member_o(aa_int_o(F,A2)),B4) ) ).

% vimage_eq
tff(fact_623_vimage__eq,axiom,
    ! [A2: int,F: fun_int_Product_unit,B4: set_Product_unit] :
      ( aa_set_int_o(member_int(A2),vimage2000912565490760292t_unit(F,B4))
    <=> aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(F,A2)),B4) ) ).

% vimage_eq
tff(fact_624_vimage__eq,axiom,
    ! [A2: $o,F: fun_o_nat,B4: set_nat] :
      ( aa_set_o_o(member_o((A2)),vimage_o_nat(F,B4))
    <=> aa_set_nat_o(member_nat(aa_o_nat(F,(A2))),B4) ) ).

% vimage_eq
tff(fact_625_vimage__eq,axiom,
    ! [A2: $o,F: fun_o_int,B4: set_int] :
      ( aa_set_o_o(member_o((A2)),vimage_o_int(F,B4))
    <=> aa_set_int_o(member_int(aa_o_int(F,(A2))),B4) ) ).

% vimage_eq
tff(fact_626_vimageI,axiom,
    ! [F: fun_nat_nat,A2: nat,B: nat,B4: set_nat] :
      ( ( aa_nat_nat(F,A2) = B )
     => ( aa_set_nat_o(member_nat(B),B4)
       => aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ) ) ).

% vimageI
tff(fact_627_vimageI,axiom,
    ! [F: fun_int_nat,A2: int,B: nat,B4: set_nat] :
      ( ( aa_int_nat(F,A2) = B )
     => ( aa_set_nat_o(member_nat(B),B4)
       => aa_set_int_o(member_int(A2),vimage_int_nat(F,B4)) ) ) ).

% vimageI
tff(fact_628_vimageI,axiom,
    ! [F: fun_o_nat,A2: $o,B: nat,B4: set_nat] :
      ( ( aa_o_nat(F,(A2)) = B )
     => ( aa_set_nat_o(member_nat(B),B4)
       => aa_set_o_o(member_o((A2)),vimage_o_nat(F,B4)) ) ) ).

% vimageI
tff(fact_629_vimageI,axiom,
    ! [F: fun_Product_unit_nat,A2: product_unit,B: nat,B4: set_nat] :
      ( ( aa_Product_unit_nat(F,A2) = B )
     => ( aa_set_nat_o(member_nat(B),B4)
       => aa_set8780210009671629607unit_o(member_Product_unit(A2),vimage6253328473476588386it_nat(F,B4)) ) ) ).

% vimageI
tff(fact_630_vimageI,axiom,
    ! [F: fun_nat_int,A2: nat,B: int,B4: set_int] :
      ( ( aa_nat_int(F,A2) = B )
     => ( aa_set_int_o(member_int(B),B4)
       => aa_set_nat_o(member_nat(A2),vimage_nat_int(F,B4)) ) ) ).

% vimageI
tff(fact_631_vimageI,axiom,
    ! [F: fun_int_int,A2: int,B: int,B4: set_int] :
      ( ( aa_int_int(F,A2) = B )
     => ( aa_set_int_o(member_int(B),B4)
       => aa_set_int_o(member_int(A2),vimage_int_int(F,B4)) ) ) ).

% vimageI
tff(fact_632_vimageI,axiom,
    ! [F: fun_o_int,A2: $o,B: int,B4: set_int] :
      ( ( aa_o_int(F,(A2)) = B )
     => ( aa_set_int_o(member_int(B),B4)
       => aa_set_o_o(member_o((A2)),vimage_o_int(F,B4)) ) ) ).

% vimageI
tff(fact_633_vimageI,axiom,
    ! [F: fun_Product_unit_int,A2: product_unit,B: int,B4: set_int] :
      ( ( aa_Product_unit_int(F,A2) = B )
     => ( aa_set_int_o(member_int(B),B4)
       => aa_set8780210009671629607unit_o(member_Product_unit(A2),vimage6250838002967538110it_int(F,B4)) ) ) ).

% vimageI
tff(fact_634_vimageI,axiom,
    ! [F: fun_nat_o,A2: nat,B: $o,B4: set_o] :
      ( ( aa_nat_o(F,A2)
      <=> (B) )
     => ( aa_set_o_o(member_o((B)),B4)
       => aa_set_nat_o(member_nat(A2),vimage_nat_o(F,B4)) ) ) ).

% vimageI
tff(fact_635_vimageI,axiom,
    ! [F: fun_int_o,A2: int,B: $o,B4: set_o] :
      ( ( aa_int_o(F,A2)
      <=> (B) )
     => ( aa_set_o_o(member_o((B)),B4)
       => aa_set_int_o(member_int(A2),vimage_int_o(F,B4)) ) ) ).

% vimageI
tff(fact_636_vimage__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit] : vimage7995052115951654139t_unit(F,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% vimage_empty
tff(fact_637_vimage__empty,axiom,
    ! [F: fun_o_Product_unit] : vimage4222377557387472176t_unit(F,bot_bo3957492148770167129t_unit) = bot_bot_set_o ).

% vimage_empty
tff(fact_638_vimage__empty,axiom,
    ! [F: fun_nat_Product_unit] : vimage4884490618288580032t_unit(F,bot_bo3957492148770167129t_unit) = bot_bot_set_nat ).

% vimage_empty
tff(fact_639_vimage__empty,axiom,
    ! [F: fun_int_Product_unit] : vimage2000912565490760292t_unit(F,bot_bo3957492148770167129t_unit) = bot_bot_set_int ).

% vimage_empty
tff(fact_640_vimage__empty,axiom,
    ! [F: fun_Product_unit_o] : vimage8808348054653608838unit_o(F,bot_bot_set_o) = bot_bo3957492148770167129t_unit ).

% vimage_empty
tff(fact_641_vimage__empty,axiom,
    ! [F: fun_o_o] : vimage_o_o(F,bot_bot_set_o) = bot_bot_set_o ).

% vimage_empty
tff(fact_642_vimage__empty,axiom,
    ! [F: fun_nat_o] : vimage_nat_o(F,bot_bot_set_o) = bot_bot_set_nat ).

% vimage_empty
tff(fact_643_vimage__empty,axiom,
    ! [F: fun_int_o] : vimage_int_o(F,bot_bot_set_o) = bot_bot_set_int ).

% vimage_empty
tff(fact_644_vimage__empty,axiom,
    ! [F: fun_Product_unit_nat] : vimage6253328473476588386it_nat(F,bot_bot_set_nat) = bot_bo3957492148770167129t_unit ).

% vimage_empty
tff(fact_645_vimage__empty,axiom,
    ! [F: fun_o_nat] : vimage_o_nat(F,bot_bot_set_nat) = bot_bot_set_o ).

% vimage_empty
tff(fact_646_relH__sym,axiom,
    ! [As: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
     => aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),H2),Ha) ) ).

% relH_sym
tff(fact_647_relH__trans,axiom,
    ! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),H12),H23)
     => ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),H23),H32)
       => aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),H12),H32) ) ) ).

% relH_trans
tff(fact_648_vimage__Collect,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,aa_nat_nat(F,X3))
        <=> aa_nat_o(Q,X3) )
     => ( aa_set_nat_set_nat(vimage_nat_nat(F),aa_fun_nat_o_set_nat(collect_nat,P)) = aa_fun_nat_o_set_nat(collect_nat,Q) ) ) ).

% vimage_Collect
tff(fact_649_vimage__Collect,axiom,
    ! [P: fun_nat_o,F: fun_int_nat,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_nat_o(P,aa_int_nat(F,X3))
        <=> aa_int_o(Q,X3) )
     => ( vimage_int_nat(F,aa_fun_nat_o_set_nat(collect_nat,P)) = aa_fun_int_o_set_int(collect_int,Q) ) ) ).

% vimage_Collect
tff(fact_650_vimage__Collect,axiom,
    ! [P: fun_nat_o,F: fun_o_nat,Q: fun_o_o] :
      ( ! [X3: $o] :
          ( aa_nat_o(P,aa_o_nat(F,(X3)))
        <=> aa_o_o(Q,(X3)) )
     => ( vimage_o_nat(F,aa_fun_nat_o_set_nat(collect_nat,P)) = aa_fun_o_o_set_o(collect_o,Q) ) ) ).

% vimage_Collect
tff(fact_651_vimage__Collect,axiom,
    ! [P: fun_nat_o,F: fun_Product_unit_nat,Q: fun_Product_unit_o] :
      ( ! [X3: product_unit] :
          ( aa_nat_o(P,aa_Product_unit_nat(F,X3))
        <=> aa_Product_unit_o(Q,X3) )
     => ( vimage6253328473476588386it_nat(F,aa_fun_nat_o_set_nat(collect_nat,P)) = aa_fun8337552879598298401t_unit(collect_Product_unit,Q) ) ) ).

% vimage_Collect
tff(fact_652_vimage__Collect,axiom,
    ! [P: fun_int_o,F: fun_nat_int,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_int_o(P,aa_nat_int(F,X3))
        <=> aa_nat_o(Q,X3) )
     => ( vimage_nat_int(F,aa_fun_int_o_set_int(collect_int,P)) = aa_fun_nat_o_set_nat(collect_nat,Q) ) ) ).

% vimage_Collect
tff(fact_653_vimage__Collect,axiom,
    ! [P: fun_int_o,F: fun_int_int,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,aa_int_int(F,X3))
        <=> aa_int_o(Q,X3) )
     => ( vimage_int_int(F,aa_fun_int_o_set_int(collect_int,P)) = aa_fun_int_o_set_int(collect_int,Q) ) ) ).

% vimage_Collect
tff(fact_654_vimage__Collect,axiom,
    ! [P: fun_int_o,F: fun_o_int,Q: fun_o_o] :
      ( ! [X3: $o] :
          ( aa_int_o(P,aa_o_int(F,(X3)))
        <=> aa_o_o(Q,(X3)) )
     => ( vimage_o_int(F,aa_fun_int_o_set_int(collect_int,P)) = aa_fun_o_o_set_o(collect_o,Q) ) ) ).

% vimage_Collect
tff(fact_655_vimage__Collect,axiom,
    ! [P: fun_int_o,F: fun_Product_unit_int,Q: fun_Product_unit_o] :
      ( ! [X3: product_unit] :
          ( aa_int_o(P,aa_Product_unit_int(F,X3))
        <=> aa_Product_unit_o(Q,X3) )
     => ( vimage6250838002967538110it_int(F,aa_fun_int_o_set_int(collect_int,P)) = aa_fun8337552879598298401t_unit(collect_Product_unit,Q) ) ) ).

% vimage_Collect
tff(fact_656_vimage__Collect,axiom,
    ! [P: fun_o_o,F: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_o_o(P,aa_nat_o(F,X3))
        <=> aa_nat_o(Q,X3) )
     => ( vimage_nat_o(F,aa_fun_o_o_set_o(collect_o,P)) = aa_fun_nat_o_set_nat(collect_nat,Q) ) ) ).

% vimage_Collect
tff(fact_657_vimage__Collect,axiom,
    ! [P: fun_o_o,F: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_o_o(P,aa_int_o(F,X3))
        <=> aa_int_o(Q,X3) )
     => ( vimage_int_o(F,aa_fun_o_o_set_o(collect_o,P)) = aa_fun_int_o_set_int(collect_int,Q) ) ) ).

% vimage_Collect
tff(fact_658_vimageI2,axiom,
    ! [F: fun_nat_nat,A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(F,A2)),A)
     => aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(vimage_nat_nat(F),A)) ) ).

% vimageI2
tff(fact_659_vimageI2,axiom,
    ! [F: fun_int_nat,A2: int,A: set_nat] :
      ( aa_set_nat_o(member_nat(aa_int_nat(F,A2)),A)
     => aa_set_int_o(member_int(A2),vimage_int_nat(F,A)) ) ).

% vimageI2
tff(fact_660_vimageI2,axiom,
    ! [F: fun_o_nat,A2: $o,A: set_nat] :
      ( aa_set_nat_o(member_nat(aa_o_nat(F,(A2))),A)
     => aa_set_o_o(member_o((A2)),vimage_o_nat(F,A)) ) ).

% vimageI2
tff(fact_661_vimageI2,axiom,
    ! [F: fun_Product_unit_nat,A2: product_unit,A: set_nat] :
      ( aa_set_nat_o(member_nat(aa_Product_unit_nat(F,A2)),A)
     => aa_set8780210009671629607unit_o(member_Product_unit(A2),vimage6253328473476588386it_nat(F,A)) ) ).

% vimageI2
tff(fact_662_vimageI2,axiom,
    ! [F: fun_nat_int,A2: nat,A: set_int] :
      ( aa_set_int_o(member_int(aa_nat_int(F,A2)),A)
     => aa_set_nat_o(member_nat(A2),vimage_nat_int(F,A)) ) ).

% vimageI2
tff(fact_663_vimageI2,axiom,
    ! [F: fun_int_int,A2: int,A: set_int] :
      ( aa_set_int_o(member_int(aa_int_int(F,A2)),A)
     => aa_set_int_o(member_int(A2),vimage_int_int(F,A)) ) ).

% vimageI2
tff(fact_664_vimageI2,axiom,
    ! [F: fun_o_int,A2: $o,A: set_int] :
      ( aa_set_int_o(member_int(aa_o_int(F,(A2))),A)
     => aa_set_o_o(member_o((A2)),vimage_o_int(F,A)) ) ).

% vimageI2
tff(fact_665_vimageI2,axiom,
    ! [F: fun_Product_unit_int,A2: product_unit,A: set_int] :
      ( aa_set_int_o(member_int(aa_Product_unit_int(F,A2)),A)
     => aa_set8780210009671629607unit_o(member_Product_unit(A2),vimage6250838002967538110it_int(F,A)) ) ).

% vimageI2
tff(fact_666_vimageI2,axiom,
    ! [F: fun_nat_o,A2: nat,A: set_o] :
      ( aa_set_o_o(member_o(aa_nat_o(F,A2)),A)
     => aa_set_nat_o(member_nat(A2),vimage_nat_o(F,A)) ) ).

% vimageI2
tff(fact_667_vimageI2,axiom,
    ! [F: fun_int_o,A2: int,A: set_o] :
      ( aa_set_o_o(member_o(aa_int_o(F,A2)),A)
     => aa_set_int_o(member_int(A2),vimage_int_o(F,A)) ) ).

% vimageI2
tff(fact_668_vimageE,axiom,
    ! [A2: nat,F: fun_nat_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(vimage_nat_nat(F),B4))
     => aa_set_nat_o(member_nat(aa_nat_nat(F,A2)),B4) ) ).

% vimageE
tff(fact_669_vimageE,axiom,
    ! [A2: nat,F: fun_nat_int,B4: set_int] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_int(F,B4))
     => aa_set_int_o(member_int(aa_nat_int(F,A2)),B4) ) ).

% vimageE
tff(fact_670_vimageE,axiom,
    ! [A2: nat,F: fun_nat_o,B4: set_o] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_o(F,B4))
     => aa_set_o_o(member_o(aa_nat_o(F,A2)),B4) ) ).

% vimageE
tff(fact_671_vimageE,axiom,
    ! [A2: nat,F: fun_nat_Product_unit,B4: set_Product_unit] :
      ( aa_set_nat_o(member_nat(A2),vimage4884490618288580032t_unit(F,B4))
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(F,A2)),B4) ) ).

% vimageE
tff(fact_672_vimageE,axiom,
    ! [A2: int,F: fun_int_nat,B4: set_nat] :
      ( aa_set_int_o(member_int(A2),vimage_int_nat(F,B4))
     => aa_set_nat_o(member_nat(aa_int_nat(F,A2)),B4) ) ).

% vimageE
tff(fact_673_vimageE,axiom,
    ! [A2: int,F: fun_int_int,B4: set_int] :
      ( aa_set_int_o(member_int(A2),vimage_int_int(F,B4))
     => aa_set_int_o(member_int(aa_int_int(F,A2)),B4) ) ).

% vimageE
tff(fact_674_vimageE,axiom,
    ! [A2: int,F: fun_int_o,B4: set_o] :
      ( aa_set_int_o(member_int(A2),vimage_int_o(F,B4))
     => aa_set_o_o(member_o(aa_int_o(F,A2)),B4) ) ).

% vimageE
tff(fact_675_vimageE,axiom,
    ! [A2: int,F: fun_int_Product_unit,B4: set_Product_unit] :
      ( aa_set_int_o(member_int(A2),vimage2000912565490760292t_unit(F,B4))
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(F,A2)),B4) ) ).

% vimageE
tff(fact_676_vimageE,axiom,
    ! [A2: $o,F: fun_o_nat,B4: set_nat] :
      ( aa_set_o_o(member_o((A2)),vimage_o_nat(F,B4))
     => aa_set_nat_o(member_nat(aa_o_nat(F,(A2))),B4) ) ).

% vimageE
tff(fact_677_vimageE,axiom,
    ! [A2: $o,F: fun_o_int,B4: set_int] :
      ( aa_set_o_o(member_o((A2)),vimage_o_int(F,B4))
     => aa_set_int_o(member_int(aa_o_int(F,(A2))),B4) ) ).

% vimageE
tff(fact_678_vimageD,axiom,
    ! [A2: nat,F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(vimage_nat_nat(F),A))
     => aa_set_nat_o(member_nat(aa_nat_nat(F,A2)),A) ) ).

% vimageD
tff(fact_679_vimageD,axiom,
    ! [A2: nat,F: fun_nat_int,A: set_int] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_int(F,A))
     => aa_set_int_o(member_int(aa_nat_int(F,A2)),A) ) ).

% vimageD
tff(fact_680_vimageD,axiom,
    ! [A2: nat,F: fun_nat_o,A: set_o] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_o(F,A))
     => aa_set_o_o(member_o(aa_nat_o(F,A2)),A) ) ).

% vimageD
tff(fact_681_vimageD,axiom,
    ! [A2: nat,F: fun_nat_Product_unit,A: set_Product_unit] :
      ( aa_set_nat_o(member_nat(A2),vimage4884490618288580032t_unit(F,A))
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(F,A2)),A) ) ).

% vimageD
tff(fact_682_vimageD,axiom,
    ! [A2: int,F: fun_int_nat,A: set_nat] :
      ( aa_set_int_o(member_int(A2),vimage_int_nat(F,A))
     => aa_set_nat_o(member_nat(aa_int_nat(F,A2)),A) ) ).

% vimageD
tff(fact_683_vimageD,axiom,
    ! [A2: int,F: fun_int_int,A: set_int] :
      ( aa_set_int_o(member_int(A2),vimage_int_int(F,A))
     => aa_set_int_o(member_int(aa_int_int(F,A2)),A) ) ).

% vimageD
tff(fact_684_vimageD,axiom,
    ! [A2: int,F: fun_int_o,A: set_o] :
      ( aa_set_int_o(member_int(A2),vimage_int_o(F,A))
     => aa_set_o_o(member_o(aa_int_o(F,A2)),A) ) ).

% vimageD
tff(fact_685_vimageD,axiom,
    ! [A2: int,F: fun_int_Product_unit,A: set_Product_unit] :
      ( aa_set_int_o(member_int(A2),vimage2000912565490760292t_unit(F,A))
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(F,A2)),A) ) ).

% vimageD
tff(fact_686_vimageD,axiom,
    ! [A2: $o,F: fun_o_nat,A: set_nat] :
      ( aa_set_o_o(member_o((A2)),vimage_o_nat(F,A))
     => aa_set_nat_o(member_nat(aa_o_nat(F,(A2))),A) ) ).

% vimageD
tff(fact_687_vimageD,axiom,
    ! [A2: $o,F: fun_o_int,A: set_int] :
      ( aa_set_o_o(member_o((A2)),vimage_o_int(F,A))
     => aa_set_int_o(member_int(aa_o_int(F,(A2))),A) ) ).

% vimageD
tff(fact_688_transD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( trans_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),R2) ) ) ) ).

% transD
tff(fact_689_transD,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
      ( trans_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),R2) ) ) ) ).

% transD
tff(fact_690_transE,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( trans_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),R2) ) ) ) ).

% transE
tff(fact_691_transE,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
      ( trans_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),R2) ) ) ) ).

% transE
tff(fact_692_transI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat,Z3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),R2)
           => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Z3),R2) ) )
     => trans_nat(R2) ) ).

% transI
tff(fact_693_transI,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y2: int,Z3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),R2)
           => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Z3),R2) ) )
     => trans_int(R2) ) ).

% transI
tff(fact_694_trans__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( trans_nat(R2)
    <=> ! [X2: nat,Y3: nat,Z2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y3),R2)
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z2),R2)
           => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Z2),R2) ) ) ) ).

% trans_def
tff(fact_695_trans__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( trans_int(R2)
    <=> ! [X2: int,Y3: int,Z2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y3),R2)
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y3),Z2),R2)
           => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Z2),R2) ) ) ) ).

% trans_def
tff(fact_696_Domain_Ocases,axiom,
    ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member242389114213480694t_unit(A2,domain2007548326387822151et_nat(R2))
     => ~ ! [B2: set_nat] : ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B2),R2) ) ).

% Domain.cases
tff(fact_697_Domain_Ocases,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),domain_nat_nat(R2))
     => ~ ! [B2: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),R2) ) ).

% Domain.cases
tff(fact_698_Domain_Ocases,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),domain_int_int(R2))
     => ~ ! [B2: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),R2) ) ).

% Domain.cases
tff(fact_699_Domain_Ocases,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain2953816653896713061et_nat(R2))
     => ~ ! [B2: produc3658429121746597890et_nat] : ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B2),R2) ) ).

% Domain.cases
tff(fact_700_Domain_Ocases,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain5137017734253780739et_nat(R2))
     => ~ ! [B2: produc8799533983703829792et_nat] : ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B2),R2) ) ).

% Domain.cases
tff(fact_701_Domain_Osimps,axiom,
    ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member242389114213480694t_unit(A2,domain2007548326387822151et_nat(R2))
    <=> ? [A5: heap_e7401611519738050253t_unit,B6: set_nat] :
          ( ( A2 = A5 )
          & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A5),B6),R2) ) ) ).

% Domain.simps
tff(fact_702_Domain_Osimps,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),domain_nat_nat(R2))
    <=> ? [A5: nat,B6: nat] :
          ( ( A2 = A5 )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),R2) ) ) ).

% Domain.simps
tff(fact_703_Domain_Osimps,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),domain_int_int(R2))
    <=> ? [A5: int,B6: int] :
          ( ( A2 = A5 )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),R2) ) ) ).

% Domain.simps
tff(fact_704_Domain_Osimps,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain2953816653896713061et_nat(R2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc3658429121746597890et_nat] :
          ( ( A2 = A5 )
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B6),R2) ) ) ).

% Domain.simps
tff(fact_705_Domain_Osimps,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain5137017734253780739et_nat(R2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc8799533983703829792et_nat] :
          ( ( A2 = A5 )
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B6),R2) ) ) ).

% Domain.simps
tff(fact_706_Domain_ODomainI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R2)
     => member242389114213480694t_unit(A2,domain2007548326387822151et_nat(R2)) ) ).

% Domain.DomainI
tff(fact_707_Domain_ODomainI,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => aa_set_nat_o(member_nat(A2),domain_nat_nat(R2)) ) ).

% Domain.DomainI
tff(fact_708_Domain_ODomainI,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => aa_set_int_o(member_int(A2),domain_int_int(R2)) ) ).

% Domain.DomainI
tff(fact_709_Domain_ODomainI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain2953816653896713061et_nat(R2)) ) ).

% Domain.DomainI
tff(fact_710_Domain_ODomainI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain5137017734253780739et_nat(R2)) ) ).

% Domain.DomainI
tff(fact_711_DomainE,axiom,
    ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member242389114213480694t_unit(A2,domain2007548326387822151et_nat(R2))
     => ~ ! [B2: set_nat] : ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B2),R2) ) ).

% DomainE
tff(fact_712_DomainE,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),domain_nat_nat(R2))
     => ~ ! [B2: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),R2) ) ).

% DomainE
tff(fact_713_DomainE,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),domain_int_int(R2))
     => ~ ! [B2: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),R2) ) ).

% DomainE
tff(fact_714_DomainE,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain2953816653896713061et_nat(R2))
     => ~ ! [B2: produc3658429121746597890et_nat] : ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B2),R2) ) ).

% DomainE
tff(fact_715_DomainE,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain5137017734253780739et_nat(R2))
     => ~ ! [B2: produc8799533983703829792et_nat] : ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B2),R2) ) ).

% DomainE
tff(fact_716_Domain__iff,axiom,
    ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member242389114213480694t_unit(A2,domain2007548326387822151et_nat(R2))
    <=> ? [Y3: set_nat] : member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),Y3),R2) ) ).

% Domain_iff
tff(fact_717_Domain__iff,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),domain_nat_nat(R2))
    <=> ? [Y3: nat] : member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y3),R2) ) ).

% Domain_iff
tff(fact_718_Domain__iff,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),domain_int_int(R2))
    <=> ? [Y3: int] : member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),Y3),R2) ) ).

% Domain_iff
tff(fact_719_Domain__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain2953816653896713061et_nat(R2))
    <=> ? [Y3: produc3658429121746597890et_nat] : member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),Y3),R2) ) ).

% Domain_iff
tff(fact_720_Domain__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),domain5137017734253780739et_nat(R2))
    <=> ? [Y3: produc8799533983703829792et_nat] : member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),Y3),R2) ) ).

% Domain_iff
tff(fact_721_Range_Ocases,axiom,
    ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member_set_nat(A2,range_9132311542094983838et_nat(R2))
     => ~ ! [A3: heap_e7401611519738050253t_unit] : ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),A2),R2) ) ).

% Range.cases
tff(fact_722_Range_Ocases,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),range_nat_nat(R2))
     => ~ ! [A3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),A2),R2) ) ).

% Range.cases
tff(fact_723_Range_Ocases,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),range_int_int(R2))
     => ~ ! [A3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),A2),R2) ) ).

% Range.cases
tff(fact_724_Range_Ocases,axiom,
    ! [A2: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member6260224972018164377et_nat(A2,range_3211142380271236156et_nat(R2))
     => ~ ! [A3: fun_Pr6460944699146269935_nat_o] : ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),A2),R2) ) ).

% Range.cases
tff(fact_725_Range_Ocases,axiom,
    ! [A2: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member4513763875858039863et_nat(A2,range_1185527156928805978et_nat(R2))
     => ~ ! [A3: fun_Pr6460944699146269935_nat_o] : ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),A2),R2) ) ).

% Range.cases
tff(fact_726_Range_Osimps,axiom,
    ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member_set_nat(A2,range_9132311542094983838et_nat(R2))
    <=> ? [A5: heap_e7401611519738050253t_unit,B6: set_nat] :
          ( ( A2 = B6 )
          & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A5),B6),R2) ) ) ).

% Range.simps
tff(fact_727_Range_Osimps,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),range_nat_nat(R2))
    <=> ? [A5: nat,B6: nat] :
          ( ( A2 = B6 )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),R2) ) ) ).

% Range.simps
tff(fact_728_Range_Osimps,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),range_int_int(R2))
    <=> ? [A5: int,B6: int] :
          ( ( A2 = B6 )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),R2) ) ) ).

% Range.simps
tff(fact_729_Range_Osimps,axiom,
    ! [A2: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member6260224972018164377et_nat(A2,range_3211142380271236156et_nat(R2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc3658429121746597890et_nat] :
          ( ( A2 = B6 )
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B6),R2) ) ) ).

% Range.simps
tff(fact_730_Range_Osimps,axiom,
    ! [A2: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member4513763875858039863et_nat(A2,range_1185527156928805978et_nat(R2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc8799533983703829792et_nat] :
          ( ( A2 = B6 )
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B6),R2) ) ) ).

% Range.simps
tff(fact_731_Range_Ointros,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R2)
     => member_set_nat(B,range_9132311542094983838et_nat(R2)) ) ).

% Range.intros
tff(fact_732_Range_Ointros,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => aa_set_nat_o(member_nat(B),range_nat_nat(R2)) ) ).

% Range.intros
tff(fact_733_Range_Ointros,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => aa_set_int_o(member_int(B),range_int_int(R2)) ) ).

% Range.intros
tff(fact_734_Range_Ointros,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)
     => member6260224972018164377et_nat(B,range_3211142380271236156et_nat(R2)) ) ).

% Range.intros
tff(fact_735_Range_Ointros,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)
     => member4513763875858039863et_nat(B,range_1185527156928805978et_nat(R2)) ) ).

% Range.intros
tff(fact_736_RangeE,axiom,
    ! [B: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member_set_nat(B,range_9132311542094983838et_nat(R2))
     => ~ ! [A3: heap_e7401611519738050253t_unit] : ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B),R2) ) ).

% RangeE
tff(fact_737_RangeE,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(B),range_nat_nat(R2))
     => ~ ! [A3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B),R2) ) ).

% RangeE
tff(fact_738_RangeE,axiom,
    ! [B: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(B),range_int_int(R2))
     => ~ ! [A3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B),R2) ) ).

% RangeE
tff(fact_739_RangeE,axiom,
    ! [B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member6260224972018164377et_nat(B,range_3211142380271236156et_nat(R2))
     => ~ ! [A3: fun_Pr6460944699146269935_nat_o] : ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B),R2) ) ).

% RangeE
tff(fact_740_RangeE,axiom,
    ! [B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member4513763875858039863et_nat(B,range_1185527156928805978et_nat(R2))
     => ~ ! [A3: fun_Pr6460944699146269935_nat_o] : ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B),R2) ) ).

% RangeE
tff(fact_741_Range__iff,axiom,
    ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member_set_nat(A2,range_9132311542094983838et_nat(R2))
    <=> ? [Y3: heap_e7401611519738050253t_unit] : member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y3),A2),R2) ) ).

% Range_iff
tff(fact_742_Range__iff,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(A2),range_nat_nat(R2))
    <=> ? [Y3: nat] : member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),A2),R2) ) ).

% Range_iff
tff(fact_743_Range__iff,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(A2),range_int_int(R2))
    <=> ? [Y3: int] : member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y3),A2),R2) ) ).

% Range_iff
tff(fact_744_Range__iff,axiom,
    ! [A2: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member6260224972018164377et_nat(A2,range_3211142380271236156et_nat(R2))
    <=> ? [Y3: fun_Pr6460944699146269935_nat_o] : member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y3),A2),R2) ) ).

% Range_iff
tff(fact_745_Range__iff,axiom,
    ! [A2: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member4513763875858039863et_nat(A2,range_1185527156928805978et_nat(R2))
    <=> ? [Y3: fun_Pr6460944699146269935_nat_o] : member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y3),A2),R2) ) ).

% Range_iff
tff(fact_746_vimage__singleton__eq,axiom,
    ! [A2: nat,F: fun_nat_Product_unit,B: product_unit] :
      ( aa_set_nat_o(member_nat(A2),vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
    <=> ( aa_nat_Product_unit(F,A2) = B ) ) ).

% vimage_singleton_eq
tff(fact_747_vimage__singleton__eq,axiom,
    ! [A2: int,F: fun_int_Product_unit,B: product_unit] :
      ( aa_set_int_o(member_int(A2),vimage2000912565490760292t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
    <=> ( aa_int_Product_unit(F,A2) = B ) ) ).

% vimage_singleton_eq
tff(fact_748_vimage__singleton__eq,axiom,
    ! [A2: $o,F: fun_o_Product_unit,B: product_unit] :
      ( aa_set_o_o(member_o((A2)),vimage4222377557387472176t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
    <=> ( aa_o_Product_unit(F,(A2)) = B ) ) ).

% vimage_singleton_eq
tff(fact_749_vimage__singleton__eq,axiom,
    ! [A2: product_unit,F: fun_Pr2191293291710260995t_unit,B: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),vimage7995052115951654139t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
    <=> ( aa_Pro8294289767544744956t_unit(F,A2) = B ) ) ).

% vimage_singleton_eq
tff(fact_750_vimage__singleton__eq,axiom,
    ! [A2: nat,F: fun_nat_o,B: $o] :
      ( aa_set_nat_o(member_nat(A2),vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
    <=> ( aa_nat_o(F,A2)
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_751_vimage__singleton__eq,axiom,
    ! [A2: int,F: fun_int_o,B: $o] :
      ( aa_set_int_o(member_int(A2),vimage_int_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
    <=> ( aa_int_o(F,A2)
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_752_vimage__singleton__eq,axiom,
    ! [A2: $o,F: fun_o_o,B: $o] :
      ( aa_set_o_o(member_o((A2)),vimage_o_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
    <=> ( aa_o_o(F,(A2))
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_753_vimage__singleton__eq,axiom,
    ! [A2: product_unit,F: fun_Product_unit_o,B: $o] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),vimage8808348054653608838unit_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
    <=> ( aa_Product_unit_o(F,A2)
      <=> (B) ) ) ).

% vimage_singleton_eq
tff(fact_754_vimage__singleton__eq,axiom,
    ! [A2: nat,F: fun_nat_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)))
    <=> ( aa_nat_nat(F,A2) = B ) ) ).

% vimage_singleton_eq
tff(fact_755_vimage__singleton__eq,axiom,
    ! [A2: int,F: fun_int_nat,B: nat] :
      ( aa_set_int_o(member_int(A2),vimage_int_nat(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)))
    <=> ( aa_int_nat(F,A2) = B ) ) ).

% vimage_singleton_eq
tff(fact_756_relH__in__rangeI_I2_J,axiom,
    ! [As: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
     => aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As)) ) ).

% relH_in_rangeI(2)
tff(fact_757_relH__in__rangeI_I1_J,axiom,
    ! [As: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
     => aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As)) ) ).

% relH_in_rangeI(1)
tff(fact_758_relH__refl,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
     => aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),Ha) ) ).

% relH_refl
tff(fact_759_Image__empty__trancl__Image__empty,axiom,
    ! [R: set_Pr5094982260447487303t_unit,V: product_unit] :
      ( ( image_8431784415008483091t_unit(R,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit )
     => ( image_8431784415008483091t_unit(transi430408012122777006t_unit(R),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_760_Image__empty__trancl__Image__empty,axiom,
    ! [R: set_Product_prod_o_o,V: $o] :
      ( ( image_o_o(R,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o)) = bot_bot_set_o )
     => ( image_o_o(transitive_trancl_o(R),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o)) = bot_bot_set_o ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_761_Image__empty__trancl__Image__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat,V: nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat)) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(image_nat_nat(transi6264000038957366511cl_nat(R)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat)) = bot_bot_set_nat ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_762_Image__empty__trancl__Image__empty,axiom,
    ! [R: set_Pr958786334691620121nt_int,V: int] :
      ( ( image_int_int(R,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int)) = bot_bot_set_int )
     => ( image_int_int(transi6261509568448316235cl_int(R),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int)) = bot_bot_set_int ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_763_Image__empty__trancl__Image__empty,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,V: fun_Pr6460944699146269935_nat_o] :
      ( ( image_2188787866847195539_nat_o(R,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o )
     => ( image_2188787866847195539_nat_o(transi8445014155877283344_nat_o(R),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_764_mod__relH,axiom,
    ! [As: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,P: assn] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
      <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As)) ) ) ).

% mod_relH
tff(fact_765_for__in__RI,axiom,
    ! [X: heap_e7401611519738050253t_unit,R: set_Pr3948176798113811640et_nat] :
      ( member242389114213480694t_unit(X,domain2007548326387822151et_nat(R))
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),fun_of5878246503291251064et_nat(R,X)),R) ) ).

% for_in_RI
tff(fact_766_for__in__RI,axiom,
    ! [X: nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(X),domain_nat_nat(R))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(fun_of_rel_nat_nat(R),X)),R) ) ).

% for_in_RI
tff(fact_767_for__in__RI,axiom,
    ! [X: int,R: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(X),domain_int_int(R))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),aa_int_int(fun_of_rel_int_int(R),X)),R) ) ).

% for_in_RI
tff(fact_768_for__in__RI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,R: set_Pr1312406793078187350et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),domain2953816653896713061et_nat(R))
     => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),fun_of7894932681850162710et_nat(R,X)),R) ) ).

% for_in_RI
tff(fact_769_for__in__RI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,R: set_Pr4040394263813967220et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),domain5137017734253780739et_nat(R))
     => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),fun_of2710119056957159732et_nat(R,X)),R) ) ).

% for_in_RI
tff(fact_770_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),transi6264000038957366511cl_nat(R2))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),R2)
       => ~ ! [B2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),B2),transi6264000038957366511cl_nat(R2))
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),A22),R2) ) ) ) ).

% trancl.cases
tff(fact_771_trancl_Ocases,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),transi6261509568448316235cl_int(R2))
     => ( ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),R2)
       => ~ ! [B2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),B2),transi6261509568448316235cl_int(R2))
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),A22),R2) ) ) ) ).

% trancl.cases
tff(fact_772_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),transi6264000038957366511cl_nat(R2))
    <=> ( ? [A5: nat,B6: nat] :
            ( ( A1 = A5 )
            & ( A22 = B6 )
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),R2) )
        | ? [A5: nat,B6: nat,C4: nat] :
            ( ( A1 = A5 )
            & ( A22 = C4 )
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),transi6264000038957366511cl_nat(R2))
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B6),C4),R2) ) ) ) ).

% trancl.simps
tff(fact_773_trancl_Osimps,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),transi6261509568448316235cl_int(R2))
    <=> ( ? [A5: int,B6: int] :
            ( ( A1 = A5 )
            & ( A22 = B6 )
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),R2) )
        | ? [A5: int,B6: int,C4: int] :
            ( ( A1 = A5 )
            & ( A22 = C4 )
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),transi6261509568448316235cl_int(R2))
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B6),C4),R2) ) ) ) ).

% trancl.simps
tff(fact_774_trancl_Or__into__trancl,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2)) ) ).

% trancl.r_into_trancl
tff(fact_775_trancl_Or__into__trancl,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2)) ) ).

% trancl.r_into_trancl
tff(fact_776_tranclE,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => ~ ! [C2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C2),transi6264000038957366511cl_nat(R2))
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(C2),B),R2) ) ) ) ).

% tranclE
tff(fact_777_tranclE,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2))
     => ( ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => ~ ! [C2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C2),transi6261509568448316235cl_int(R2))
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(C2),B),R2) ) ) ) ).

% tranclE
tff(fact_778_trancl__trans,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),transi6264000038957366511cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),transi6264000038957366511cl_nat(R2)) ) ) ).

% trancl_trans
tff(fact_779_trancl__trans,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z),transi6261509568448316235cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),transi6261509568448316235cl_int(R2)) ) ) ).

% trancl_trans
tff(fact_780_trancl__induct,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2))
     => ( ! [Y2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y2),R2)
           => aa_nat_o(P,Y2) )
       => ( ! [Y2: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y2),transi6264000038957366511cl_nat(R2))
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),R2)
               => ( aa_nat_o(P,Y2)
                 => aa_nat_o(P,Z3) ) ) )
         => aa_nat_o(P,B) ) ) ) ).

% trancl_induct
tff(fact_781_trancl__induct,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2))
     => ( ! [Y2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),Y2),R2)
           => aa_int_o(P,Y2) )
       => ( ! [Y2: int,Z3: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),Y2),transi6261509568448316235cl_int(R2))
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),R2)
               => ( aa_int_o(P,Y2)
                 => aa_int_o(P,Z3) ) ) )
         => aa_int_o(P,B) ) ) ) ).

% trancl_induct
tff(fact_782_r__r__into__trancl,axiom,
    ! [A2: nat,B: nat,R: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),R)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi6264000038957366511cl_nat(R)) ) ) ).

% r_r_into_trancl
tff(fact_783_r__r__into__trancl,axiom,
    ! [A2: int,B: int,R: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),R)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi6261509568448316235cl_int(R)) ) ) ).

% r_r_into_trancl
tff(fact_784_converse__tranclE,axiom,
    ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),transi6264000038957366511cl_nat(R2))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),R2)
       => ~ ! [Y2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y2),R2)
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z),transi6264000038957366511cl_nat(R2)) ) ) ) ).

% converse_tranclE
tff(fact_785_converse__tranclE,axiom,
    ! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),transi6261509568448316235cl_int(R2))
     => ( ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),R2)
       => ~ ! [Y2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y2),R2)
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z),transi6261509568448316235cl_int(R2)) ) ) ) ).

% converse_tranclE
tff(fact_786_trancl__induct2,axiom,
    ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: fun_he5746958392356810273_nat_o] :
      ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ax),Ay),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Bx),By)),transi4137310414928362273et_nat(R2))
     => ( ! [A3: heap_e7401611519738050253t_unit,B2: set_nat] :
            ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ax),Ay),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2)),R2)
           => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A3),B2) )
       => ( ! [A3: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
              ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ax),Ay),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2)),transi4137310414928362273et_nat(R2))
             => ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Aa),Ba)),R2)
               => ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A3),B2)
                 => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Aa),Ba) ) ) )
         => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_787_trancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: fun_nat_fun_nat_o] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Bx),By)),transi243908449541399842at_nat(R2))
     => ( ! [A3: nat,B2: nat] :
            ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2)),R2)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
       => ( ! [A3: nat,B2: nat,Aa: nat,Ba: nat] :
              ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2)),transi243908449541399842at_nat(R2))
             => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),Ba)),R2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_788_trancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: fun_int_fun_int_o] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Ax),Ay),aa_int7462565939943104719nt_int(product_Pair_int_int(Bx),By)),transi6288783178788033498nt_int(R2))
     => ( ! [A3: int,B2: int] :
            ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Ax),Ay),aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2)),R2)
           => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
       => ( ! [A3: int,B2: int,Aa: int,Ba: int] :
              ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Ax),Ay),aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2)),transi6288783178788033498nt_int(R2))
             => ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Aa),Ba)),R2)
               => ( aa_int_o(aa_int_fun_int_o(P,A3),B2)
                 => aa_int_o(aa_int_fun_int_o(P,Aa),Ba) ) ) )
         => aa_int_o(aa_int_fun_int_o(P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_789_trancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc3658429121746597890et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc3658429121746597890et_nat,R2: set_Pr9066344408807579265et_nat,P: fun_fu8419763957888848259_nat_o] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Ax),Ay),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Bx),By)),transi7952567189410240703et_nat(R2))
     => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
            ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Ax),Ay),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2)),R2)
           => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B2) )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc3658429121746597890et_nat] :
              ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Ax),Ay),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2)),transi7952567189410240703et_nat(R2))
             => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Aa),Ba)),R2)
               => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B2)
                 => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Aa),Ba) ) ) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_790_trancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc8799533983703829792et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc8799533983703829792et_nat,R2: set_Pr7057080801110549565et_nat,P: fun_fu2611741329380808165_nat_o] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Ax),Ay),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Bx),By)),transi2167408805420631517et_nat(R2))
     => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
            ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Ax),Ay),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2)),R2)
           => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B2) )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc8799533983703829792et_nat] :
              ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Ax),Ay),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2)),transi2167408805420631517et_nat(R2))
             => ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Aa),Ba)),R2)
               => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B2)
                 => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Aa),Ba) ) ) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_791_converse__trancl__induct,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2))
     => ( ! [Y2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),B),R2)
           => aa_nat_o(P,Y2) )
       => ( ! [Y2: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),R2)
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z3),B),transi6264000038957366511cl_nat(R2))
               => ( aa_nat_o(P,Z3)
                 => aa_nat_o(P,Y2) ) ) )
         => aa_nat_o(P,A2) ) ) ) ).

% converse_trancl_induct
tff(fact_792_converse__trancl__induct,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2))
     => ( ! [Y2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),B),R2)
           => aa_int_o(P,Y2) )
       => ( ! [Y2: int,Z3: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),R2)
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Z3),B),transi6261509568448316235cl_int(R2))
               => ( aa_int_o(P,Z3)
                 => aa_int_o(P,Y2) ) ) )
         => aa_int_o(P,A2) ) ) ) ).

% converse_trancl_induct
tff(fact_793_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(R2))
     => ( ! [X3: nat,Y2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
           => aa_nat_o(aa_nat_fun_nat_o(P,X3),Y2) )
       => ( ! [X3: nat,Y2: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),transi6264000038957366511cl_nat(R2))
             => ( aa_nat_o(aa_nat_fun_nat_o(P,X3),Y2)
               => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),transi6264000038957366511cl_nat(R2))
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,Y2),Z3)
                   => aa_nat_o(aa_nat_fun_nat_o(P,X3),Z3) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,X),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_794_trancl__trans__induct,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,P: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(R2))
     => ( ! [X3: int,Y2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
           => aa_int_o(aa_int_fun_int_o(P,X3),Y2) )
       => ( ! [X3: int,Y2: int,Z3: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),transi6261509568448316235cl_int(R2))
             => ( aa_int_o(aa_int_fun_int_o(P,X3),Y2)
               => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),transi6261509568448316235cl_int(R2))
                 => ( aa_int_o(aa_int_fun_int_o(P,Y2),Z3)
                   => aa_int_o(aa_int_fun_int_o(P,X3),Z3) ) ) ) )
         => aa_int_o(aa_int_fun_int_o(P,X),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_795_trancl__into__trancl2,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),transi6264000038957366511cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi6264000038957366511cl_nat(R2)) ) ) ).

% trancl_into_trancl2
tff(fact_796_trancl__into__trancl2,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),transi6261509568448316235cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi6261509568448316235cl_int(R2)) ) ) ).

% trancl_into_trancl2
tff(fact_797_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),R2)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi6264000038957366511cl_nat(R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_798_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),R2)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi6261509568448316235cl_int(R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_799_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3),transi6264000038957366511cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_800_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ! [X3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),X3),transi6261509568448316235cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_801_trancl__Image__advance__ss,axiom,
    ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(U),V),E)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(transi430408012122777006t_unit(E),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit))),image_8431784415008483091t_unit(transi430408012122777006t_unit(E),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,U),bot_bo3957492148770167129t_unit))) ) ).

% trancl_Image_advance_ss
tff(fact_802_trancl__Image__advance__ss,axiom,
    ! [U: $o,V: $o,E: set_Product_prod_o_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((U)),(V)),E)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(transitive_trancl_o(E),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o))),image_o_o(transitive_trancl_o(E),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(U)),bot_bot_set_o))) ) ).

% trancl_Image_advance_ss
tff(fact_803_trancl__Image__advance__ss,axiom,
    ! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),V),E)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(transi6261509568448316235cl_int(E),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int))),image_int_int(transi6261509568448316235cl_int(E),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,U),bot_bot_set_int))) ) ).

% trancl_Image_advance_ss
tff(fact_804_trancl__Image__advance__ss,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o,V: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(U,V),E)
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(transi8445014155877283344_nat_o(E),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o))),image_2188787866847195539_nat_o(transi8445014155877283344_nat_o(E),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(U),bot_bo8632255151191677115_nat_o))) ) ).

% trancl_Image_advance_ss
tff(fact_805_trancl__Image__advance__ss,axiom,
    ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V),E)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(transi6264000038957366511cl_nat(E)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(transi6264000038957366511cl_nat(E)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,U),bot_bot_set_nat))) ) ).

% trancl_Image_advance_ss
tff(fact_806_wand__assnI,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,R: assn] :
      ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
     => ( ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As4) = bot_bot_set_nat )
           => ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H5)
             => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As))
               => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As4))
                 => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,R),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4))) ) ) ) )
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(wand_assn(Q),R)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As)) ) ) ).

% wand_assnI
tff(fact_807_trancl__Image__in__Range,axiom,
    ! [R: set_Pr1261947904930325089at_nat,V2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(transi6264000038957366511cl_nat(R)),V2)),range_nat_nat(R)) ).

% trancl_Image_in_Range
tff(fact_808_Pair__vimage__Sigma,axiom,
    ! [X: nat,A: set_nat,F: fun_na896991816813928310t_unit] :
      vimage7305857734363866308t_unit(produc3434450474625521892t_unit(X),produc5506539147108004339t_unit(A,F)) = $ite(aa_set_nat_o(member_nat(X),A),aa_nat2714212862977473631t_unit(F,X),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_809_Pair__vimage__Sigma,axiom,
    ! [X: int,A: set_int,F: fun_in4731230917697954842t_unit] :
      vimage4422279681566046568t_unit(produc550872421827702152t_unit(X),produc2622961094310184599t_unit(A,F)) = $ite(aa_set_int_o(member_int(X),A),aa_int6548451963861500163t_unit(F,X),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_810_Pair__vimage__Sigma,axiom,
    ! [X: $o,A: set_o,F: fun_o_1833192751255691672t_unit] :
      vimage508909887551888138t_unit(produc3901946521647900940t_unit((X)),produc411009820408249917t_unit(A,F)) = $ite(aa_set_o_o(member_o((X)),A),aa_o_s3492205133957624465t_unit(F,(X)),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_811_Pair__vimage__Sigma,axiom,
    ! [X: product_unit,A: set_Product_unit,F: fun_Pr3013102738158121955t_unit] :
      vimage8114940889160087125t_unit(produc5692694829384537815t_unit(X),produc6121607364029222536t_unit(A,F)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_Pro234974305641502940t_unit(F,X),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_812_Pair__vimage__Sigma,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_set_o] :
      vimage6125388955539849434_nat_o(product_Pair_nat_o(X),product_Sigma_nat_o(A,F)) = $ite(aa_set_nat_o(member_nat(X),A),aa_nat_set_o(F,X),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_813_Pair__vimage__Sigma,axiom,
    ! [X: int,A: set_int,F: fun_int_set_o] :
      vimage4304346610107161470_int_o(product_Pair_int_o(X),product_Sigma_int_o(A,F)) = $ite(aa_set_int_o(member_int(X),A),aa_int_set_o(F,X),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_814_Pair__vimage__Sigma,axiom,
    ! [X: $o,A: set_o,F: fun_o_set_o] :
      vimage8945963521958007626od_o_o(product_Pair_o_o((X)),product_Sigma_o_o(A,F)) = $ite(aa_set_o_o(member_o((X)),A),aa_o_set_o(F,(X)),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_815_Pair__vimage__Sigma,axiom,
    ! [X: product_unit,A: set_Product_unit,F: fun_Pr6739565440344758894_set_o] :
      vimage9156843197228287893unit_o(produc8487917018914037602unit_o(X),produc4996980317674386579unit_o(A,F)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_Pro536174410435844839_set_o(F,X),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_816_Pair__vimage__Sigma,axiom,
    ! [X: int,A: set_int,F: fun_int_set_nat] :
      vimage9012682431555523728nt_nat(product_Pair_int_nat(X),produc1456381018704787142nt_nat(A,F)) = $ite(aa_set_int_o(member_int(X),A),aa_int_set_nat(F,X),bot_bot_set_nat) ).

% Pair_vimage_Sigma
tff(fact_817_Pair__vimage__Sigma,axiom,
    ! [X: $o,A: set_o,F: fun_o_set_nat] :
      vimage5656379182745576100_o_nat(product_Pair_o_nat((X)),product_Sigma_o_nat(A,F)) = $ite(aa_set_o_o(member_o((X)),A),aa_o_set_nat(F,(X)),bot_bot_set_nat) ).

% Pair_vimage_Sigma
tff(fact_818_relH__def,axiom,
    ! [As: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
        & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H2),As))
        & ! [T: typerep,X2: nat] :
            ( aa_set_nat_o(member_nat(X2),As)
           => ( ( aa_nat_nat(refs_Product_unit(Ha,T),X2) = aa_nat_nat(refs_Product_unit(H2,T),X2) )
              & ( arrays_Product_unit(Ha,T,X2) = arrays_Product_unit(H2,T,X2) ) ) ) ) ) ).

% relH_def
tff(fact_819_vimage__insert,axiom,
    ! [F: fun_nat_Product_unit,A2: product_unit,B4: set_Product_unit] : vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))),vimage4884490618288580032t_unit(F,B4)) ).

% vimage_insert
tff(fact_820_vimage__insert,axiom,
    ! [F: fun_nat_o,A2: $o,B4: set_o] : vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))),vimage_nat_o(F,B4)) ).

% vimage_insert
tff(fact_821_vimage__insert,axiom,
    ! [F: fun_nat_nat,A2: nat,B4: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ).

% vimage_insert
tff(fact_822_vimage__insert,axiom,
    ! [F: fun_nat_int,A2: int,B4: set_int] : vimage_nat_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage_nat_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))),vimage_nat_int(F,B4)) ).

% vimage_insert
tff(fact_823_vimage__insert,axiom,
    ! [F: fun_na5781657914532371832_nat_o,A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] : vimage984285786009958690_nat_o(F,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,vimage984285786009958690_nat_o(F,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))),vimage984285786009958690_nat_o(F,B4)) ).

% vimage_insert
tff(fact_824_trancl__over__edgeE,axiom,
    ! [U: nat,W: nat,V1: nat,V22: nat,E: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),W),transi6264000038957366511cl_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V1),V22),E)))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),W),transi6264000038957366511cl_nat(E))
       => ~ ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V1),transi2905341329935302413cl_nat(E))
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V22),W),transi2905341329935302413cl_nat(E)) ) ) ) ).

% trancl_over_edgeE
tff(fact_825_trancl__over__edgeE,axiom,
    ! [U: int,W: int,V1: int,V22: int,E: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),W),transi6261509568448316235cl_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V1),V22),E)))
     => ( ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),W),transi6261509568448316235cl_int(E))
       => ~ ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),V1),transi2902850859426252137cl_int(E))
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V22),W),transi2902850859426252137cl_int(E)) ) ) ) ).

% trancl_over_edgeE
tff(fact_826_partial__order__on__empty,axiom,
    order_2505591618835473410t_unit(bot_bo3957492148770167129t_unit,bot_bo1654022762305692595t_unit) ).

% partial_order_on_empty
tff(fact_827_partial__order__on__empty,axiom,
    order_8538205470752950541r_on_o(bot_bot_set_o,bot_bo7073875226086086771od_o_o) ).

% partial_order_on_empty
tff(fact_828_partial__order__on__empty,axiom,
    order_5251275573222108571on_nat(bot_bot_set_nat,bot_bo2099793752762293965at_nat) ).

% partial_order_on_empty
tff(fact_829_partial__order__on__empty,axiom,
    order_5248785102713058295on_int(bot_bot_set_int,bot_bo1796632182523588997nt_int) ).

% partial_order_on_empty
tff(fact_830_partial__order__on__empty,axiom,
    order_3152091435976458852_nat_o(bot_bo8632255151191677115_nat_o,bot_bo6763312114214544691_nat_o) ).

% partial_order_on_empty
tff(fact_831_prod__set__simps_I2_J,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat] : basic_2698842327037663054et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = insert_set_nat(Y,bot_bot_set_set_nat) ).

% prod_set_simps(2)
tff(fact_832_prod__set__simps_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : basic_7091812037573673196et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = insert1271470778893645362et_nat(Y,bot_bo3771712512152092492et_nat) ).

% prod_set_simps(2)
tff(fact_833_prod__set__simps_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : basic_6089933144288473354et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = insert663245765287195472et_nat(Y,bot_bo1493351289776401898et_nat) ).

% prod_set_simps(2)
tff(fact_834_prod__set__simps_I2_J,axiom,
    ! [X: nat,Y: nat] : basic_snds_nat_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat) ).

% prod_set_simps(2)
tff(fact_835_prod__set__simps_I2_J,axiom,
    ! [X: int,Y: int] : basic_snds_int_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int) ).

% prod_set_simps(2)
tff(fact_836_dual__order_Orefl,axiom,
    ! [A2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),A2) ).

% dual_order.refl
tff(fact_837_dual__order_Orefl,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),A2) ).

% dual_order.refl
tff(fact_838_dual__order_Orefl,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),A2) ).

% dual_order.refl
tff(fact_839_dual__order_Orefl,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A2) ).

% dual_order.refl
tff(fact_840_dual__order_Orefl,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A2) ).

% dual_order.refl
tff(fact_841_order__refl,axiom,
    ! [X: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),X) ).

% order_refl
tff(fact_842_order__refl,axiom,
    ! [X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X) ).

% order_refl
tff(fact_843_order__refl,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X) ).

% order_refl
tff(fact_844_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_845_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_846_subset__antisym,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
       => ( A = B4 ) ) ) ).

% subset_antisym
tff(fact_847_subsetI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),B4) )
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4) ) ).

% subsetI
tff(fact_848_subsetI,axiom,
    ! [A: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_set_int_o(member_int(X3),B4) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4) ) ).

% subsetI
tff(fact_849_subsetI,axiom,
    ! [A: set_o,B4: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => aa_set_o_o(member_o((X3)),B4) )
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4) ) ).

% subsetI
tff(fact_850_subsetI,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(X3),B4) )
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4) ) ).

% subsetI
tff(fact_851_subsetI,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_set_nat_o(member_nat(X3),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ).

% subsetI
tff(fact_852_Int__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A,B4))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% Int_iff
tff(fact_853_Int__iff,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A,B4))
    <=> ( aa_set_int_o(member_int(C),A)
        & aa_set_int_o(member_int(C),B4) ) ) ).

% Int_iff
tff(fact_854_Int__iff,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A,B4))
    <=> ( aa_set_o_o(member_o((C)),A)
        & aa_set_o_o(member_o((C)),B4) ) ) ).

% Int_iff
tff(fact_855_Int__iff,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),inf_in4660618365625256667t_unit(A,B4))
    <=> ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
        & aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% Int_iff
tff(fact_856_Int__iff,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
    <=> ( aa_set_nat_o(member_nat(C),A)
        & aa_set_nat_o(member_nat(C),B4) ) ) ).

% Int_iff
tff(fact_857_IntI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A,B4)) ) ) ).

% IntI
tff(fact_858_IntI,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A)
     => ( aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),inf_inf_set_int(A,B4)) ) ) ).

% IntI
tff(fact_859_IntI,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),A)
     => ( aa_set_o_o(member_o((C)),B4)
       => aa_set_o_o(member_o((C)),inf_inf_set_o(A,B4)) ) ) ).

% IntI
tff(fact_860_IntI,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(C),B4)
       => aa_set8780210009671629607unit_o(member_Product_unit(C),inf_in4660618365625256667t_unit(A,B4)) ) ) ).

% IntI
tff(fact_861_IntI,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A)
     => ( aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) ) ) ).

% IntI
tff(fact_862_Un__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
        | aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% Un_iff
tff(fact_863_Un__iff,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4))
    <=> ( aa_set_int_o(member_int(C),A)
        | aa_set_int_o(member_int(C),B4) ) ) ).

% Un_iff
tff(fact_864_Un__iff,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
    <=> ( aa_set_o_o(member_o((C)),A)
        | aa_set_o_o(member_o((C)),B4) ) ) ).

% Un_iff
tff(fact_865_Un__iff,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))
    <=> ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
        | aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% Un_iff
tff(fact_866_Un__iff,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(member_nat(C),A)
        | aa_set_nat_o(member_nat(C),B4) ) ) ).

% Un_iff
tff(fact_867_UnCI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A) )
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) ) ).

% UnCI
tff(fact_868_UnCI,axiom,
    ! [C: int,B4: set_int,A: set_int] :
      ( ( ~ aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),A) )
     => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) ) ).

% UnCI
tff(fact_869_UnCI,axiom,
    ! [C: $o,B4: set_o,A: set_o] :
      ( ( ~ aa_set_o_o(member_o((C)),B4)
       => aa_set_o_o(member_o((C)),A) )
     => aa_set_o_o(member_o((C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) ) ).

% UnCI
tff(fact_870_UnCI,axiom,
    ! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
      ( ( ~ aa_set8780210009671629607unit_o(member_Product_unit(C),B4)
       => aa_set8780210009671629607unit_o(member_Product_unit(C),A) )
     => aa_set8780210009671629607unit_o(member_Product_unit(C),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) ) ).

% UnCI
tff(fact_871_UnCI,axiom,
    ! [C: nat,B4: set_nat,A: set_nat] :
      ( ( ~ aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),A) )
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) ) ).

% UnCI
tff(fact_872_empty__subsetI,axiom,
    ! [A: set_Product_unit] : aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,bot_bo3957492148770167129t_unit),A) ).

% empty_subsetI
tff(fact_873_empty__subsetI,axiom,
    ! [A: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,bot_bot_set_o),A) ).

% empty_subsetI
tff(fact_874_empty__subsetI,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A) ).

% empty_subsetI
tff(fact_875_empty__subsetI,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(bot_bo8632255151191677115_nat_o),A) ).

% empty_subsetI
tff(fact_876_empty__subsetI,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% empty_subsetI
tff(fact_877_subset__empty,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),bot_bo3957492148770167129t_unit)
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% subset_empty
tff(fact_878_subset__empty,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_879_subset__empty,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 ) ) ).

% subset_empty
tff(fact_880_subset__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),bot_bo8632255151191677115_nat_o)
    <=> ( A = bot_bo8632255151191677115_nat_o ) ) ).

% subset_empty
tff(fact_881_subset__empty,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 ) ) ).

% subset_empty
tff(fact_882_insert__subset,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)),B4)
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4)
        & aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4) ) ) ).

% insert_subset
tff(fact_883_insert__subset,axiom,
    ! [X: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)),B4)
    <=> ( aa_set_int_o(member_int(X),B4)
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4) ) ) ).

% insert_subset
tff(fact_884_insert__subset,axiom,
    ! [X: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)),B4)
    <=> ( aa_set_o_o(member_o((X)),B4)
        & aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4) ) ) ).

% insert_subset
tff(fact_885_insert__subset,axiom,
    ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)),B4)
    <=> ( aa_set8780210009671629607unit_o(member_Product_unit(X),B4)
        & aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4) ) ) ).

% insert_subset
tff(fact_886_insert__subset,axiom,
    ! [X: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)),B4)
    <=> ( aa_set_nat_o(member_nat(X),B4)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ) ).

% insert_subset
tff(fact_887_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B: int,B4: fun_nat_set_int] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( aa_set_int_o(member_int(B),aa_nat_set_int(B4,A2))
       => member4262671552274231302at_int(product_Pair_nat_int(A2,B),produc454536836294682310at_int(A,B4)) ) ) ).

% SigmaI
tff(fact_888_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B: $o,B4: fun_nat_set_o] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( aa_set_o_o(member_o((B)),aa_nat_set_o(B4,A2))
       => member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A2),(B)),product_Sigma_nat_o(A,B4)) ) ) ).

% SigmaI
tff(fact_889_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B: product_unit,B4: fun_na896991816813928310t_unit] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(B),aa_nat2714212862977473631t_unit(B4,A2))
       => member348563419383025609t_unit(aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(A2),B),produc5506539147108004339t_unit(A,B4)) ) ) ).

% SigmaI
tff(fact_890_SigmaI,axiom,
    ! [A2: int,A: set_int,B: nat,B4: fun_int_set_nat] :
      ( aa_set_int_o(member_int(A2),A)
     => ( aa_set_nat_o(member_nat(B),aa_int_set_nat(B4,A2))
       => member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A2),B),produc1456381018704787142nt_nat(A,B4)) ) ) ).

% SigmaI
tff(fact_891_SigmaI,axiom,
    ! [A2: int,A: set_int,B: $o,B4: fun_int_set_o] :
      ( aa_set_int_o(member_int(A2),A)
     => ( aa_set_o_o(member_o((B)),aa_int_set_o(B4,A2))
       => member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A2),(B)),product_Sigma_int_o(A,B4)) ) ) ).

% SigmaI
tff(fact_892_SigmaI,axiom,
    ! [A2: int,A: set_int,B: product_unit,B4: fun_in4731230917697954842t_unit] :
      ( aa_set_int_o(member_int(A2),A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(B),aa_int6548451963861500163t_unit(B4,A2))
       => member6688357403439981677t_unit(aa_Pro399436582157352231t_unit(produc550872421827702152t_unit(A2),B),produc2622961094310184599t_unit(A,B4)) ) ) ).

% SigmaI
tff(fact_893_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: nat,B4: fun_o_set_nat] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( aa_set_nat_o(member_nat(B),aa_o_set_nat(B4,(A2)))
       => member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A2)),B),product_Sigma_o_nat(A,B4)) ) ) ).

% SigmaI
tff(fact_894_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: int,B4: fun_o_set_int] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( aa_set_int_o(member_int(B),aa_o_set_int(B4,(A2)))
       => member7847949116333733898_o_int(product_Pair_o_int((A2),B),product_Sigma_o_int(A,B4)) ) ) ).

% SigmaI
tff(fact_895_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: $o,B4: fun_o_set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( aa_set_o_o(member_o((B)),aa_o_set_o(B4,(A2)))
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),product_Sigma_o_o(A,B4)) ) ) ).

% SigmaI
tff(fact_896_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: product_unit,B4: fun_o_1833192751255691672t_unit] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(B),aa_o_s3492205133957624465t_unit(B4,(A2)))
       => member9215336023991391429t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A2)),B),produc411009820408249917t_unit(A,B4)) ) ) ).

% SigmaI
tff(fact_897_mem__Sigma__iff,axiom,
    ! [A2: nat,B: int,A: set_nat,B4: fun_nat_set_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(A2,B),produc454536836294682310at_int(A,B4))
    <=> ( aa_set_nat_o(member_nat(A2),A)
        & aa_set_int_o(member_int(B),aa_nat_set_int(B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_898_mem__Sigma__iff,axiom,
    ! [A2: nat,B: $o,A: set_nat,B4: fun_nat_set_o] :
      ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A2),(B)),product_Sigma_nat_o(A,B4))
    <=> ( aa_set_nat_o(member_nat(A2),A)
        & aa_set_o_o(member_o((B)),aa_nat_set_o(B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_899_mem__Sigma__iff,axiom,
    ! [A2: nat,B: product_unit,A: set_nat,B4: fun_na896991816813928310t_unit] :
      ( member348563419383025609t_unit(aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(A2),B),produc5506539147108004339t_unit(A,B4))
    <=> ( aa_set_nat_o(member_nat(A2),A)
        & aa_set8780210009671629607unit_o(member_Product_unit(B),aa_nat2714212862977473631t_unit(B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_900_mem__Sigma__iff,axiom,
    ! [A2: int,B: nat,A: set_int,B4: fun_int_set_nat] :
      ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A2),B),produc1456381018704787142nt_nat(A,B4))
    <=> ( aa_set_int_o(member_int(A2),A)
        & aa_set_nat_o(member_nat(B),aa_int_set_nat(B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_901_mem__Sigma__iff,axiom,
    ! [A2: int,B: $o,A: set_int,B4: fun_int_set_o] :
      ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A2),(B)),product_Sigma_int_o(A,B4))
    <=> ( aa_set_int_o(member_int(A2),A)
        & aa_set_o_o(member_o((B)),aa_int_set_o(B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_902_mem__Sigma__iff,axiom,
    ! [A2: int,B: product_unit,A: set_int,B4: fun_in4731230917697954842t_unit] :
      ( member6688357403439981677t_unit(aa_Pro399436582157352231t_unit(produc550872421827702152t_unit(A2),B),produc2622961094310184599t_unit(A,B4))
    <=> ( aa_set_int_o(member_int(A2),A)
        & aa_set8780210009671629607unit_o(member_Product_unit(B),aa_int6548451963861500163t_unit(B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_903_mem__Sigma__iff,axiom,
    ! [A2: $o,B: nat,A: set_o,B4: fun_o_set_nat] :
      ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A2)),B),product_Sigma_o_nat(A,B4))
    <=> ( aa_set_o_o(member_o((A2)),A)
        & aa_set_nat_o(member_nat(B),aa_o_set_nat(B4,(A2))) ) ) ).

% mem_Sigma_iff
tff(fact_904_mem__Sigma__iff,axiom,
    ! [A2: $o,B: int,A: set_o,B4: fun_o_set_int] :
      ( member7847949116333733898_o_int(product_Pair_o_int((A2),B),product_Sigma_o_int(A,B4))
    <=> ( aa_set_o_o(member_o((A2)),A)
        & aa_set_int_o(member_int(B),aa_o_set_int(B4,(A2))) ) ) ).

% mem_Sigma_iff
tff(fact_905_mem__Sigma__iff,axiom,
    ! [A2: $o,B: $o,A: set_o,B4: fun_o_set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),product_Sigma_o_o(A,B4))
    <=> ( aa_set_o_o(member_o((A2)),A)
        & aa_set_o_o(member_o((B)),aa_o_set_o(B4,(A2))) ) ) ).

% mem_Sigma_iff
tff(fact_906_mem__Sigma__iff,axiom,
    ! [A2: $o,B: product_unit,A: set_o,B4: fun_o_1833192751255691672t_unit] :
      ( member9215336023991391429t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A2)),B),produc411009820408249917t_unit(A,B4))
    <=> ( aa_set_o_o(member_o((A2)),A)
        & aa_set8780210009671629607unit_o(member_Product_unit(B),aa_o_s3492205133957624465t_unit(B4,(A2))) ) ) ).

% mem_Sigma_iff
tff(fact_907_Un__empty,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4) = bot_bo3957492148770167129t_unit )
    <=> ( ( A = bot_bo3957492148770167129t_unit )
        & ( B4 = bot_bo3957492148770167129t_unit ) ) ) ).

% Un_empty
tff(fact_908_Un__empty,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4) = bot_bot_set_o )
    <=> ( ( A = bot_bot_set_o )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Un_empty
tff(fact_909_Un__empty,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_910_Un__empty,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B4 = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_911_Un__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4) = bot_bo8632255151191677115_nat_o )
    <=> ( ( A = bot_bo8632255151191677115_nat_o )
        & ( B4 = bot_bo8632255151191677115_nat_o ) ) ) ).

% Un_empty
tff(fact_912_Int__subset__iff,axiom,
    ! [C5: set_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),B4) ) ) ).

% Int_subset_iff
tff(fact_913_Int__insert__right__if1,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ( inf_in3376754634429691197_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),inf_in3376754634429691197_nat_o(A,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_914_Int__insert__right__if1,axiom,
    ! [A2: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ( inf_inf_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),inf_inf_set_int(A,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_915_Int__insert__right__if1,axiom,
    ! [A2: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( inf_inf_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),inf_inf_set_o(A,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_916_Int__insert__right__if1,axiom,
    ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ( inf_in4660618365625256667t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),inf_in4660618365625256667t_unit(A,B4)) ) ) ).

% Int_insert_right_if1
tff(fact_917_Int__insert__right__if1,axiom,
    ! [A2: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) ) ) ).

% Int_insert_right_if1
tff(fact_918_Int__insert__right__if0,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ( inf_in3376754634429691197_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) = inf_in3376754634429691197_nat_o(A,B4) ) ) ).

% Int_insert_right_if0
tff(fact_919_Int__insert__right__if0,axiom,
    ! [A2: int,A: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A2),A)
     => ( inf_inf_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = inf_inf_set_int(A,B4) ) ) ).

% Int_insert_right_if0
tff(fact_920_Int__insert__right__if0,axiom,
    ! [A2: $o,A: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A2)),A)
     => ( inf_inf_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = inf_inf_set_o(A,B4) ) ) ).

% Int_insert_right_if0
tff(fact_921_Int__insert__right__if0,axiom,
    ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ( inf_in4660618365625256667t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = inf_in4660618365625256667t_unit(A,B4) ) ) ).

% Int_insert_right_if0
tff(fact_922_Int__insert__right__if0,axiom,
    ! [A2: nat,A: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A2),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ) ) ).

% Int_insert_right_if0
tff(fact_923_insert__inter__insert,axiom,
    ! [A2: int,A: set_int,B4: set_int] : inf_inf_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),inf_inf_set_int(A,B4)) ).

% insert_inter_insert
tff(fact_924_insert__inter__insert,axiom,
    ! [A2: $o,A: set_o,B4: set_o] : inf_inf_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),inf_inf_set_o(A,B4)) ).

% insert_inter_insert
tff(fact_925_insert__inter__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] : inf_in4660618365625256667t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),inf_in4660618365625256667t_unit(A,B4)) ).

% insert_inter_insert
tff(fact_926_insert__inter__insert,axiom,
    ! [A2: nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) ).

% insert_inter_insert
tff(fact_927_Int__insert__left__if1,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C5: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),C5)
     => ( inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4),C5) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),inf_in3376754634429691197_nat_o(B4,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_928_Int__insert__left__if1,axiom,
    ! [A2: int,C5: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A2),C5)
     => ( inf_inf_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4),C5) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),inf_inf_set_int(B4,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_929_Int__insert__left__if1,axiom,
    ! [A2: $o,C5: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A2)),C5)
     => ( inf_inf_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4),C5) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),inf_inf_set_o(B4,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_930_Int__insert__left__if1,axiom,
    ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),C5)
     => ( inf_in4660618365625256667t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4),C5) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),inf_in4660618365625256667t_unit(B4,C5)) ) ) ).

% Int_insert_left_if1
tff(fact_931_Int__insert__left__if1,axiom,
    ! [A2: nat,C5: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A2),C5)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)),C5) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) ) ) ).

% Int_insert_left_if1
tff(fact_932_Int__insert__left__if0,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C5: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),C5)
     => ( inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4),C5) = inf_in3376754634429691197_nat_o(B4,C5) ) ) ).

% Int_insert_left_if0
tff(fact_933_Int__insert__left__if0,axiom,
    ! [A2: int,C5: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A2),C5)
     => ( inf_inf_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4),C5) = inf_inf_set_int(B4,C5) ) ) ).

% Int_insert_left_if0
tff(fact_934_Int__insert__left__if0,axiom,
    ! [A2: $o,C5: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A2)),C5)
     => ( inf_inf_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4),C5) = inf_inf_set_o(B4,C5) ) ) ).

% Int_insert_left_if0
tff(fact_935_Int__insert__left__if0,axiom,
    ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),C5)
     => ( inf_in4660618365625256667t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4),C5) = inf_in4660618365625256667t_unit(B4,C5) ) ) ).

% Int_insert_left_if0
tff(fact_936_Int__insert__left__if0,axiom,
    ! [A2: nat,C5: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A2),C5)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)),C5) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5) ) ) ).

% Int_insert_left_if0
tff(fact_937_Un__subset__iff,axiom,
    ! [A: set_nat,B4: set_nat,C5: 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),B4)),C5)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C5) ) ) ).

% Un_subset_iff
tff(fact_938_Un__insert__right,axiom,
    ! [A: set_int,A2: int,B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) ).

% Un_insert_right
tff(fact_939_Un__insert__right,axiom,
    ! [A: set_o,A2: $o,B4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) ).

% Un_insert_right
tff(fact_940_Un__insert__right,axiom,
    ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) ).

% Un_insert_right
tff(fact_941_Un__insert__right,axiom,
    ! [A: set_nat,A2: nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) ).

% Un_insert_right
tff(fact_942_Un__insert__left,axiom,
    ! [A2: int,B4: set_int,C5: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)),C5) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B4),C5)) ).

% Un_insert_left
tff(fact_943_Un__insert__left,axiom,
    ! [A2: $o,B4: set_o,C5: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)),C5) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,B4),C5)) ).

% Un_insert_left
tff(fact_944_Un__insert__left,axiom,
    ! [A2: product_unit,B4: set_Product_unit,C5: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)),C5) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,B4),C5)) ).

% Un_insert_left
tff(fact_945_Un__insert__left,axiom,
    ! [A2: nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)),C5) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C5)) ).

% Un_insert_left
tff(fact_946_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T2: 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),T2)),S) = S ).

% Un_Int_eq(1)
tff(fact_947_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T2: 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),T2)),T2) = T2 ).

% Un_Int_eq(2)
tff(fact_948_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T2: 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),T2)) = S ).

% Un_Int_eq(3)
tff(fact_949_Un__Int__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T2)) = T2 ).

% Un_Int_eq(4)
tff(fact_950_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T2: 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),T2)),S) = S ).

% Int_Un_eq(1)
tff(fact_951_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T2: 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),T2)),T2) = T2 ).

% Int_Un_eq(2)
tff(fact_952_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T2: 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),T2)) = S ).

% Int_Un_eq(3)
tff(fact_953_Int__Un__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T2)) = T2 ).

% Int_Un_eq(4)
tff(fact_954_vimage__Int,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),A)),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ).

% vimage_Int
tff(fact_955_vimage__Un,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),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,aa_set_nat_set_nat(vimage_nat_nat(F),A)),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ).

% vimage_Un
tff(fact_956_relH__dist__union,axiom,
    ! [As: set_nat,As5: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5)),Ha),H2)
    <=> ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2)
        & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As5),Ha),H2) ) ) ).

% relH_dist_union
tff(fact_957_singleton__insert__inj__eq,axiom,
    ! [B: product_unit,A2: product_unit,A: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) )
    <=> ( ( A2 = B )
        & aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) ) ) ).

% singleton_insert_inj_eq
tff(fact_958_singleton__insert__inj__eq,axiom,
    ! [B: $o,A2: $o,A: set_o] :
      ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) )
    <=> ( ( (A2)
        <=> (B) )
        & aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_959_singleton__insert__inj__eq,axiom,
    ! [B: int,A2: int,A: set_int] :
      ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) )
    <=> ( ( A2 = B )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_960_singleton__insert__inj__eq,axiom,
    ! [B: fun_Pr6460944699146269935_nat_o,A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) )
    <=> ( ( A2 = B )
        & aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_961_singleton__insert__inj__eq,axiom,
    ! [B: nat,A2: nat,A: set_nat] :
      ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) )
    <=> ( ( A2 = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_962_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_unit,A: set_Product_unit,B: product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit) )
    <=> ( ( A2 = B )
        & aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_963_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B: $o] :
      ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o) )
    <=> ( ( (A2)
        <=> (B) )
        & aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_964_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B: int] :
      ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int) )
    <=> ( ( A2 = B )
        & aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_965_singleton__insert__inj__eq_H,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o) )
    <=> ( ( A2 = B )
        & aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_966_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B: nat] :
      ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat) )
    <=> ( ( A2 = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_967_disjoint__insert_I2_J,axiom,
    ! [A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = inf_in4660618365625256667t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),B4)) )
    <=> ( ~ aa_set8780210009671629607unit_o(member_Product_unit(B),A)
        & ( bot_bo3957492148770167129t_unit = inf_in4660618365625256667t_unit(A,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_968_disjoint__insert_I2_J,axiom,
    ! [A: set_o,B: $o,B4: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),B4)) )
    <=> ( ~ aa_set_o_o(member_o((B)),A)
        & ( bot_bot_set_o = inf_inf_set_o(A,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_969_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B: nat,B4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),B4)) )
    <=> ( ~ aa_set_nat_o(member_nat(B),A)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_970_disjoint__insert_I2_J,axiom,
    ! [A: set_int,B: int,B4: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),B4)) )
    <=> ( ~ aa_set_int_o(member_int(B),A)
        & ( bot_bot_set_int = inf_inf_set_int(A,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_971_disjoint__insert_I2_J,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),B4)) )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A,B4) ) ) ) ).

% disjoint_insert(2)
tff(fact_972_disjoint__insert_I1_J,axiom,
    ! [B4: set_Product_unit,A2: product_unit,A: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(B4,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) = bot_bo3957492148770167129t_unit )
    <=> ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
        & ( inf_in4660618365625256667t_unit(B4,A) = bot_bo3957492148770167129t_unit ) ) ) ).

% disjoint_insert(1)
tff(fact_973_disjoint__insert_I1_J,axiom,
    ! [B4: set_o,A2: $o,A: set_o] :
      ( ( inf_inf_set_o(B4,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A)) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A2)),B4)
        & ( inf_inf_set_o(B4,A) = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
tff(fact_974_disjoint__insert_I1_J,axiom,
    ! [B4: set_nat,A2: nat,A: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A2),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_975_disjoint__insert_I1_J,axiom,
    ! [B4: set_int,A2: int,A: set_int] :
      ( ( inf_inf_set_int(B4,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A)) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A2),B4)
        & ( inf_inf_set_int(B4,A) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_976_disjoint__insert_I1_J,axiom,
    ! [B4: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(B4,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A)) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
        & ( inf_in3376754634429691197_nat_o(B4,A) = bot_bo8632255151191677115_nat_o ) ) ) ).

% disjoint_insert(1)
tff(fact_977_insert__disjoint_I2_J,axiom,
    ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = inf_in4660618365625256667t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A),B4) )
    <=> ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
        & ( bot_bo3957492148770167129t_unit = inf_in4660618365625256667t_unit(A,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_978_insert__disjoint_I2_J,axiom,
    ! [A2: $o,A: set_o,B4: set_o] :
      ( ( bot_bot_set_o = inf_inf_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A),B4) )
    <=> ( ~ aa_set_o_o(member_o((A2)),B4)
        & ( bot_bot_set_o = inf_inf_set_o(A,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_979_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)),B4) )
    <=> ( ~ aa_set_nat_o(member_nat(A2),B4)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_980_insert__disjoint_I2_J,axiom,
    ! [A2: int,A: set_int,B4: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A),B4) )
    <=> ( ~ aa_set_int_o(member_int(A2),B4)
        & ( bot_bot_set_int = inf_inf_set_int(A,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_981_insert__disjoint_I2_J,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A),B4) )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A,B4) ) ) ) ).

% insert_disjoint(2)
tff(fact_982_insert__disjoint_I1_J,axiom,
    ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A),B4) = bot_bo3957492148770167129t_unit )
    <=> ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
        & ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit ) ) ) ).

% insert_disjoint(1)
tff(fact_983_insert__disjoint_I1_J,axiom,
    ! [A2: $o,A: set_o,B4: set_o] :
      ( ( inf_inf_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A),B4) = bot_bot_set_o )
    <=> ( ~ aa_set_o_o(member_o((A2)),B4)
        & ( inf_inf_set_o(A,B4) = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
tff(fact_984_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)),B4) = bot_bot_set_nat )
    <=> ( ~ aa_set_nat_o(member_nat(A2),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_985_insert__disjoint_I1_J,axiom,
    ! [A2: int,A: set_int,B4: set_int] :
      ( ( inf_inf_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A),B4) = bot_bot_set_int )
    <=> ( ~ aa_set_int_o(member_int(A2),B4)
        & ( inf_inf_set_int(A,B4) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_986_insert__disjoint_I1_J,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A),B4) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
        & ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o ) ) ) ).

% insert_disjoint(1)
tff(fact_987_Image__Id__on,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(image_nat_nat(id_on_nat(A)),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ).

% Image_Id_on
tff(fact_988_in__range__dist__union,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,As: set_nat,As5: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5)))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
        & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As5)) ) ) ).

% in_range_dist_union
tff(fact_989_reachable__mono,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat,X5: set_nat,X6: set_nat] :
      ( ord_le3146513528884898305at_nat(R,R3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),X6)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),X5)),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R3)),X6)) ) ) ).

% reachable_mono
tff(fact_990_order__antisym__conv,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_991_order__antisym__conv,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_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_992_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_993_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_994_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_995_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_996_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_997_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_998_ord__le__eq__subst,axiom,
    ! [A2: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_999_ord__le__eq__subst,axiom,
    ! [A2: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1000_ord__le__eq__subst,axiom,
    ! [A2: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1001_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1002_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( ( aa_nat_nat(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1003_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( ( aa_nat_int(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1004_ord__le__eq__subst,axiom,
    ! [A2: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( ( aa_int_rat(F,B) = C )
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1005_ord__le__eq__subst,axiom,
    ! [A2: int,B: int,F: fun_int_nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( ( aa_int_nat(F,B) = C )
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1006_ord__le__eq__subst,axiom,
    ! [A2: int,B: int,F: fun_int_int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( ( aa_int_int(F,B) = C )
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1007_ord__le__eq__subst,axiom,
    ! [A2: filter_nat,B: filter_nat,F: fun_filter_nat_rat,C: rat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( ( aa_filter_nat_rat(F,B) = C )
       => ( ! [X3: filter_nat,Y2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,X3)),aa_filter_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,A2)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1008_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A2 = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1009_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A2 = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1010_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A2 = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1011_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A2 = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1012_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( ( A2 = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1013_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_nat_int,B: nat,C: nat] :
      ( ( A2 = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1014_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_int_rat,B: int,C: int] :
      ( ( A2 = aa_int_rat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_int_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1015_ord__eq__le__subst,axiom,
    ! [A2: nat,F: fun_int_nat,B: int,C: int] :
      ( ( A2 = aa_int_nat(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_int_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1016_ord__eq__le__subst,axiom,
    ! [A2: int,F: fun_int_int,B: int,C: int] :
      ( ( A2 = aa_int_int(F,B) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1017_ord__eq__le__subst,axiom,
    ! [A2: rat,F: fun_filter_nat_rat,B: filter_nat,C: filter_nat] :
      ( ( A2 = aa_filter_nat_rat(F,B) )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => ( ! [X3: filter_nat,Y2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,X3)),aa_filter_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_filter_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1018_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_1019_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_1020_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_1021_subset__Collect__conv,axiom,
    ! [S: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(S),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P))
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S)
         => aa_fun1675921005893230629at_o_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1022_subset__Collect__conv,axiom,
    ! [S: set_int,P: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),aa_fun_int_o_set_int(collect_int,P))
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),S)
         => aa_int_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1023_subset__Collect__conv,axiom,
    ! [S: set_o,P: fun_o_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),aa_fun_o_o_set_o(collect_o,P))
    <=> ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),S)
         => aa_o_o(P,(X2)) ) ) ).

% subset_Collect_conv
tff(fact_1024_subset__Collect__conv,axiom,
    ! [S: set_Product_unit,P: fun_Product_unit_o] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,S),aa_fun8337552879598298401t_unit(collect_Product_unit,P))
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),S)
         => aa_Product_unit_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1025_subset__Collect__conv,axiom,
    ! [S: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),aa_fun_nat_o_set_nat(collect_nat,P))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S)
         => aa_nat_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1026_order__eq__refl,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y) ) ).

% order_eq_refl
tff(fact_1027_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% order_eq_refl
tff(fact_1028_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_1029_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_1030_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_1031_ord__eq__le__eq__trans,axiom,
    ! [A2: filter_nat,B: filter_nat,C: filter_nat,D2: filter_nat] :
      ( ( A2 = B )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => ( ( C = D2 )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1032_ord__eq__le__eq__trans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( A2 = B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => ( ( C = D2 )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1033_ord__eq__le__eq__trans,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( ( A2 = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ( C = D2 )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1034_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( A2 = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ( C = D2 )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1035_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( A2 = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ( C = D2 )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1036_order__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1037_order__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1038_order__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1039_order__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1040_order__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1041_order__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1042_order__subst2,axiom,
    ! [A2: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1043_order__subst2,axiom,
    ! [A2: int,B: int,F: fun_int_nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,B)),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1044_order__subst2,axiom,
    ! [A2: int,B: int,F: fun_int_int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,B)),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1045_order__subst2,axiom,
    ! [A2: filter_nat,B: filter_nat,F: fun_filter_nat_rat,C: rat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,B)),C)
       => ( ! [X3: filter_nat,Y2: filter_nat] :
              ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,X3)),aa_filter_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_filter_nat_rat(F,A2)),C) ) ) ) ).

% order_subst2
tff(fact_1046_order__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1047_order__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1048_order__subst1,axiom,
    ! [A2: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1049_order__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1050_order__subst1,axiom,
    ! [A2: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1051_order__subst1,axiom,
    ! [A2: nat,F: fun_int_nat,B: int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_int_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1052_order__subst1,axiom,
    ! [A2: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_rat_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_1053_order__subst1,axiom,
    ! [A2: int,F: fun_nat_int,B: nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_1054_order__subst1,axiom,
    ! [A2: int,F: fun_int_int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(F,C)) ) ) ) ).

% order_subst1
tff(fact_1055_order__subst1,axiom,
    ! [A2: filter_nat,F: fun_rat_filter_nat,B: rat,C: rat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),aa_rat_filter_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_rat_filter_nat(F,X3)),aa_rat_filter_nat(F,Y2)) )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),aa_rat_filter_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1056_Orderings_Oorder__eq__iff,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( ( A2 = B )
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1057_Orderings_Oorder__eq__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1058_Orderings_Oorder__eq__iff,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1059_Orderings_Oorder__eq__iff,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1060_Orderings_Oorder__eq__iff,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1061_inter__eq__subsetI,axiom,
    ! [S: set_nat,S4: set_nat,A: 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,A),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,A),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_1062_Int__left__commute,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) = 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,A),C5)) ).

% Int_left_commute
tff(fact_1063_Int__Collect__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o,P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
           => ( aa_fun1675921005893230629at_o_o(P,X3)
             => aa_fun1675921005893230629at_o_o(Q,X3) ) )
       => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(inf_in3376754634429691197_nat_o(A,aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P))),inf_in3376754634429691197_nat_o(B4,aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,Q))) ) ) ).

% Int_Collect_mono
tff(fact_1064_Int__Collect__mono,axiom,
    ! [A: set_int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => ( aa_int_o(P,X3)
             => aa_int_o(Q,X3) ) )
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,inf_inf_set_int(A,aa_fun_int_o_set_int(collect_int,P))),inf_inf_set_int(B4,aa_fun_int_o_set_int(collect_int,Q))) ) ) ).

% Int_Collect_mono
tff(fact_1065_Int__Collect__mono,axiom,
    ! [A: set_o,B4: set_o,P: fun_o_o,Q: fun_o_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ( aa_o_o(P,(X3))
             => aa_o_o(Q,(X3)) ) )
       => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,inf_inf_set_o(A,aa_fun_o_o_set_o(collect_o,P))),inf_inf_set_o(B4,aa_fun_o_o_set_o(collect_o,Q))) ) ) ).

% Int_Collect_mono
tff(fact_1066_Int__Collect__mono,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,P: fun_Product_unit_o,Q: fun_Product_unit_o] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => ( aa_Product_unit_o(P,X3)
             => aa_Product_unit_o(Q,X3) ) )
       => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,inf_in4660618365625256667t_unit(A,aa_fun8337552879598298401t_unit(collect_Product_unit,P))),inf_in4660618365625256667t_unit(B4,aa_fun8337552879598298401t_unit(collect_Product_unit,Q))) ) ) ).

% Int_Collect_mono
tff(fact_1067_Int__Collect__mono,axiom,
    ! [A: set_nat,B4: set_nat,P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_nat_o(P,X3)
             => aa_nat_o(Q,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,A),aa_fun_nat_o_set_nat(collect_nat,P))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),aa_fun_nat_o_set_nat(collect_nat,Q))) ) ) ).

% Int_Collect_mono
tff(fact_1068_Collect__mono__iff,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P)),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,Q))
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(P,X2)
         => aa_fun1675921005893230629at_o_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1069_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_fun_int_o_set_int(collect_int,P)),aa_fun_int_o_set_int(collect_int,Q))
    <=> ! [X2: int] :
          ( aa_int_o(P,X2)
         => aa_int_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1070_Collect__mono__iff,axiom,
    ! [P: fun_o_o,Q: fun_o_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_fun_o_o_set_o(collect_o,P)),aa_fun_o_o_set_o(collect_o,Q))
    <=> ! [X2: $o] :
          ( aa_o_o(P,(X2))
         => aa_o_o(Q,(X2)) ) ) ).

% Collect_mono_iff
tff(fact_1071_Collect__mono__iff,axiom,
    ! [P: fun_Product_unit_o,Q: fun_Product_unit_o] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_fun8337552879598298401t_unit(collect_Product_unit,P)),aa_fun8337552879598298401t_unit(collect_Product_unit,Q))
    <=> ! [X2: product_unit] :
          ( aa_Product_unit_o(P,X2)
         => aa_Product_unit_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1072_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_fun_nat_o_set_nat(collect_nat,P)),aa_fun_nat_o_set_nat(collect_nat,Q))
    <=> ! [X2: nat] :
          ( aa_nat_o(P,X2)
         => aa_nat_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1073_Un__left__commute,axiom,
    ! [A: set_nat,B4: set_nat,C5: 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,B4),C5)) = 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,A),C5)) ).

% Un_left_commute
tff(fact_1074_Un__Int__distrib2,axiom,
    ! [B4: set_nat,C5: set_nat,A: 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),C5)),A) = 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),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C5),A)) ).

% Un_Int_distrib2
tff(fact_1075_Un__Int__assoc__eq,axiom,
    ! [A: set_nat,B4: set_nat,C5: 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,A),B4)),C5) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C5)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A) ) ).

% Un_Int_assoc_eq
tff(fact_1076_Int__left__absorb,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ).

% Int_left_absorb
tff(fact_1077_Int__Un__distrib2,axiom,
    ! [B4: set_nat,C5: set_nat,A: 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),C5)),A) = 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),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C5),A)) ).

% Int_Un_distrib2
tff(fact_1078_Un__left__absorb,axiom,
    ! [A: set_nat,B4: 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),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) ).

% Un_left_absorb
tff(fact_1079_Un__Int__distrib,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,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,A),C5)) ).

% Un_Int_distrib
tff(fact_1080_Int__Un__distrib,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C5)) ).

% Int_Un_distrib
tff(fact_1081_set__eq__subset,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A = B4 )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A) ) ) ).

% set_eq_subset
tff(fact_1082_antisym,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
       => ( A2 = B ) ) ) ).

% antisym
tff(fact_1083_antisym,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
       => ( A2 = B ) ) ) ).

% antisym
tff(fact_1084_antisym,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
       => ( A2 = B ) ) ) ).

% antisym
tff(fact_1085_antisym,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
       => ( A2 = B ) ) ) ).

% antisym
tff(fact_1086_antisym,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
       => ( A2 = B ) ) ) ).

% antisym
tff(fact_1087_subset__trans,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C5)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5) ) ) ).

% subset_trans
tff(fact_1088_subset__Un__eq,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = B4 ) ) ).

% subset_Un_eq
tff(fact_1089_Un__Int__crazy,axiom,
    ! [A: set_nat,B4: set_nat,C5: 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,A),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C5),A)) = 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,A),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C5))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C5),A)) ).

% Un_Int_crazy
tff(fact_1090_Int__greatest,axiom,
    ! [C5: set_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),B4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) ) ) ).

% Int_greatest
tff(fact_1091_Collect__mono,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,Q: fun_fu5135969500543783852at_o_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(P,X3)
         => aa_fun1675921005893230629at_o_o(Q,X3) )
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,P)),aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,Q)) ) ).

% Collect_mono
tff(fact_1092_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_fun_int_o_set_int(collect_int,P)),aa_fun_int_o_set_int(collect_int,Q)) ) ).

% Collect_mono
tff(fact_1093_Collect__mono,axiom,
    ! [P: fun_o_o,Q: fun_o_o] :
      ( ! [X3: $o] :
          ( aa_o_o(P,(X3))
         => aa_o_o(Q,(X3)) )
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_fun_o_o_set_o(collect_o,P)),aa_fun_o_o_set_o(collect_o,Q)) ) ).

% Collect_mono
tff(fact_1094_Collect__mono,axiom,
    ! [P: fun_Product_unit_o,Q: fun_Product_unit_o] :
      ( ! [X3: product_unit] :
          ( aa_Product_unit_o(P,X3)
         => aa_Product_unit_o(Q,X3) )
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_fun8337552879598298401t_unit(collect_Product_unit,P)),aa_fun8337552879598298401t_unit(collect_Product_unit,Q)) ) ).

% Collect_mono
tff(fact_1095_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_fun_nat_o_set_nat(collect_nat,P)),aa_fun_nat_o_set_nat(collect_nat,Q)) ) ).

% Collect_mono
tff(fact_1096_subset__refl,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A) ).

% subset_refl
tff(fact_1097_Int__commute,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),A) ).

% Int_commute
tff(fact_1098_Int__absorb2,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = A ) ) ).

% Int_absorb2
tff(fact_1099_Int__absorb1,axiom,
    ! [B4: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = B4 ) ) ).

% Int_absorb1
tff(fact_1100_subset__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
    <=> ! [T: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(T),A)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(T),B4) ) ) ).

% subset_iff
tff(fact_1101_subset__iff,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A)
         => aa_set_int_o(member_int(T),B4) ) ) ).

% subset_iff
tff(fact_1102_subset__iff,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
    <=> ! [T: $o] :
          ( aa_set_o_o(member_o((T)),A)
         => aa_set_o_o(member_o((T)),B4) ) ) ).

% subset_iff
tff(fact_1103_subset__iff,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
    <=> ! [T: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(T),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(T),B4) ) ) ).

% subset_iff
tff(fact_1104_subset__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A)
         => aa_set_nat_o(member_nat(T),B4) ) ) ).

% subset_iff
tff(fact_1105_subset__UnE,axiom,
    ! [C5: set_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
     => ~ ! [A6: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A6),A)
           => ! [B7: set_nat] :
                ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B7),B4)
               => ( C5 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A6),B7) ) ) ) ) ).

% subset_UnE
tff(fact_1106_equalityD2,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A = B4 )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A) ) ).

% equalityD2
tff(fact_1107_equalityD1,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A = B4 )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ).

% equalityD1
tff(fact_1108_Un__commute,axiom,
    ! [A: set_nat,B4: set_nat] : 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,B4),A) ).

% Un_commute
tff(fact_1109_Un__absorb2,axiom,
    ! [B4: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = A ) ) ).

% Un_absorb2
tff(fact_1110_Un__absorb1,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = B4 ) ) ).

% Un_absorb1
tff(fact_1111_Int__lower2,axiom,
    ! [A: 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,A),B4)),B4) ).

% Int_lower2
tff(fact_1112_Int__lower1,axiom,
    ! [A: 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,A),B4)),A) ).

% Int_lower1
tff(fact_1113_Int__absorb,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),A) = A ).

% Int_absorb
tff(fact_1114_subset__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B4) ) ) ).

% subset_eq
tff(fact_1115_subset__eq,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => aa_set_int_o(member_int(X2),B4) ) ) ).

% subset_eq
tff(fact_1116_subset__eq,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
    <=> ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A)
         => aa_set_o_o(member_o((X2)),B4) ) ) ).

% subset_eq
tff(fact_1117_subset__eq,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(X2),B4) ) ) ).

% subset_eq
tff(fact_1118_subset__eq,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_nat_o(member_nat(X2),B4) ) ) ).

% subset_eq
tff(fact_1119_equalityE,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A = B4 )
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A) ) ) ).

% equalityE
tff(fact_1120_Un__upper2,axiom,
    ! [B4: set_nat,A: 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,A),B4)) ).

% Un_upper2
tff(fact_1121_Un__upper1,axiom,
    ! [A: set_nat,B4: 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),B4)) ).

% Un_upper1
tff(fact_1122_Un__absorb,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).

% Un_absorb
tff(fact_1123_Int__assoc,axiom,
    ! [A: set_nat,B4: set_nat,C5: 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,A),B4)),C5) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) ).

% Int_assoc
tff(fact_1124_Un__least,axiom,
    ! [A: set_nat,C5: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C5)
       => 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),B4)),C5) ) ) ).

% Un_least
tff(fact_1125_Un__assoc,axiom,
    ! [A: set_nat,B4: set_nat,C5: 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),B4)),C5) = 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,B4),C5)) ).

% Un_assoc
tff(fact_1126_Int__mono,axiom,
    ! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),D3)
       => 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),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C5),D3)) ) ) ).

% Int_mono
tff(fact_1127_subsetD,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o,C: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% subsetD
tff(fact_1128_subsetD,axiom,
    ! [A: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( aa_set_int_o(member_int(C),A)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% subsetD
tff(fact_1129_subsetD,axiom,
    ! [A: set_o,B4: set_o,C: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( aa_set_o_o(member_o((C)),A)
       => aa_set_o_o(member_o((C)),B4) ) ) ).

% subsetD
tff(fact_1130_subsetD,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% subsetD
tff(fact_1131_subsetD,axiom,
    ! [A: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_o(member_nat(C),A)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% subsetD
tff(fact_1132_in__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4) ) ) ).

% in_mono
tff(fact_1133_in__mono,axiom,
    ! [A: set_int,B4: set_int,X: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( aa_set_int_o(member_int(X),A)
       => aa_set_int_o(member_int(X),B4) ) ) ).

% in_mono
tff(fact_1134_in__mono,axiom,
    ! [A: set_o,B4: set_o,X: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( aa_set_o_o(member_o((X)),A)
       => aa_set_o_o(member_o((X)),B4) ) ) ).

% in_mono
tff(fact_1135_in__mono,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(X),B4) ) ) ).

% in_mono
tff(fact_1136_in__mono,axiom,
    ! [A: set_nat,B4: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_set_nat_o(member_nat(X),B4) ) ) ).

% in_mono
tff(fact_1137_ball__Un,axiom,
    ! [A: set_nat,B4: set_nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
         => aa_nat_o(P,X2) )
    <=> ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
           => aa_nat_o(P,X2) )
        & ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),B4)
           => aa_nat_o(P,X2) ) ) ) ).

% ball_Un
tff(fact_1138_Un__mono,axiom,
    ! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),D3)
       => 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),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C5),D3)) ) ) ).

% Un_mono
tff(fact_1139_bex__Un,axiom,
    ! [A: set_nat,B4: set_nat,P: fun_nat_o] :
      ( ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
          & aa_nat_o(P,X2) )
    <=> ( ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A)
            & aa_nat_o(P,X2) )
        | ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),B4)
            & aa_nat_o(P,X2) ) ) ) ).

% bex_Un
tff(fact_1140_IntD2,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A,B4))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ).

% IntD2
tff(fact_1141_IntD2,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A,B4))
     => aa_set_int_o(member_int(C),B4) ) ).

% IntD2
tff(fact_1142_IntD2,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A,B4))
     => aa_set_o_o(member_o((C)),B4) ) ).

% IntD2
tff(fact_1143_IntD2,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),inf_in4660618365625256667t_unit(A,B4))
     => aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ).

% IntD2
tff(fact_1144_IntD2,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
     => aa_set_nat_o(member_nat(C),B4) ) ).

% IntD2
tff(fact_1145_IntD1,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A,B4))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A) ) ).

% IntD1
tff(fact_1146_IntD1,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A,B4))
     => aa_set_int_o(member_int(C),A) ) ).

% IntD1
tff(fact_1147_IntD1,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A,B4))
     => aa_set_o_o(member_o((C)),A) ) ).

% IntD1
tff(fact_1148_IntD1,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),inf_in4660618365625256667t_unit(A,B4))
     => aa_set8780210009671629607unit_o(member_Product_unit(C),A) ) ).

% IntD1
tff(fact_1149_IntD1,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
     => aa_set_nat_o(member_nat(C),A) ) ).

% IntD1
tff(fact_1150_UnI2,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) ) ).

% UnI2
tff(fact_1151_UnI2,axiom,
    ! [C: int,B4: set_int,A: set_int] :
      ( aa_set_int_o(member_int(C),B4)
     => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) ) ).

% UnI2
tff(fact_1152_UnI2,axiom,
    ! [C: $o,B4: set_o,A: set_o] :
      ( aa_set_o_o(member_o((C)),B4)
     => aa_set_o_o(member_o((C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) ) ).

% UnI2
tff(fact_1153_UnI2,axiom,
    ! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),B4)
     => aa_set8780210009671629607unit_o(member_Product_unit(C),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) ) ).

% UnI2
tff(fact_1154_UnI2,axiom,
    ! [C: nat,B4: set_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(C),B4)
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) ) ).

% UnI2
tff(fact_1155_UnI1,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) ) ).

% UnI1
tff(fact_1156_UnI1,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A)
     => aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) ) ).

% UnI1
tff(fact_1157_UnI1,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),A)
     => aa_set_o_o(member_o((C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) ) ).

% UnI1
tff(fact_1158_UnI1,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
     => aa_set8780210009671629607unit_o(member_Product_unit(C),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) ) ).

% UnI1
tff(fact_1159_UnI1,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A)
     => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) ) ).

% UnI1
tff(fact_1160_IntE,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),inf_in3376754634429691197_nat_o(A,B4))
     => ~ ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% IntE
tff(fact_1161_IntE,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),inf_inf_set_int(A,B4))
     => ~ ( aa_set_int_o(member_int(C),A)
         => ~ aa_set_int_o(member_int(C),B4) ) ) ).

% IntE
tff(fact_1162_IntE,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),inf_inf_set_o(A,B4))
     => ~ ( aa_set_o_o(member_o((C)),A)
         => ~ aa_set_o_o(member_o((C)),B4) ) ) ).

% IntE
tff(fact_1163_IntE,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),inf_in4660618365625256667t_unit(A,B4))
     => ~ ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% IntE
tff(fact_1164_IntE,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
     => ~ ( aa_set_nat_o(member_nat(C),A)
         => ~ aa_set_nat_o(member_nat(C),B4) ) ) ).

% IntE
tff(fact_1165_UnE,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4))
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% UnE
tff(fact_1166_UnE,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4))
     => ( ~ aa_set_int_o(member_int(C),A)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% UnE
tff(fact_1167_UnE,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
     => ( ~ aa_set_o_o(member_o((C)),A)
       => aa_set_o_o(member_o((C)),B4) ) ) ).

% UnE
tff(fact_1168_UnE,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(C),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% UnE
tff(fact_1169_UnE,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
     => ( ~ aa_set_nat_o(member_nat(C),A)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% UnE
tff(fact_1170_dual__order_Otrans,axiom,
    ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),A2) ) ) ).

% dual_order.trans
tff(fact_1171_dual__order_Otrans,axiom,
    ! [B: set_nat,A2: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A2) ) ) ).

% dual_order.trans
tff(fact_1172_dual__order_Otrans,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2) ) ) ).

% dual_order.trans
tff(fact_1173_dual__order_Otrans,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2) ) ) ).

% dual_order.trans
tff(fact_1174_dual__order_Otrans,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2) ) ) ).

% dual_order.trans
tff(fact_1175_dual__order_Oantisym,axiom,
    ! [B: filter_nat,A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
       => ( A2 = B ) ) ) ).

% dual_order.antisym
tff(fact_1176_dual__order_Oantisym,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => ( A2 = B ) ) ) ).

% dual_order.antisym
tff(fact_1177_dual__order_Oantisym,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
       => ( A2 = B ) ) ) ).

% dual_order.antisym
tff(fact_1178_dual__order_Oantisym,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
       => ( A2 = B ) ) ) ).

% dual_order.antisym
tff(fact_1179_dual__order_Oantisym,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
       => ( A2 = B ) ) ) ).

% dual_order.antisym
tff(fact_1180_dual__order_Oeq__iff,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( ( A2 = B )
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B) ) ) ).

% dual_order.eq_iff
tff(fact_1181_dual__order_Oeq__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 = B )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ) ).

% dual_order.eq_iff
tff(fact_1182_dual__order_Oeq__iff,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ) ).

% dual_order.eq_iff
tff(fact_1183_dual__order_Oeq__iff,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ) ).

% dual_order.eq_iff
tff(fact_1184_dual__order_Oeq__iff,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ) ).

% dual_order.eq_iff
tff(fact_1185_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A2: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
     => ( ! [A3: rat,B2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A2),B) ) ) ).

% linorder_wlog
tff(fact_1186_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A2: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
     => ( ! [A3: nat,B2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A2),B) ) ) ).

% linorder_wlog
tff(fact_1187_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A2: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B2)
         => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
     => ( ! [A3: int,B2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B2),A3)
           => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
       => aa_int_o(aa_int_fun_int_o(P,A2),B) ) ) ).

% linorder_wlog
tff(fact_1188_order__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Z) ) ) ).

% order_trans
tff(fact_1189_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: 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,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).

% order_trans
tff(fact_1190_order__trans,axiom,
    ! [X: rat,Y: rat,Z: 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),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) ) ) ).

% order_trans
tff(fact_1191_order__trans,axiom,
    ! [X: nat,Y: nat,Z: 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),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).

% order_trans
tff(fact_1192_order__trans,axiom,
    ! [X: int,Y: int,Z: 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),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).

% order_trans
tff(fact_1193_order_Otrans,axiom,
    ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),C) ) ) ).

% order.trans
tff(fact_1194_order_Otrans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C) ) ) ).

% order.trans
tff(fact_1195_order_Otrans,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),C) ) ) ).

% order.trans
tff(fact_1196_order_Otrans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C) ) ) ).

% order.trans
tff(fact_1197_order_Otrans,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),C) ) ) ).

% order.trans
tff(fact_1198_order__antisym,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1199_order__antisym,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_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1200_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_1201_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_1202_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_1203_ord__le__eq__trans,axiom,
    ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( ( B = C )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),C) ) ) ).

% ord_le_eq_trans
tff(fact_1204_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( ( B = C )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C) ) ) ).

% ord_le_eq_trans
tff(fact_1205_ord__le__eq__trans,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),C) ) ) ).

% ord_le_eq_trans
tff(fact_1206_ord__le__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C) ) ) ).

% ord_le_eq_trans
tff(fact_1207_ord__le__eq__trans,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),C) ) ) ).

% ord_le_eq_trans
tff(fact_1208_ord__eq__le__trans,axiom,
    ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( A2 = B )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),C) ) ) ).

% ord_eq_le_trans
tff(fact_1209_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( ( A2 = B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C) ) ) ).

% ord_eq_le_trans
tff(fact_1210_ord__eq__le__trans,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( ( A2 = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),C) ) ) ).

% ord_eq_le_trans
tff(fact_1211_ord__eq__le__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C) ) ) ).

% ord_eq_le_trans
tff(fact_1212_ord__eq__le__trans,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( A2 = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),C) ) ) ).

% ord_eq_le_trans
tff(fact_1213_order__class_Oorder__eq__iff,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1214_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( 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,Y),X) ) ) ).

% order_class.order_eq_iff
tff(fact_1215_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_1216_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_1217_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_1218_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: 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),Z) )
     => ( ( 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),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),X)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1219_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: 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),Z) )
     => ( ( 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),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1220_le__cases3,axiom,
    ! [X: int,Y: int,Z: 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),Z) )
     => ( ( 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),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1221_nle__le,axiom,
    ! [A2: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
        & ( B != A2 ) ) ) ).

% nle_le
tff(fact_1222_nle__le,axiom,
    ! [A2: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
        & ( B != A2 ) ) ) ).

% nle_le
tff(fact_1223_nle__le,axiom,
    ! [A2: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
        & ( B != A2 ) ) ) ).

% nle_le
tff(fact_1224_Sigma__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,C5: set_fu3555266933746557007_nat_o,B4: fun_fu2098174926252251144et_nat,D3: fun_fu2098174926252251144et_nat] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),C5)
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_fun7872747175131507769et_nat(B4,X3)),aa_fun7872747175131507769et_nat(D3,X3)) )
       => ord_le4092436198336188292_o_nat(produc2348080924541952247_o_nat(A,B4),produc2348080924541952247_o_nat(C5,D3)) ) ) ).

% Sigma_mono
tff(fact_1225_Sigma__mono,axiom,
    ! [A: set_int,C5: set_int,B4: fun_int_set_nat,D3: fun_int_set_nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_int_set_nat(B4,X3)),aa_int_set_nat(D3,X3)) )
       => ord_le5333435103577920093nt_nat(produc1456381018704787142nt_nat(A,B4),produc1456381018704787142nt_nat(C5,D3)) ) ) ).

% Sigma_mono
tff(fact_1226_Sigma__mono,axiom,
    ! [A: set_o,C5: set_o,B4: fun_o_set_nat,D3: fun_o_set_nat] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_o_set_nat(B4,(X3))),aa_o_set_nat(D3,(X3))) )
       => ord_le6472378536984071597_o_nat(product_Sigma_o_nat(A,B4),product_Sigma_o_nat(C5,D3)) ) ) ).

% Sigma_mono
tff(fact_1227_Sigma__mono,axiom,
    ! [A: set_Product_unit,C5: set_Product_unit,B4: fun_Pr1628945989756059814et_nat,D3: fun_Pr1628945989756059814et_nat] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),C5)
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_Pro6064029160526250199et_nat(B4,X3)),aa_Pro6064029160526250199et_nat(D3,X3)) )
       => ord_le8121013740822022818it_nat(produc6875377002296012693it_nat(A,B4),produc6875377002296012693it_nat(C5,D3)) ) ) ).

% Sigma_mono
tff(fact_1228_Sigma__mono,axiom,
    ! [A: set_nat,C5: set_nat,B4: fun_nat_set_nat,D3: fun_nat_set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(B4,X3)),aa_nat_set_nat(D3,X3)) )
       => ord_le3146513528884898305at_nat(produc457027306803732586at_nat(A,B4),produc457027306803732586at_nat(C5,D3)) ) ) ).

% Sigma_mono
tff(fact_1229_rtrancl__Un__separator__converseE,axiom,
    ! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(sup_su6327502436637775413at_nat(P,Q)))
     => ( ! [X3: nat,Y2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),B),transi2905341329935302413cl_nat(P))
           => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),X3),Q)
             => ( Y2 = X3 ) ) )
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(P)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_1230_rtrancl__Un__separator__converseE,axiom,
    ! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(sup_su6024340866399070445nt_int(P,Q)))
     => ( ! [X3: int,Y2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),B),transi2902850859426252137cl_int(P))
           => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),X3),Q)
             => ( Y2 = X3 ) ) )
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(P)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_1231_rtrancl__Un__separatorE,axiom,
    ! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(sup_su6327502436637775413at_nat(P,Q)))
     => ( ! [X3: nat,Y2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),X3),transi2905341329935302413cl_nat(P))
           => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),Q)
             => ( X3 = Y2 ) ) )
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(P)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_1232_rtrancl__Un__separatorE,axiom,
    ! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(sup_su6024340866399070445nt_int(P,Q)))
     => ( ! [X3: int,Y2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),X3),transi2902850859426252137cl_int(P))
           => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),Q)
             => ( X3 = Y2 ) ) )
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(P)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_1233_disjoint__mono,axiom,
    ! [A2: set_Product_unit,A4: set_Product_unit,B: set_Product_unit,B3: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A2),A4)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B),B3)
       => ( ( inf_in4660618365625256667t_unit(A4,B3) = bot_bo3957492148770167129t_unit )
         => ( inf_in4660618365625256667t_unit(A2,B) = bot_bo3957492148770167129t_unit ) ) ) ) ).

% disjoint_mono
tff(fact_1234_disjoint__mono,axiom,
    ! [A2: set_o,A4: set_o,B: set_o,B3: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),A4)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B),B3)
       => ( ( inf_inf_set_o(A4,B3) = bot_bot_set_o )
         => ( inf_inf_set_o(A2,B) = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
tff(fact_1235_disjoint__mono,axiom,
    ! [A2: set_int,A4: set_int,B: set_int,B3: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),A4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B),B3)
       => ( ( inf_inf_set_int(A4,B3) = bot_bot_set_int )
         => ( inf_inf_set_int(A2,B) = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
tff(fact_1236_disjoint__mono,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A4: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o,B3: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),A4)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B),B3)
       => ( ( inf_in3376754634429691197_nat_o(A4,B3) = bot_bo8632255151191677115_nat_o )
         => ( inf_in3376754634429691197_nat_o(A2,B) = bot_bo8632255151191677115_nat_o ) ) ) ) ).

% disjoint_mono
tff(fact_1237_disjoint__mono,axiom,
    ! [A2: set_nat,A4: set_nat,B: set_nat,B3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B3)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A4),B3) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_1238_rtrancl__image__unfold__right,axiom,
    ! [E: set_Pr1261947904930325089at_nat,V2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(E),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),V2))),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),V2)) ).

% rtrancl_image_unfold_right
tff(fact_1239_rtrancl__reachable__induct,axiom,
    ! [I: set_nat,INV: set_nat,E: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),INV)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(E),INV)),INV)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),I)),INV) ) ) ).

% rtrancl_reachable_induct
tff(fact_1240_Image__Int__subset,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(image_nat_nat(R),A)),aa_set_nat_set_nat(image_nat_nat(R),B4))) ).

% Image_Int_subset
tff(fact_1241_Image__mono,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,A7: set_nat,A: set_nat] :
      ( ord_le3146513528884898305at_nat(R4,R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A7),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R4),A7)),aa_set_nat_set_nat(image_nat_nat(R2),A)) ) ) ).

% Image_mono
tff(fact_1242_refl__on__Un,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
      ( refl_on_nat(A,R2)
     => ( refl_on_nat(B4,S2)
       => refl_on_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4),sup_su6327502436637775413at_nat(R2,S2)) ) ) ).

% refl_on_Un
tff(fact_1243_refl__on__Int,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
      ( refl_on_nat(A,R2)
     => ( refl_on_nat(B4,S2)
       => refl_on_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4),inf_in2572325071724192079at_nat(R2,S2)) ) ) ).

% refl_on_Int
tff(fact_1244_trancl__union__outside,axiom,
    ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,U2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V),W),transi6264000038957366511cl_nat(sup_su6327502436637775413at_nat(E,U2)))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V),W),transi6264000038957366511cl_nat(E))
       => ? [X3: nat,Y2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V),X3),transi2905341329935302413cl_nat(sup_su6327502436637775413at_nat(E,U2)))
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),U2)
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),W),transi2905341329935302413cl_nat(sup_su6327502436637775413at_nat(E,U2))) ) ) ) ).

% trancl_union_outside
tff(fact_1245_trancl__union__outside,axiom,
    ! [V: int,W: int,E: set_Pr958786334691620121nt_int,U2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V),W),transi6261509568448316235cl_int(sup_su6024340866399070445nt_int(E,U2)))
     => ( ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V),W),transi6261509568448316235cl_int(E))
       => ? [X3: int,Y2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V),X3),transi2902850859426252137cl_int(sup_su6024340866399070445nt_int(E,U2)))
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),U2)
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),W),transi2902850859426252137cl_int(sup_su6024340866399070445nt_int(E,U2))) ) ) ) ).

% trancl_union_outside
tff(fact_1246_SigmaE,axiom,
    ! [C: product_prod_nat_int,A: set_nat,B4: fun_nat_set_int] :
      ( member4262671552274231302at_int(C,produc454536836294682310at_int(A,B4))
     => ~ ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ! [Y2: int] :
                ( aa_set_int_o(member_int(Y2),aa_nat_set_int(B4,X3))
               => ( C != product_Pair_nat_int(X3,Y2) ) ) ) ) ).

% SigmaE
tff(fact_1247_SigmaE,axiom,
    ! [C: product_prod_nat_o,A: set_nat,B4: fun_nat_set_o] :
      ( member6310962623043647828_nat_o(C,product_Sigma_nat_o(A,B4))
     => ~ ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ! [Y2: $o] :
                ( aa_set_o_o(member_o((Y2)),aa_nat_set_o(B4,X3))
               => ( C != aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(X3),(Y2)) ) ) ) ) ).

% SigmaE
tff(fact_1248_SigmaE,axiom,
    ! [C: produc526117512892780850t_unit,A: set_nat,B4: fun_na896991816813928310t_unit] :
      ( member348563419383025609t_unit(C,produc5506539147108004339t_unit(A,B4))
     => ~ ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ! [Y2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),aa_nat2714212862977473631t_unit(B4,X3))
               => ( C != aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(X3),Y2) ) ) ) ) ).

% SigmaE
tff(fact_1249_SigmaE,axiom,
    ! [C: product_prod_int_nat,A: set_int,B4: fun_int_set_nat] :
      ( member216504246829706758nt_nat(C,produc1456381018704787142nt_nat(A,B4))
     => ~ ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => ! [Y2: nat] :
                ( aa_set_nat_o(member_nat(Y2),aa_int_set_nat(B4,X3))
               => ( C != aa_nat8602574874988167631nt_nat(product_Pair_int_nat(X3),Y2) ) ) ) ) ).

% SigmaE
tff(fact_1250_SigmaE,axiom,
    ! [C: product_prod_int_o,A: set_int,B4: fun_int_set_o] :
      ( member4489920277610959864_int_o(C,product_Sigma_int_o(A,B4))
     => ~ ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => ! [Y2: $o] :
                ( aa_set_o_o(member_o((Y2)),aa_int_set_o(B4,X3))
               => ( C != aa_o_P5078366980852708413_int_o(product_Pair_int_o(X3),(Y2)) ) ) ) ) ).

% SigmaE
tff(fact_1251_SigmaE,axiom,
    ! [C: produc6865911496949736918t_unit,A: set_int,B4: fun_in4731230917697954842t_unit] :
      ( member6688357403439981677t_unit(C,produc2622961094310184599t_unit(A,B4))
     => ~ ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => ! [Y2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),aa_int6548451963861500163t_unit(B4,X3))
               => ( C != aa_Pro399436582157352231t_unit(produc550872421827702152t_unit(X3),Y2) ) ) ) ) ).

% SigmaE
tff(fact_1252_SigmaE,axiom,
    ! [C: product_prod_o_nat,A: set_o,B4: fun_o_set_nat] :
      ( member2802428098988154798_o_nat(C,product_Sigma_o_nat(A,B4))
     => ~ ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ! [Y2: nat] :
                ( aa_set_nat_o(member_nat(Y2),aa_o_set_nat(B4,(X3)))
               => ( C != aa_nat139401504800077733_o_nat(product_Pair_o_nat((X3)),Y2) ) ) ) ) ).

% SigmaE
tff(fact_1253_SigmaE,axiom,
    ! [C: product_prod_o_int,A: set_o,B4: fun_o_set_int] :
      ( member7847949116333733898_o_int(C,product_Sigma_o_int(A,B4))
     => ~ ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ! [Y2: int] :
                ( aa_set_int_o(member_int(Y2),aa_o_set_int(B4,(X3)))
               => ( C != product_Pair_o_int((X3),Y2) ) ) ) ) ).

% SigmaE
tff(fact_1254_SigmaE,axiom,
    ! [C: product_prod_o_o,A: set_o,B4: fun_o_set_o] :
      ( member7466972457876170832od_o_o(C,product_Sigma_o_o(A,B4))
     => ~ ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ! [Y2: $o] :
                ( aa_set_o_o(member_o((Y2)),aa_o_set_o(B4,(X3)))
               => ( C != aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X3)),(Y2)) ) ) ) ) ).

% SigmaE
tff(fact_1255_SigmaE,axiom,
    ! [C: produc3788275437394581276t_unit,A: set_o,B4: fun_o_1833192751255691672t_unit] :
      ( member9215336023991391429t_unit(C,produc411009820408249917t_unit(A,B4))
     => ~ ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => ! [Y2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),aa_o_s3492205133957624465t_unit(B4,(X3)))
               => ( C != aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((X3)),Y2) ) ) ) ) ).

% SigmaE
tff(fact_1256_SigmaD1,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: fun_he8038253169496836892et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),produc5260988393495149579et_nat(A,B4))
     => member242389114213480694t_unit(A2,A) ) ).

% SigmaD1
tff(fact_1257_SigmaD1,axiom,
    ! [A2: nat,B: nat,A: set_nat,B4: fun_nat_set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),produc457027306803732586at_nat(A,B4))
     => aa_set_nat_o(member_nat(A2),A) ) ).

% SigmaD1
tff(fact_1258_SigmaD1,axiom,
    ! [A2: int,B: int,A: set_int,B4: fun_int_set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),produc1453890548195736866nt_int(A,B4))
     => aa_set_int_o(member_int(A2),A) ) ).

% SigmaD1
tff(fact_1259_SigmaD1,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A: set_fu3555266933746557007_nat_o,B4: fun_fu2890809920214373946et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),produc6971759541029228329et_nat(A,B4))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A) ) ).

% SigmaD1
tff(fact_1260_SigmaD1,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A: set_fu3555266933746557007_nat_o,B4: fun_fu8357309440898100312et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),produc5673588578087545031et_nat(A,B4))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A) ) ).

% SigmaD1
tff(fact_1261_SigmaD2,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: fun_he8038253169496836892et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),produc5260988393495149579et_nat(A,B4))
     => member_set_nat(B,aa_hea431243932251723341et_nat(B4,A2)) ) ).

% SigmaD2
tff(fact_1262_SigmaD2,axiom,
    ! [A2: nat,B: nat,A: set_nat,B4: fun_nat_set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),produc457027306803732586at_nat(A,B4))
     => aa_set_nat_o(member_nat(B),aa_nat_set_nat(B4,A2)) ) ).

% SigmaD2
tff(fact_1263_SigmaD2,axiom,
    ! [A2: int,B: int,A: set_int,B4: fun_int_set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),produc1453890548195736866nt_int(A,B4))
     => aa_set_int_o(member_int(B),aa_int_set_int(B4,A2)) ) ).

% SigmaD2
tff(fact_1264_SigmaD2,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A: set_fu3555266933746557007_nat_o,B4: fun_fu2890809920214373946et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),produc6971759541029228329et_nat(A,B4))
     => member6260224972018164377et_nat(B,aa_fun786139119071955819et_nat(B4,A2)) ) ).

% SigmaD2
tff(fact_1265_SigmaD2,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A: set_fu3555266933746557007_nat_o,B4: fun_fu8357309440898100312et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),produc5673588578087545031et_nat(A,B4))
     => member4513763875858039863et_nat(B,aa_fun4924015400185905801et_nat(B4,A2)) ) ).

% SigmaD2
tff(fact_1266_SigmaE2,axiom,
    ! [A2: nat,B: int,A: set_nat,B4: fun_nat_set_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(A2,B),produc454536836294682310at_int(A,B4))
     => ~ ( aa_set_nat_o(member_nat(A2),A)
         => ~ aa_set_int_o(member_int(B),aa_nat_set_int(B4,A2)) ) ) ).

% SigmaE2
tff(fact_1267_SigmaE2,axiom,
    ! [A2: nat,B: $o,A: set_nat,B4: fun_nat_set_o] :
      ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A2),(B)),product_Sigma_nat_o(A,B4))
     => ~ ( aa_set_nat_o(member_nat(A2),A)
         => ~ aa_set_o_o(member_o((B)),aa_nat_set_o(B4,A2)) ) ) ).

% SigmaE2
tff(fact_1268_SigmaE2,axiom,
    ! [A2: nat,B: product_unit,A: set_nat,B4: fun_na896991816813928310t_unit] :
      ( member348563419383025609t_unit(aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(A2),B),produc5506539147108004339t_unit(A,B4))
     => ~ ( aa_set_nat_o(member_nat(A2),A)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(B),aa_nat2714212862977473631t_unit(B4,A2)) ) ) ).

% SigmaE2
tff(fact_1269_SigmaE2,axiom,
    ! [A2: int,B: nat,A: set_int,B4: fun_int_set_nat] :
      ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A2),B),produc1456381018704787142nt_nat(A,B4))
     => ~ ( aa_set_int_o(member_int(A2),A)
         => ~ aa_set_nat_o(member_nat(B),aa_int_set_nat(B4,A2)) ) ) ).

% SigmaE2
tff(fact_1270_SigmaE2,axiom,
    ! [A2: int,B: $o,A: set_int,B4: fun_int_set_o] :
      ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A2),(B)),product_Sigma_int_o(A,B4))
     => ~ ( aa_set_int_o(member_int(A2),A)
         => ~ aa_set_o_o(member_o((B)),aa_int_set_o(B4,A2)) ) ) ).

% SigmaE2
tff(fact_1271_SigmaE2,axiom,
    ! [A2: int,B: product_unit,A: set_int,B4: fun_in4731230917697954842t_unit] :
      ( member6688357403439981677t_unit(aa_Pro399436582157352231t_unit(produc550872421827702152t_unit(A2),B),produc2622961094310184599t_unit(A,B4))
     => ~ ( aa_set_int_o(member_int(A2),A)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(B),aa_int6548451963861500163t_unit(B4,A2)) ) ) ).

% SigmaE2
tff(fact_1272_SigmaE2,axiom,
    ! [A2: $o,B: nat,A: set_o,B4: fun_o_set_nat] :
      ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A2)),B),product_Sigma_o_nat(A,B4))
     => ~ ( aa_set_o_o(member_o((A2)),A)
         => ~ aa_set_nat_o(member_nat(B),aa_o_set_nat(B4,(A2))) ) ) ).

% SigmaE2
tff(fact_1273_SigmaE2,axiom,
    ! [A2: $o,B: int,A: set_o,B4: fun_o_set_int] :
      ( member7847949116333733898_o_int(product_Pair_o_int((A2),B),product_Sigma_o_int(A,B4))
     => ~ ( aa_set_o_o(member_o((A2)),A)
         => ~ aa_set_int_o(member_int(B),aa_o_set_int(B4,(A2))) ) ) ).

% SigmaE2
tff(fact_1274_SigmaE2,axiom,
    ! [A2: $o,B: $o,A: set_o,B4: fun_o_set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),product_Sigma_o_o(A,B4))
     => ~ ( aa_set_o_o(member_o((A2)),A)
         => ~ aa_set_o_o(member_o((B)),aa_o_set_o(B4,(A2))) ) ) ).

% SigmaE2
tff(fact_1275_SigmaE2,axiom,
    ! [A2: $o,B: product_unit,A: set_o,B4: fun_o_1833192751255691672t_unit] :
      ( member9215336023991391429t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A2)),B),produc411009820408249917t_unit(A,B4))
     => ~ ( aa_set_o_o(member_o((A2)),A)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(B),aa_o_s3492205133957624465t_unit(B4,(A2))) ) ) ).

% SigmaE2
tff(fact_1276_trancl__image__by__rtrancl,axiom,
    ! [E: set_Pr1261947904930325089at_nat,Vi: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(image_nat_nat(transi6264000038957366511cl_nat(E)),Vi)),Vi) = aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),Vi) ).

% trancl_image_by_rtrancl
tff(fact_1277_converse__rtranclE_H,axiom,
    ! [U: nat,V: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V),transi2905341329935302413cl_nat(R))
     => ( ( U != V )
       => ~ ! [Vh: nat] :
              ( ( U != Vh )
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),Vh),R)
               => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Vh),V),transi2905341329935302413cl_nat(R)) ) ) ) ) ).

% converse_rtranclE'
tff(fact_1278_converse__rtranclE_H,axiom,
    ! [U: int,V: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),V),transi2902850859426252137cl_int(R))
     => ( ( U != V )
       => ~ ! [Vh: int] :
              ( ( U != Vh )
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),Vh),R)
               => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Vh),V),transi2902850859426252137cl_int(R)) ) ) ) ) ).

% converse_rtranclE'
tff(fact_1279_rtrancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),transi2905341329935302413cl_nat(R2))
     => ( ( A22 != A1 )
       => ~ ! [B2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),B2),transi2905341329935302413cl_nat(R2))
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),A22),R2) ) ) ) ).

% rtrancl.cases
tff(fact_1280_rtrancl_Ocases,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),transi2902850859426252137cl_int(R2))
     => ( ( A22 != A1 )
       => ~ ! [B2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),B2),transi2902850859426252137cl_int(R2))
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),A22),R2) ) ) ) ).

% rtrancl.cases
tff(fact_1281_rtrancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),transi2905341329935302413cl_nat(R2))
    <=> ( ? [A5: nat] :
            ( ( A1 = A5 )
            & ( A22 = A5 ) )
        | ? [A5: nat,B6: nat,C4: nat] :
            ( ( A1 = A5 )
            & ( A22 = C4 )
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),transi2905341329935302413cl_nat(R2))
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B6),C4),R2) ) ) ) ).

% rtrancl.simps
tff(fact_1282_rtrancl_Osimps,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),transi2902850859426252137cl_int(R2))
    <=> ( ? [A5: int] :
            ( ( A1 = A5 )
            & ( A22 = A5 ) )
        | ? [A5: int,B6: int,C4: int] :
            ( ( A1 = A5 )
            & ( A22 = C4 )
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),transi2902850859426252137cl_int(R2))
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B6),C4),R2) ) ) ) ).

% rtrancl.simps
tff(fact_1283_rtrancl_Ortrancl__refl,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] : member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),transi2905341329935302413cl_nat(R2)) ).

% rtrancl.rtrancl_refl
tff(fact_1284_rtrancl_Ortrancl__refl,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] : member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),transi2902850859426252137cl_int(R2)) ).

% rtrancl.rtrancl_refl
tff(fact_1285_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),R2)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi2905341329935302413cl_nat(R2)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_1286_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),R2)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi2902850859426252137cl_int(R2)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_1287_rtranclE,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R2))
     => ( ( A2 != B )
       => ~ ! [Y2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y2),transi2905341329935302413cl_nat(R2))
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),B),R2) ) ) ) ).

% rtranclE
tff(fact_1288_rtranclE,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R2))
     => ( ( A2 != B )
       => ~ ! [Y2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),Y2),transi2902850859426252137cl_int(R2))
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),B),R2) ) ) ) ).

% rtranclE
tff(fact_1289_rtrancl__trans,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),transi2905341329935302413cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),transi2905341329935302413cl_nat(R2)) ) ) ).

% rtrancl_trans
tff(fact_1290_rtrancl__trans,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z),transi2902850859426252137cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),transi2902850859426252137cl_int(R2)) ) ) ).

% rtrancl_trans
tff(fact_1291_rtrancl__induct,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R2))
     => ( aa_nat_o(P,A2)
       => ( ! [Y2: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y2),transi2905341329935302413cl_nat(R2))
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),R2)
               => ( aa_nat_o(P,Y2)
                 => aa_nat_o(P,Z3) ) ) )
         => aa_nat_o(P,B) ) ) ) ).

% rtrancl_induct
tff(fact_1292_rtrancl__induct,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R2))
     => ( aa_int_o(P,A2)
       => ( ! [Y2: int,Z3: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),Y2),transi2902850859426252137cl_int(R2))
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),R2)
               => ( aa_int_o(P,Y2)
                 => aa_int_o(P,Z3) ) ) )
         => aa_int_o(P,B) ) ) ) ).

% rtrancl_induct
tff(fact_1293_converse__rtranclE,axiom,
    ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),transi2905341329935302413cl_nat(R2))
     => ( ( X != Z )
       => ~ ! [Y2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y2),R2)
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z),transi2905341329935302413cl_nat(R2)) ) ) ) ).

% converse_rtranclE
tff(fact_1294_converse__rtranclE,axiom,
    ! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),transi2902850859426252137cl_int(R2))
     => ( ( X != Z )
       => ~ ! [Y2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y2),R2)
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z),transi2902850859426252137cl_int(R2)) ) ) ) ).

% converse_rtranclE
tff(fact_1295_converse__rtrancl__induct,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R2))
     => ( aa_nat_o(P,B)
       => ( ! [Y2: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),R2)
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z3),B),transi2905341329935302413cl_nat(R2))
               => ( aa_nat_o(P,Z3)
                 => aa_nat_o(P,Y2) ) ) )
         => aa_nat_o(P,A2) ) ) ) ).

% converse_rtrancl_induct
tff(fact_1296_converse__rtrancl__induct,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R2))
     => ( aa_int_o(P,B)
       => ( ! [Y2: int,Z3: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),R2)
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Z3),B),transi2902850859426252137cl_int(R2))
               => ( aa_int_o(P,Z3)
                 => aa_int_o(P,Y2) ) ) )
         => aa_int_o(P,A2) ) ) ) ).

% converse_rtrancl_induct
tff(fact_1297_converse__rtrancl__into__rtrancl,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),transi2905341329935302413cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi2905341329935302413cl_nat(R2)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_1298_converse__rtrancl__into__rtrancl,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),transi2902850859426252137cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi2902850859426252137cl_int(R2)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_1299_rtrancl__induct2,axiom,
    ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: fun_he5746958392356810273_nat_o] :
      ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ax),Ay),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Bx),By)),transi6919426408697524543et_nat(R2))
     => ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Ax),Ay)
       => ( ! [A3: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
              ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ax),Ay),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2)),transi6919426408697524543et_nat(R2))
             => ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Aa),Ba)),R2)
               => ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A3),B2)
                 => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Aa),Ba) ) ) )
         => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_1300_rtrancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: fun_nat_fun_nat_o] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Bx),By)),transi8609417484261754244at_nat(R2))
     => ( aa_nat_o(aa_nat_fun_nat_o(P,Ax),Ay)
       => ( ! [A3: nat,B2: nat,Aa: nat,Ba: nat] :
              ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2)),transi8609417484261754244at_nat(R2))
             => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),Ba)),R2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
                 => aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_1301_rtrancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: fun_int_fun_int_o] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Ax),Ay),aa_int7462565939943104719nt_int(product_Pair_int_int(Bx),By)),transi5430920176653612092nt_int(R2))
     => ( aa_int_o(aa_int_fun_int_o(P,Ax),Ay)
       => ( ! [A3: int,B2: int,Aa: int,Ba: int] :
              ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Ax),Ay),aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2)),transi5430920176653612092nt_int(R2))
             => ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Aa),Ba)),R2)
               => ( aa_int_o(aa_int_fun_int_o(P,A3),B2)
                 => aa_int_o(aa_int_fun_int_o(P,Aa),Ba) ) ) )
         => aa_int_o(aa_int_fun_int_o(P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_1302_rtrancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc3658429121746597890et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc3658429121746597890et_nat,R2: set_Pr9066344408807579265et_nat,P: fun_fu8419763957888848259_nat_o] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Ax),Ay),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Bx),By)),transi722511410275069917et_nat(R2))
     => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Ax),Ay)
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc3658429121746597890et_nat] :
              ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Ax),Ay),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2)),transi722511410275069917et_nat(R2))
             => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Aa),Ba)),R2)
               => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B2)
                 => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Aa),Ba) ) ) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_1303_rtrancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc8799533983703829792et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc8799533983703829792et_nat,R2: set_Pr7057080801110549565et_nat,P: fun_fu2611741329380808165_nat_o] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Ax),Ay),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Bx),By)),transi2528772592005381627et_nat(R2))
     => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Ax),Ay)
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc8799533983703829792et_nat] :
              ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Ax),Ay),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2)),transi2528772592005381627et_nat(R2))
             => ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Aa),Ba)),R2)
               => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B2)
                 => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Aa),Ba) ) ) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_1304_converse__rtranclE2,axiom,
    ! [Xa: heap_e7401611519738050253t_unit,Xb: set_nat,Za: heap_e7401611519738050253t_unit,Zb: set_nat,R2: set_Pr719794911490849221et_nat] :
      ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xa),Xb),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Za),Zb)),transi6919426408697524543et_nat(R2))
     => ( ( aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xa),Xb) != aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Za),Zb) )
       => ~ ! [A3: heap_e7401611519738050253t_unit,B2: set_nat] :
              ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Xa),Xb),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2)),R2)
             => ~ member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Za),Zb)),transi6919426408697524543et_nat(R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_1305_converse__rtranclE2,axiom,
    ! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R2: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Za),Zb)),transi8609417484261754244at_nat(R2))
     => ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xb) != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Za),Zb) )
       => ~ ! [A3: nat,B2: nat] :
              ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2)),R2)
             => ~ member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Za),Zb)),transi8609417484261754244at_nat(R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_1306_converse__rtranclE2,axiom,
    ! [Xa: int,Xb: int,Za: int,Zb: int,R2: set_Pr2560585780119916871nt_int] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa),Xb),aa_int7462565939943104719nt_int(product_Pair_int_int(Za),Zb)),transi5430920176653612092nt_int(R2))
     => ( ( aa_int7462565939943104719nt_int(product_Pair_int_int(Xa),Xb) != aa_int7462565939943104719nt_int(product_Pair_int_int(Za),Zb) )
       => ~ ! [A3: int,B2: int] :
              ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa),Xb),aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2)),R2)
             => ~ member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Za),Zb)),transi5430920176653612092nt_int(R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_1307_converse__rtranclE2,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Za: fun_Pr6460944699146269935_nat_o,Zb: produc3658429121746597890et_nat,R2: set_Pr9066344408807579265et_nat] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Za),Zb)),transi722511410275069917et_nat(R2))
     => ( ( aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb) != aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Za),Zb) )
       => ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
              ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2)),R2)
             => ~ member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Za),Zb)),transi722511410275069917et_nat(R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_1308_converse__rtranclE2,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xb: produc8799533983703829792et_nat,Za: fun_Pr6460944699146269935_nat_o,Zb: produc8799533983703829792et_nat,R2: set_Pr7057080801110549565et_nat] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Xa),Xb),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Za),Zb)),transi2528772592005381627et_nat(R2))
     => ( ( aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Xa),Xb) != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Za),Zb) )
       => ~ ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
              ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Xa),Xb),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2)),R2)
             => ~ member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Za),Zb)),transi2528772592005381627et_nat(R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_1309_converse__rtrancl__induct2,axiom,
    ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: fun_he5746958392356810273_nat_o] :
      ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ax),Ay),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Bx),By)),transi6919426408697524543et_nat(R2))
     => ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Bx),By)
       => ( ! [A3: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
              ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A3),B2),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Aa),Ba)),R2)
             => ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Aa),Ba),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Bx),By)),transi6919426408697524543et_nat(R2))
               => ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Aa),Ba)
                 => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A3),B2) ) ) )
         => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_1310_converse__rtrancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: fun_nat_fun_nat_o] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Bx),By)),transi8609417484261754244at_nat(R2))
     => ( aa_nat_o(aa_nat_fun_nat_o(P,Bx),By)
       => ( ! [A3: nat,B2: nat,Aa: nat,Ba: nat] :
              ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),Ba)),R2)
             => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),Ba),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Bx),By)),transi8609417484261754244at_nat(R2))
               => ( aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba)
                 => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_1311_converse__rtrancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: fun_int_fun_int_o] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Ax),Ay),aa_int7462565939943104719nt_int(product_Pair_int_int(Bx),By)),transi5430920176653612092nt_int(R2))
     => ( aa_int_o(aa_int_fun_int_o(P,Bx),By)
       => ( ! [A3: int,B2: int,Aa: int,Ba: int] :
              ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Aa),Ba)),R2)
             => ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Aa),Ba),aa_int7462565939943104719nt_int(product_Pair_int_int(Bx),By)),transi5430920176653612092nt_int(R2))
               => ( aa_int_o(aa_int_fun_int_o(P,Aa),Ba)
                 => aa_int_o(aa_int_fun_int_o(P,A3),B2) ) ) )
         => aa_int_o(aa_int_fun_int_o(P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_1312_converse__rtrancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc3658429121746597890et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc3658429121746597890et_nat,R2: set_Pr9066344408807579265et_nat,P: fun_fu8419763957888848259_nat_o] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Ax),Ay),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Bx),By)),transi722511410275069917et_nat(R2))
     => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Bx),By)
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc3658429121746597890et_nat] :
              ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A3),B2),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Aa),Ba)),R2)
             => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Aa),Ba),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Bx),By)),transi722511410275069917et_nat(R2))
               => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Aa),Ba)
                 => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,A3),B2) ) ) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_1313_converse__rtrancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc8799533983703829792et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc8799533983703829792et_nat,R2: set_Pr7057080801110549565et_nat,P: fun_fu2611741329380808165_nat_o] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Ax),Ay),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Bx),By)),transi2528772592005381627et_nat(R2))
     => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Bx),By)
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc8799533983703829792et_nat] :
              ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A3),B2),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Aa),Ba)),R2)
             => ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Aa),Ba),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Bx),By)),transi2528772592005381627et_nat(R2))
               => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Aa),Ba)
                 => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,A3),B2) ) ) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_1314_Un__empty__right,axiom,
    ! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),bot_bo3957492148770167129t_unit) = A ).

% Un_empty_right
tff(fact_1315_Un__empty__right,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),bot_bot_set_o) = A ).

% Un_empty_right
tff(fact_1316_Un__empty__right,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% Un_empty_right
tff(fact_1317_Un__empty__right,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ).

% Un_empty_right
tff(fact_1318_Un__empty__right,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),bot_bo8632255151191677115_nat_o) = A ).

% Un_empty_right
tff(fact_1319_Un__empty__left,axiom,
    ! [B4: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,bot_bo3957492148770167129t_unit),B4) = B4 ).

% Un_empty_left
tff(fact_1320_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_1321_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_1322_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_1323_Un__empty__left,axiom,
    ! [B4: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,bot_bo8632255151191677115_nat_o),B4) = B4 ).

% Un_empty_left
tff(fact_1324_disjoint__iff__not__equal,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
         => ! [Xa2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),B4)
             => ( X2 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1325_disjoint__iff__not__equal,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
    <=> ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A)
         => ! [Xa2: $o] :
              ( aa_set_o_o(member_o((Xa2)),B4)
             => ( (X2)
              <=> ~ (Xa2) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1326_disjoint__iff__not__equal,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => ! [Xa2: nat] :
              ( aa_set_nat_o(member_nat(Xa2),B4)
             => ( X2 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1327_disjoint__iff__not__equal,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),B4)
             => ( X2 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1328_disjoint__iff__not__equal,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
         => ! [Xa2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa2),B4)
             => ( X2 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1329_Int__empty__right,axiom,
    ! [A: set_Product_unit] : inf_in4660618365625256667t_unit(A,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% Int_empty_right
tff(fact_1330_Int__empty__right,axiom,
    ! [A: set_o] : inf_inf_set_o(A,bot_bot_set_o) = bot_bot_set_o ).

% Int_empty_right
tff(fact_1331_Int__empty__right,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_1332_Int__empty__right,axiom,
    ! [A: set_int] : inf_inf_set_int(A,bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_1333_Int__empty__right,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(A,bot_bo8632255151191677115_nat_o) = bot_bo8632255151191677115_nat_o ).

% Int_empty_right
tff(fact_1334_Int__empty__left,axiom,
    ! [B4: set_Product_unit] : inf_in4660618365625256667t_unit(bot_bo3957492148770167129t_unit,B4) = bot_bo3957492148770167129t_unit ).

% Int_empty_left
tff(fact_1335_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_1336_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_1337_Int__empty__left,axiom,
    ! [B4: set_int] : inf_inf_set_int(bot_bot_set_int,B4) = bot_bot_set_int ).

% Int_empty_left
tff(fact_1338_Int__empty__left,axiom,
    ! [B4: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(bot_bo8632255151191677115_nat_o,B4) = bot_bo8632255151191677115_nat_o ).

% Int_empty_left
tff(fact_1339_disjoint__iff,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(X2),B4) ) ) ).

% disjoint_iff
tff(fact_1340_disjoint__iff,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
    <=> ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),A)
         => ~ aa_set_o_o(member_o((X2)),B4) ) ) ).

% disjoint_iff
tff(fact_1341_disjoint__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => ~ aa_set_nat_o(member_nat(X2),B4) ) ) ).

% disjoint_iff
tff(fact_1342_disjoint__iff,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => ~ aa_set_int_o(member_int(X2),B4) ) ) ).

% disjoint_iff
tff(fact_1343_disjoint__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B4) ) ) ).

% disjoint_iff
tff(fact_1344_Int__emptyI,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),B4) )
     => ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit ) ) ).

% Int_emptyI
tff(fact_1345_Int__emptyI,axiom,
    ! [A: set_o,B4: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => ~ aa_set_o_o(member_o((X3)),B4) )
     => ( inf_inf_set_o(A,B4) = bot_bot_set_o ) ) ).

% Int_emptyI
tff(fact_1346_Int__emptyI,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ~ aa_set_nat_o(member_nat(X3),B4) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_1347_Int__emptyI,axiom,
    ! [A: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ~ aa_set_int_o(member_int(X3),B4) )
     => ( inf_inf_set_int(A,B4) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_1348_Int__emptyI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),B4) )
     => ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o ) ) ).

% Int_emptyI
tff(fact_1349_disjointI,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A2)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),B) )
     => ( inf_in4660618365625256667t_unit(A2,B) = bot_bo3957492148770167129t_unit ) ) ).

% disjointI
tff(fact_1350_disjointI,axiom,
    ! [A2: set_o,B: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A2)
         => ~ aa_set_o_o(member_o((X3)),B) )
     => ( inf_inf_set_o(A2,B) = bot_bot_set_o ) ) ).

% disjointI
tff(fact_1351_disjointI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => ~ aa_set_nat_o(member_nat(X3),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_1352_disjointI,axiom,
    ! [A2: set_int,B: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A2)
         => ~ aa_set_int_o(member_int(X3),B) )
     => ( inf_inf_set_int(A2,B) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_1353_disjointI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A2)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),B) )
     => ( inf_in3376754634429691197_nat_o(A2,B) = bot_bo8632255151191677115_nat_o ) ) ).

% disjointI
tff(fact_1354_Int__insert__right,axiom,
    ! [A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] :
      inf_in3376754634429691197_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),inf_in3376754634429691197_nat_o(A,B4)),inf_in3376754634429691197_nat_o(A,B4)) ).

% Int_insert_right
tff(fact_1355_Int__insert__right,axiom,
    ! [A: set_int,A2: int,B4: set_int] :
      inf_inf_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = $ite(aa_set_int_o(member_int(A2),A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),inf_inf_set_int(A,B4)),inf_inf_set_int(A,B4)) ).

% Int_insert_right
tff(fact_1356_Int__insert__right,axiom,
    ! [A: set_o,A2: $o,B4: set_o] :
      inf_inf_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = $ite(aa_set_o_o(member_o((A2)),A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),inf_inf_set_o(A,B4)),inf_inf_set_o(A,B4)) ).

% Int_insert_right
tff(fact_1357_Int__insert__right,axiom,
    ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
      inf_in4660618365625256667t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(A2),A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),inf_in4660618365625256667t_unit(A,B4)),inf_in4660618365625256667t_unit(A,B4)) ).

% Int_insert_right
tff(fact_1358_Int__insert__right,axiom,
    ! [A: set_nat,A2: nat,B4: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = $ite(aa_set_nat_o(member_nat(A2),A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) ).

% Int_insert_right
tff(fact_1359_Int__insert__left,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o,C5: set_fu3555266933746557007_nat_o] :
      inf_in3376754634429691197_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4),C5) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),C5),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),inf_in3376754634429691197_nat_o(B4,C5)),inf_in3376754634429691197_nat_o(B4,C5)) ).

% Int_insert_left
tff(fact_1360_Int__insert__left,axiom,
    ! [A2: int,B4: set_int,C5: set_int] :
      inf_inf_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4),C5) = $ite(aa_set_int_o(member_int(A2),C5),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),inf_inf_set_int(B4,C5)),inf_inf_set_int(B4,C5)) ).

% Int_insert_left
tff(fact_1361_Int__insert__left,axiom,
    ! [A2: $o,B4: set_o,C5: set_o] :
      inf_inf_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4),C5) = $ite(aa_set_o_o(member_o((A2)),C5),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),inf_inf_set_o(B4,C5)),inf_inf_set_o(B4,C5)) ).

% Int_insert_left
tff(fact_1362_Int__insert__left,axiom,
    ! [A2: product_unit,B4: set_Product_unit,C5: set_Product_unit] :
      inf_in4660618365625256667t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4),C5) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(A2),C5),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),inf_in4660618365625256667t_unit(B4,C5)),inf_in4660618365625256667t_unit(B4,C5)) ).

% Int_insert_left
tff(fact_1363_Int__insert__left,axiom,
    ! [A2: nat,B4: set_nat,C5: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)),C5) = $ite(aa_set_nat_o(member_nat(A2),C5),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) ).

% Int_insert_left
tff(fact_1364_subrelI,axiom,
    ! [R2: set_Pr3948176798113811640et_nat,S2: set_Pr3948176798113811640et_nat] :
      ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
          ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R2)
         => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),S2) )
     => ord_le8090042543978450200et_nat(R2,S2) ) ).

% subrelI
tff(fact_1365_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),S2) )
     => ord_le3146513528884898305at_nat(R2,S2) ) ).

% subrelI
tff(fact_1366_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),S2) )
     => ord_le2843351958646193337nt_int(R2,S2) ) ).

% subrelI
tff(fact_1367_subrelI,axiom,
    ! [R2: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),R2)
         => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),S2) )
     => ord_le3919382008627836342et_nat(R2,S2) ) ).

% subrelI
tff(fact_1368_subrelI,axiom,
    ! [R2: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2),R2)
         => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2),S2) )
     => ord_le5189643068899110356et_nat(R2,S2) ) ).

% subrelI
tff(fact_1369_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A2),bot_bo3957492148770167129t_unit)
     => ( A2 = bot_bo3957492148770167129t_unit ) ) ).

% bot.extremum_uniqueI
tff(fact_1370_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),bot_bot_set_o)
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_1371_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),bot_bot_set_int)
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1372_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),bot_bo8632255151191677115_nat_o)
     => ( A2 = bot_bo8632255151191677115_nat_o ) ) ).

% bot.extremum_uniqueI
tff(fact_1373_bot_Oextremum__uniqueI,axiom,
    ! [A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),bot_bot_filter_nat)
     => ( A2 = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1374_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat)
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1375_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),bot_bot_nat)
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1376_bot_Oextremum__unique,axiom,
    ! [A2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A2),bot_bo3957492148770167129t_unit)
    <=> ( A2 = bot_bo3957492148770167129t_unit ) ) ).

% bot.extremum_unique
tff(fact_1377_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_1378_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1379_bot_Oextremum__unique,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A2),bot_bo8632255151191677115_nat_o)
    <=> ( A2 = bot_bo8632255151191677115_nat_o ) ) ).

% bot.extremum_unique
tff(fact_1380_bot_Oextremum__unique,axiom,
    ! [A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),bot_bot_filter_nat)
    <=> ( A2 = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_1381_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1382_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),bot_bot_nat)
    <=> ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1383_bot_Oextremum,axiom,
    ! [A2: set_Product_unit] : aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,bot_bo3957492148770167129t_unit),A2) ).

% bot.extremum
tff(fact_1384_bot_Oextremum,axiom,
    ! [A2: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,bot_bot_set_o),A2) ).

% bot.extremum
tff(fact_1385_bot_Oextremum,axiom,
    ! [A2: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,bot_bot_set_int),A2) ).

% bot.extremum
tff(fact_1386_bot_Oextremum,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(bot_bo8632255151191677115_nat_o),A2) ).

% bot.extremum
tff(fact_1387_bot_Oextremum,axiom,
    ! [A2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,bot_bot_filter_nat),A2) ).

% bot.extremum
tff(fact_1388_bot_Oextremum,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A2) ).

% bot.extremum
tff(fact_1389_bot_Oextremum,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A2) ).

% bot.extremum
tff(fact_1390_subset__emptyI,axiom,
    ! [A: set_Product_unit] :
      ( ! [X3: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),bot_bo3957492148770167129t_unit) ) ).

% subset_emptyI
tff(fact_1391_subset__emptyI,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] : ~ aa_set_o_o(member_o((X3)),A)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_1392_subset__emptyI,axiom,
    ! [A: set_int] :
      ( ! [X3: int] : ~ aa_set_int_o(member_int(X3),A)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1393_subset__emptyI,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),bot_bo8632255151191677115_nat_o) ) ).

% subset_emptyI
tff(fact_1394_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1395_insert__subsetI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,X5: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(X5),A)
       => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),X5)),A) ) ) ).

% insert_subsetI
tff(fact_1396_insert__subsetI,axiom,
    ! [X: int,A: set_int,X5: set_int] :
      ( aa_set_int_o(member_int(X),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X5),A)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),X5)),A) ) ) ).

% insert_subsetI
tff(fact_1397_insert__subsetI,axiom,
    ! [X: $o,A: set_o,X5: set_o] :
      ( aa_set_o_o(member_o((X)),A)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X5),A)
       => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),X5)),A) ) ) ).

% insert_subsetI
tff(fact_1398_insert__subsetI,axiom,
    ! [X: product_unit,A: set_Product_unit,X5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,X5),A)
       => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),X5)),A) ) ) ).

% insert_subsetI
tff(fact_1399_insert__subsetI,axiom,
    ! [X: nat,A: set_nat,X5: set_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),X5)),A) ) ) ).

% insert_subsetI
tff(fact_1400_subset__insertI2,axiom,
    ! [A: set_int,B4: set_int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),B4)) ) ).

% subset_insertI2
tff(fact_1401_subset__insertI2,axiom,
    ! [A: set_o,B4: set_o,B: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),B4)) ) ).

% subset_insertI2
tff(fact_1402_subset__insertI2,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,B: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),B4)) ) ).

% subset_insertI2
tff(fact_1403_subset__insertI2,axiom,
    ! [A: set_nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),B4)) ) ).

% subset_insertI2
tff(fact_1404_subset__insertI,axiom,
    ! [B4: set_int,A2: int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) ).

% subset_insertI
tff(fact_1405_subset__insertI,axiom,
    ! [B4: set_o,A2: $o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) ).

% subset_insertI
tff(fact_1406_subset__insertI,axiom,
    ! [B4: set_Product_unit,A2: product_unit] : aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) ).

% subset_insertI
tff(fact_1407_subset__insertI,axiom,
    ! [B4: set_nat,A2: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) ).

% subset_insertI
tff(fact_1408_subset__insert,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B4))
      <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4) ) ) ).

% subset_insert
tff(fact_1409_subset__insert,axiom,
    ! [X: int,A: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4))
      <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4) ) ) ).

% subset_insert
tff(fact_1410_subset__insert,axiom,
    ! [X: $o,A: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4))
      <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4) ) ) ).

% subset_insert
tff(fact_1411_subset__insert,axiom,
    ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4))
      <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4) ) ) ).

% subset_insert
tff(fact_1412_subset__insert,axiom,
    ! [X: nat,A: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ) ).

% subset_insert
tff(fact_1413_insert__mono,axiom,
    ! [C5: set_int,D3: set_int,A2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,C5),D3)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),C5)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),D3)) ) ).

% insert_mono
tff(fact_1414_insert__mono,axiom,
    ! [C5: set_o,D3: set_o,A2: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,C5),D3)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),C5)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),D3)) ) ).

% insert_mono
tff(fact_1415_insert__mono,axiom,
    ! [C5: set_Product_unit,D3: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,C5),D3)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),C5)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),D3)) ) ).

% insert_mono
tff(fact_1416_insert__mono,axiom,
    ! [C5: set_nat,D3: set_nat,A2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),D3)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),C5)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),D3)) ) ).

% insert_mono
tff(fact_1417_Image__Un,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(image_nat_nat(R),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,aa_set_nat_set_nat(image_nat_nat(R),A)),aa_set_nat_set_nat(image_nat_nat(R),B4)) ).

% Image_Un
tff(fact_1418_subset__vimage__iff,axiom,
    ! [A: set_nat,F: fun_na5781657914532371832_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),vimage984285786009958690_nat_o(F,B4))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_nat2198927446035044705_nat_o(F,X2)),B4) ) ) ).

% subset_vimage_iff
tff(fact_1419_subset__vimage__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(vimage_nat_nat(F),B4))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X2)),B4) ) ) ).

% subset_vimage_iff
tff(fact_1420_subset__vimage__iff,axiom,
    ! [A: set_nat,F: fun_nat_int,B4: set_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),vimage_nat_int(F,B4))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_int_o(member_int(aa_nat_int(F,X2)),B4) ) ) ).

% subset_vimage_iff
tff(fact_1421_subset__vimage__iff,axiom,
    ! [A: set_nat,F: fun_nat_o,B4: set_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),vimage_nat_o(F,B4))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_o_o(member_o(aa_nat_o(F,X2)),B4) ) ) ).

% subset_vimage_iff
tff(fact_1422_subset__vimage__iff,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit,B4: set_Product_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),vimage4884490618288580032t_unit(F,B4))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(F,X2)),B4) ) ) ).

% subset_vimage_iff
tff(fact_1423_vimage__mono,axiom,
    ! [A: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),A)),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ) ).

% vimage_mono
tff(fact_1424_relH__subset,axiom,
    ! [Bs: set_nat,Ha: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(Bs),Ha),H2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As),Bs)
       => aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As),Ha),H2) ) ) ).

% relH_subset
tff(fact_1425_rtrancl__apply__insert,axiom,
    ! [R: set_Pr5094982260447487303t_unit,X: product_unit,S: set_Product_unit] : image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),S)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,S),image_8431784415008483091t_unit(R,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ).

% rtrancl_apply_insert
tff(fact_1426_rtrancl__apply__insert,axiom,
    ! [R: set_Product_prod_o_o,X: $o,S: set_o] : image_o_o(transitive_rtrancl_o(R),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),S)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),image_o_o(transitive_rtrancl_o(R),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,S),image_o_o(R,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ).

% rtrancl_apply_insert
tff(fact_1427_rtrancl__apply__insert,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,S: set_nat] : aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),S)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ).

% rtrancl_apply_insert
tff(fact_1428_rtrancl__apply__insert,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int,S: set_int] : image_int_int(transi2902850859426252137cl_int(R),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),S)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),image_int_int(transi2902850859426252137cl_int(R),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,S),image_int_int(R,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ).

% rtrancl_apply_insert
tff(fact_1429_rtrancl__apply__insert,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] : image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),S)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,S),image_2188787866847195539_nat_o(R,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ).

% rtrancl_apply_insert
tff(fact_1430_antisymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( antisym_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),R2)
         => ( A2 = B ) ) ) ) ).

% antisymD
tff(fact_1431_antisymD,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( antisym_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),R2)
         => ( A2 = B ) ) ) ) ).

% antisymD
tff(fact_1432_antisymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),X3),R2)
           => ( X3 = Y2 ) ) )
     => antisym_nat(R2) ) ).

% antisymI
tff(fact_1433_antisymI,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),X3),R2)
           => ( X3 = Y2 ) ) )
     => antisym_int(R2) ) ).

% antisymI
tff(fact_1434_antisym__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( antisym_nat(R2)
    <=> ! [X2: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y3),R2)
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),X2),R2)
           => ( X2 = Y3 ) ) ) ) ).

% antisym_def
tff(fact_1435_antisym__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( antisym_int(R2)
    <=> ! [X2: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y3),R2)
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y3),X2),R2)
           => ( X2 = Y3 ) ) ) ) ).

% antisym_def
tff(fact_1436_Image__absorb__rtrancl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,C5: set_nat] :
      ( trans_nat(A)
     => ( refl_on_nat(B4,A)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),B4)
         => ( aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(A)),C5) = aa_set_nat_set_nat(image_nat_nat(A),C5) ) ) ) ) ).

% Image_absorb_rtrancl
tff(fact_1437_times__assn__raw_Osimps,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(P),Q),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
    <=> ? [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(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As1))
          & aa_Pro653082453696054616_nat_o(Q,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As22)) ) ) ).

% times_assn_raw.simps
tff(fact_1438_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,As2: set_nat] :
            ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( (Y)
            <=> ~ ? [As1: set_nat,As22: set_nat] :
                    ( ( As2 = 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,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As1))
                    & aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As22)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_1439_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,As2: set_nat] :
            ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ~ ? [As12: set_nat,As23: set_nat] :
                  ( ( As2 = 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,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As12))
                  & aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As23)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_1440_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,As2: set_nat] :
            ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ? [As13: set_nat,As24: set_nat] :
                ( ( As2 = 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,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As13))
                & aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As24)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_1441_rtrancl__Image__advance__ss,axiom,
    ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(U),V),E)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(transi6481887417741755664t_unit(E),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit))),image_8431784415008483091t_unit(transi6481887417741755664t_unit(E),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,U),bot_bo3957492148770167129t_unit))) ) ).

% rtrancl_Image_advance_ss
tff(fact_1442_rtrancl__Image__advance__ss,axiom,
    ! [U: $o,V: $o,E: set_Product_prod_o_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((U)),(V)),E)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(transitive_rtrancl_o(E),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o))),image_o_o(transitive_rtrancl_o(E),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(U)),bot_bot_set_o))) ) ).

% rtrancl_Image_advance_ss
tff(fact_1443_rtrancl__Image__advance__ss,axiom,
    ! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),V),E)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(transi2902850859426252137cl_int(E),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int))),image_int_int(transi2902850859426252137cl_int(E),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,U),bot_bot_set_int))) ) ).

% rtrancl_Image_advance_ss
tff(fact_1444_rtrancl__Image__advance__ss,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o,V: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(U,V),E)
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(E),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o))),image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(E),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(U),bot_bo8632255151191677115_nat_o))) ) ).

% rtrancl_Image_advance_ss
tff(fact_1445_rtrancl__Image__advance__ss,axiom,
    ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V),E)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,U),bot_bot_set_nat))) ) ).

% rtrancl_Image_advance_ss
tff(fact_1446_rtrancl__image__advance__rtrancl,axiom,
    ! [Q3: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o,Q0: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Q3),image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),Q0))
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Q3,X),transi4955142523599900786_nat_o(R))
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_1447_rtrancl__image__advance__rtrancl,axiom,
    ! [Q3: $o,R: set_Product_prod_o_o,Q0: set_o,X: $o] :
      ( aa_set_o_o(member_o((Q3)),image_o_o(transitive_rtrancl_o(R),Q0))
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Q3)),(X)),transitive_rtrancl_o(R))
       => aa_set_o_o(member_o((X)),image_o_o(transitive_rtrancl_o(R),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_1448_rtrancl__image__advance__rtrancl,axiom,
    ! [Q3: product_unit,R: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(Q3),image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),Q0))
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Q3),X),transi6481887417741755664t_unit(R))
       => aa_set8780210009671629607unit_o(member_Product_unit(X),image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_1449_rtrancl__image__advance__rtrancl,axiom,
    ! [Q3: nat,R: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
      ( aa_set_nat_o(member_nat(Q3),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),Q0))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),X),transi2905341329935302413cl_nat(R))
       => aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_1450_rtrancl__image__advance__rtrancl,axiom,
    ! [Q3: int,R: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
      ( aa_set_int_o(member_int(Q3),image_int_int(transi2902850859426252137cl_int(R),Q0))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),X),transi2902850859426252137cl_int(R))
       => aa_set_int_o(member_int(X),image_int_int(transi2902850859426252137cl_int(R),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_1451_rtrancl__image__advance,axiom,
    ! [Q3: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o,Q0: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Q3),image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),Q0))
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Q3,X),R)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_1452_rtrancl__image__advance,axiom,
    ! [Q3: $o,R: set_Product_prod_o_o,Q0: set_o,X: $o] :
      ( aa_set_o_o(member_o((Q3)),image_o_o(transitive_rtrancl_o(R),Q0))
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Q3)),(X)),R)
       => aa_set_o_o(member_o((X)),image_o_o(transitive_rtrancl_o(R),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_1453_rtrancl__image__advance,axiom,
    ! [Q3: product_unit,R: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(Q3),image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),Q0))
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Q3),X),R)
       => aa_set8780210009671629607unit_o(member_Product_unit(X),image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_1454_rtrancl__image__advance,axiom,
    ! [Q3: nat,R: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
      ( aa_set_nat_o(member_nat(Q3),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),Q0))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),X),R)
       => aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_1455_rtrancl__image__advance,axiom,
    ! [Q3: int,R: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
      ( aa_set_int_o(member_int(Q3),image_int_int(transi2902850859426252137cl_int(R),Q0))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),X),R)
       => aa_set_int_o(member_int(X),image_int_int(transi2902850859426252137cl_int(R),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_1456_trancl__rtrancl__trancl,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),transi2905341329935302413cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi6264000038957366511cl_nat(R2)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_1457_trancl__rtrancl__trancl,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),transi2902850859426252137cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi6261509568448316235cl_int(R2)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_1458_rtrancl__trancl__trancl,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),transi6264000038957366511cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),transi6264000038957366511cl_nat(R2)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_1459_rtrancl__trancl__trancl,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z),transi6261509568448316235cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),transi6261509568448316235cl_int(R2)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_1460_rtrancl__into__trancl2,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),transi2905341329935302413cl_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi6264000038957366511cl_nat(R2)) ) ) ).

% rtrancl_into_trancl2
tff(fact_1461_rtrancl__into__trancl2,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),transi2902850859426252137cl_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi6261509568448316235cl_int(R2)) ) ) ).

% rtrancl_into_trancl2
tff(fact_1462_rtrancl__into__trancl1,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R2))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),R2)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),transi6264000038957366511cl_nat(R2)) ) ) ).

% rtrancl_into_trancl1
tff(fact_1463_rtrancl__into__trancl1,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R2))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),R2)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),transi6261509568448316235cl_int(R2)) ) ) ).

% rtrancl_into_trancl1
tff(fact_1464_rtrancl__eq__or__trancl,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(R)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_1465_rtrancl__eq__or__trancl,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(R)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_1466_trancl__into__rtrancl,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R2))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R2)) ) ).

% trancl_into_rtrancl
tff(fact_1467_trancl__into__rtrancl,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R2))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R2)) ) ).

% trancl_into_rtrancl
tff(fact_1468_tranclD2,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(R))
     => ? [Z3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z3),transi2905341329935302413cl_nat(R))
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z3),Y),R) ) ) ).

% tranclD2
tff(fact_1469_tranclD2,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(R))
     => ? [Z3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z3),transi2902850859426252137cl_int(R))
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Z3),Y),R) ) ) ).

% tranclD2
tff(fact_1470_rtranclD,axiom,
    ! [A2: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(R))
     => ( ( A2 = B )
        | ( ( A2 != B )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(R)) ) ) ) ).

% rtranclD
tff(fact_1471_rtranclD,axiom,
    ! [A2: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(R))
     => ( ( A2 = B )
        | ( ( A2 != B )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(R)) ) ) ) ).

% rtranclD
tff(fact_1472_tranclD,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(R))
     => ? [Z3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z3),R)
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z3),Y),transi2905341329935302413cl_nat(R)) ) ) ).

% tranclD
tff(fact_1473_tranclD,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(R))
     => ? [Z3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z3),R)
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Z3),Y),transi2902850859426252137cl_int(R)) ) ) ).

% tranclD
tff(fact_1474_insert__is__Un,axiom,
    ! [A2: product_unit,A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)),A) ).

% insert_is_Un
tff(fact_1475_insert__is__Un,axiom,
    ! [A2: $o,A: set_o] : aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)),A) ).

% insert_is_Un
tff(fact_1476_insert__is__Un,axiom,
    ! [A2: nat,A: set_nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)),A) ).

% insert_is_Un
tff(fact_1477_insert__is__Un,axiom,
    ! [A2: int,A: set_int] : aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)),A) ).

% insert_is_Un
tff(fact_1478_insert__is__Un,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) = aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)),A) ).

% insert_is_Un
tff(fact_1479_Un__singleton__iff,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) )
    <=> ( ( ( A = bot_bo3957492148770167129t_unit )
          & ( B4 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) ) )
        | ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) )
          & ( B4 = bot_bo3957492148770167129t_unit ) )
        | ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) )
          & ( B4 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1480_Un__singleton__iff,axiom,
    ! [A: set_o,B4: set_o,X: $o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) )
    <=> ( ( ( A = bot_bot_set_o )
          & ( B4 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) ) )
        | ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) )
          & ( B4 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1481_Un__singleton__iff,axiom,
    ! [A: set_nat,B4: set_nat,X: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) )
    <=> ( ( ( A = bot_bot_set_nat )
          & ( B4 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ) )
        | ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) )
          & ( B4 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1482_Un__singleton__iff,axiom,
    ! [A: set_int,B4: set_int,X: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) )
    <=> ( ( ( A = bot_bot_set_int )
          & ( B4 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ) )
        | ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) )
          & ( B4 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1483_Un__singleton__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) )
    <=> ( ( ( A = bot_bo8632255151191677115_nat_o )
          & ( B4 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) )
        | ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) )
          & ( B4 = bot_bo8632255151191677115_nat_o ) )
        | ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) )
          & ( B4 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) ) ) ) ).

% Un_singleton_iff
tff(fact_1484_singleton__Un__iff,axiom,
    ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4) )
    <=> ( ( ( A = bot_bo3957492148770167129t_unit )
          & ( B4 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) ) )
        | ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) )
          & ( B4 = bot_bo3957492148770167129t_unit ) )
        | ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) )
          & ( B4 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1485_singleton__Un__iff,axiom,
    ! [X: $o,A: set_o,B4: set_o] :
      ( ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4) )
    <=> ( ( ( A = bot_bot_set_o )
          & ( B4 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) ) )
        | ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) )
          & ( B4 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1486_singleton__Un__iff,axiom,
    ! [X: nat,A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) )
    <=> ( ( ( A = bot_bot_set_nat )
          & ( B4 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ) )
        | ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) )
          & ( B4 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1487_singleton__Un__iff,axiom,
    ! [X: int,A: set_int,B4: set_int] :
      ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4) )
    <=> ( ( ( A = bot_bot_set_int )
          & ( B4 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ) )
        | ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) )
          & ( B4 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1488_singleton__Un__iff,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) = aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4) )
    <=> ( ( ( A = bot_bo8632255151191677115_nat_o )
          & ( B4 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) )
        | ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) )
          & ( B4 = bot_bo8632255151191677115_nat_o ) )
        | ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) )
          & ( B4 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) ) ) ) ).

% singleton_Un_iff
tff(fact_1489_Not__Domain__rtrancl,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),domain5580505327626972364_nat_o(R))
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),transi4955142523599900786_nat_o(R))
      <=> ( X = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_1490_Not__Domain__rtrancl,axiom,
    ! [X: $o,R: set_Product_prod_o_o,Y: $o] :
      ( ~ aa_set_o_o(member_o((X)),domain_o_o(R))
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),transitive_rtrancl_o(R))
      <=> ( (X)
        <=> (Y) ) ) ) ).

% Not_Domain_rtrancl
tff(fact_1491_Not__Domain__rtrancl,axiom,
    ! [X: product_unit,R: set_Pr5094982260447487303t_unit,Y: product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),domain4292024440186670924t_unit(R))
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),transi6481887417741755664t_unit(R))
      <=> ( X = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_1492_Not__Domain__rtrancl,axiom,
    ! [X: nat,R: set_Pr1261947904930325089at_nat,Y: nat] :
      ( ~ aa_set_nat_o(member_nat(X),domain_nat_nat(R))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R))
      <=> ( X = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_1493_Not__Domain__rtrancl,axiom,
    ! [X: int,R: set_Pr958786334691620121nt_int,Y: int] :
      ( ~ aa_set_int_o(member_int(X),domain_int_int(R))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R))
      <=> ( X = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_1494_subset__singletonD,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))
     => ( ( A = bot_bo3957492148770167129t_unit )
        | ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) ) ) ) ).

% subset_singletonD
tff(fact_1495_subset__singletonD,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
     => ( ( A = bot_bot_set_o )
        | ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_1496_subset__singletonD,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))
     => ( ( A = bot_bot_set_int )
        | ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_1497_subset__singletonD,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))
     => ( ( A = bot_bo8632255151191677115_nat_o )
        | ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) ) ) ).

% subset_singletonD
tff(fact_1498_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))
     => ( ( A = bot_bot_set_nat )
        | ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_1499_subset__singleton__iff,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,X5),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))
    <=> ( ( X5 = bot_bo3957492148770167129t_unit )
        | ( X5 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit) ) ) ) ).

% subset_singleton_iff
tff(fact_1500_subset__singleton__iff,axiom,
    ! [X5: set_o,A2: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X5),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))
    <=> ( ( X5 = bot_bot_set_o )
        | ( X5 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_1501_subset__singleton__iff,axiom,
    ! [X5: set_int,A2: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,X5),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))
    <=> ( ( X5 = bot_bot_set_int )
        | ( X5 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_1502_subset__singleton__iff,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(X5),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))
    <=> ( ( X5 = bot_bo8632255151191677115_nat_o )
        | ( X5 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o) ) ) ) ).

% subset_singleton_iff
tff(fact_1503_subset__singleton__iff,axiom,
    ! [X5: set_nat,A2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X5),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))
    <=> ( ( X5 = bot_bot_set_nat )
        | ( X5 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_1504_mod__star__conv,axiom,
    ! [A: assn,B4: assn,Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(times_times_assn(A),B4)),Ha)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
          ( ( Ha = aa_set3979848162462698438et_nat(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(aa_ass4243947775541419017_nat_o(rep_assn,A),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Hr),As1))
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B4),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Hr),As22)) ) ) ).

% mod_star_conv
tff(fact_1505_star__assnI,axiom,
    ! [P: assn,Ha: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,As5: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,P),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Q),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As5))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(times_times_assn(P),Q)),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ).

% star_assnI
tff(fact_1506_in__range__subset,axiom,
    ! [As: set_nat,As5: set_nat,Ha: heap_e7401611519738050253t_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As),As5)
     => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As5))
       => aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As)) ) ) ).

% in_range_subset
tff(fact_1507_wand__raw_Osimps,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(wand_raw(P),Q),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
        & ! [H4: 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(aa_hea3972832001280910815unit_o(relH(As),Ha),H4)
              & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As))
              & aa_Pro653082453696054616_nat_o(P,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As6)) )
           => aa_Pro653082453696054616_nat_o(Q,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) ) ).

% wand_raw.simps
tff(fact_1508_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,As2: set_nat] :
            ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( (Y)
            <=> ~ ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
                  & ! [H4: heap_e7401611519738050253t_unit,As6: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As6) = bot_bot_set_nat )
                        & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H4)
                        & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
                        & aa_Pro653082453696054616_nat_o(X,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As6)) )
                     => aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As6))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_1509_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,As2: set_nat] :
            ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ~ ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
                & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As7) = bot_bot_set_nat )
                      & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H6)
                      & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H6),As2))
                      & aa_Pro653082453696054616_nat_o(X,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H6),As7)) )
                   => aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H6),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As7))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_1510_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,As2: set_nat] :
            ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
              & ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
                  ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As4) = bot_bot_set_nat )
                    & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H5)
                    & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As2))
                    & aa_Pro653082453696054616_nat_o(X,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As4)) )
                 => aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As4))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_1511_Image__empty__rtrancl__Image__id,axiom,
    ! [R: set_Pr5094982260447487303t_unit,V: product_unit] :
      ( ( image_8431784415008483091t_unit(R,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit )
     => ( image_8431784415008483091t_unit(transi6481887417741755664t_unit(R),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_1512_Image__empty__rtrancl__Image__id,axiom,
    ! [R: set_Product_prod_o_o,V: $o] :
      ( ( image_o_o(R,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o)) = bot_bot_set_o )
     => ( image_o_o(transitive_rtrancl_o(R),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_1513_Image__empty__rtrancl__Image__id,axiom,
    ! [R: set_Pr1261947904930325089at_nat,V: nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat)) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_1514_Image__empty__rtrancl__Image__id,axiom,
    ! [R: set_Pr958786334691620121nt_int,V: int] :
      ( ( image_int_int(R,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int)) = bot_bot_set_int )
     => ( image_int_int(transi2902850859426252137cl_int(R),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_1515_Image__empty__rtrancl__Image__id,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,V: fun_Pr6460944699146269935_nat_o] :
      ( ( image_2188787866847195539_nat_o(R,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o )
     => ( image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(R),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_1516_sup__bot_Oright__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),bot_bot_assn) = A2 ).

% sup_bot.right_neutral
tff(fact_1517_sup__bot_Oright__neutral,axiom,
    ! [A2: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),bot_bot_Product_unit) = A2 ).

% sup_bot.right_neutral
tff(fact_1518_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A2),bot_bo3957492148770167129t_unit) = A2 ).

% sup_bot.right_neutral
tff(fact_1519_sup__bot_Oright__neutral,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),bot_bot_set_o) = A2 ).

% sup_bot.right_neutral
tff(fact_1520_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),bot_bot_set_nat) = A2 ).

% sup_bot.right_neutral
tff(fact_1521_sup__bot_Oright__neutral,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),bot_bot_set_int) = A2 ).

% sup_bot.right_neutral
tff(fact_1522_sup__bot_Oright__neutral,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A2),bot_bo8632255151191677115_nat_o) = A2 ).

% sup_bot.right_neutral
tff(fact_1523_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: assn,B: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B) )
    <=> ( ( A2 = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1524_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: product_unit,B: product_unit] :
      ( ( bot_bot_Product_unit = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),B) )
    <=> ( ( A2 = bot_bot_Product_unit )
        & ( B = bot_bot_Product_unit ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1525_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A2),B) )
    <=> ( ( A2 = bot_bo3957492148770167129t_unit )
        & ( B = bot_bo3957492148770167129t_unit ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1526_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) )
    <=> ( ( A2 = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1527_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1528_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1529_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A2),B) )
    <=> ( ( A2 = bot_bo8632255151191677115_nat_o )
        & ( B = bot_bo8632255151191677115_nat_o ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_1530_sup__bot_Oleft__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1531_sup__bot_Oleft__neutral,axiom,
    ! [A2: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,bot_bot_Product_unit),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1532_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,bot_bo3957492148770167129t_unit),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1533_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,bot_bot_set_o),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1534_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1535_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1536_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,bot_bo8632255151191677115_nat_o),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_1537_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: assn,B: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B) = bot_bot_assn )
    <=> ( ( A2 = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1538_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: product_unit,B: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),B) = bot_bot_Product_unit )
    <=> ( ( A2 = bot_bot_Product_unit )
        & ( B = bot_bot_Product_unit ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1539_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Product_unit,B: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A2),B) = bot_bo3957492148770167129t_unit )
    <=> ( ( A2 = bot_bo3957492148770167129t_unit )
        & ( B = bot_bo3957492148770167129t_unit ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1540_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),B) = bot_bot_set_o )
    <=> ( ( A2 = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1541_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1542_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1543_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A2),B) = bot_bo8632255151191677115_nat_o )
    <=> ( ( A2 = bot_bo8632255151191677115_nat_o )
        & ( B = bot_bo8632255151191677115_nat_o ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_1544_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = bot_bot_assn )
    <=> ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
tff(fact_1545_sup__eq__bot__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y) = bot_bot_Product_unit )
    <=> ( ( X = bot_bot_Product_unit )
        & ( Y = bot_bot_Product_unit ) ) ) ).

% sup_eq_bot_iff
tff(fact_1546_sup__eq__bot__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y) = bot_bo3957492148770167129t_unit )
    <=> ( ( X = bot_bo3957492148770167129t_unit )
        & ( Y = bot_bo3957492148770167129t_unit ) ) ) ).

% sup_eq_bot_iff
tff(fact_1547_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_1548_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_1549_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_1550_sup__eq__bot__iff,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] :
      ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,X),Y) = bot_bo8632255151191677115_nat_o )
    <=> ( ( X = bot_bo8632255151191677115_nat_o )
        & ( Y = bot_bo8632255151191677115_nat_o ) ) ) ).

% sup_eq_bot_iff
tff(fact_1551_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) )
    <=> ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
tff(fact_1552_bot__eq__sup__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( bot_bot_Product_unit = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y) )
    <=> ( ( X = bot_bot_Product_unit )
        & ( Y = bot_bot_Product_unit ) ) ) ).

% bot_eq_sup_iff
tff(fact_1553_bot__eq__sup__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y) )
    <=> ( ( X = bot_bo3957492148770167129t_unit )
        & ( Y = bot_bo3957492148770167129t_unit ) ) ) ).

% bot_eq_sup_iff
tff(fact_1554_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_1555_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_1556_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_1557_bot__eq__sup__iff,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,X),Y) )
    <=> ( ( X = bot_bo8632255151191677115_nat_o )
        & ( Y = bot_bo8632255151191677115_nat_o ) ) ) ).

% bot_eq_sup_iff
tff(fact_1558_sup__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% sup_bot_right
tff(fact_1559_sup__bot__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),bot_bot_Product_unit) = X ).

% sup_bot_right
tff(fact_1560_sup__bot__right,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),bot_bo3957492148770167129t_unit) = X ).

% sup_bot_right
tff(fact_1561_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_1562_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_1563_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_1564_sup__bot__right,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,X),bot_bo8632255151191677115_nat_o) = X ).

% sup_bot_right
tff(fact_1565_sup__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).

% sup_bot_left
tff(fact_1566_sup__bot__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,bot_bot_Product_unit),X) = X ).

% sup_bot_left
tff(fact_1567_sup__bot__left,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,bot_bo3957492148770167129t_unit),X) = X ).

% sup_bot_left
tff(fact_1568_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_1569_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_1570_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_1571_sup__bot__left,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,bot_bo8632255151191677115_nat_o),X) = X ).

% sup_bot_left
tff(fact_1572_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% boolean_algebra.conj_zero_right
tff(fact_1573_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),bot_bot_Product_unit) = bot_bot_Product_unit ).

% boolean_algebra.conj_zero_right
tff(fact_1574_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Product_unit] : inf_in4660618365625256667t_unit(X,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% boolean_algebra.conj_zero_right
tff(fact_1575_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% boolean_algebra.conj_zero_right
tff(fact_1576_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_1577_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] : inf_inf_set_int(X,bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_1578_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(X,bot_bo8632255151191677115_nat_o) = bot_bo8632255151191677115_nat_o ).

% boolean_algebra.conj_zero_right
tff(fact_1579_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% boolean_algebra.conj_zero_left
tff(fact_1580_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,bot_bot_Product_unit),X) = bot_bot_Product_unit ).

% boolean_algebra.conj_zero_left
tff(fact_1581_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Product_unit] : inf_in4660618365625256667t_unit(bot_bo3957492148770167129t_unit,X) = bot_bo3957492148770167129t_unit ).

% boolean_algebra.conj_zero_left
tff(fact_1582_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] : inf_inf_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% boolean_algebra.conj_zero_left
tff(fact_1583_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_1584_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] : inf_inf_set_int(bot_bot_set_int,X) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_1585_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(bot_bo8632255151191677115_nat_o,X) = bot_bo8632255151191677115_nat_o ).

% boolean_algebra.conj_zero_left
tff(fact_1586_inf__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% inf_bot_right
tff(fact_1587_inf__bot__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),bot_bot_Product_unit) = bot_bot_Product_unit ).

% inf_bot_right
tff(fact_1588_inf__bot__right,axiom,
    ! [X: set_Product_unit] : inf_in4660618365625256667t_unit(X,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% inf_bot_right
tff(fact_1589_inf__bot__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,bot_bot_set_o) = bot_bot_set_o ).

% inf_bot_right
tff(fact_1590_inf__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_1591_inf__bot__right,axiom,
    ! [X: set_int] : inf_inf_set_int(X,bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_1592_inf__bot__right,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(X,bot_bo8632255151191677115_nat_o) = bot_bo8632255151191677115_nat_o ).

% inf_bot_right
tff(fact_1593_merge__pure__or,axiom,
    ! [A2: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_o_assn(pure_assn,(A2))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A2)
        | (B) )) ).

% merge_pure_or
tff(fact_1594_merge__pure__and,axiom,
    ! [A2: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_o_assn(pure_assn,(A2))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
        ( (A2)
        & (B) )) ).

% merge_pure_and
tff(fact_1595_inf__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% inf_bot_left
tff(fact_1596_inf__bot__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,bot_bot_Product_unit),X) = bot_bot_Product_unit ).

% inf_bot_left
tff(fact_1597_inf__bot__left,axiom,
    ! [X: set_Product_unit] : inf_in4660618365625256667t_unit(bot_bo3957492148770167129t_unit,X) = bot_bo3957492148770167129t_unit ).

% inf_bot_left
tff(fact_1598_inf__bot__left,axiom,
    ! [X: set_o] : inf_inf_set_o(bot_bot_set_o,X) = bot_bot_set_o ).

% inf_bot_left
tff(fact_1599_inf__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_1600_inf__bot__left,axiom,
    ! [X: set_int] : inf_inf_set_int(bot_bot_set_int,X) = bot_bot_set_int ).

% inf_bot_left
tff(fact_1601_inf__bot__left,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(bot_bo8632255151191677115_nat_o,X) = bot_bo8632255151191677115_nat_o ).

% inf_bot_left
tff(fact_1602_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1603_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),bot_bot_Product_unit) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1604_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),bot_bo3957492148770167129t_unit) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1605_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_1606_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_1607_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_1608_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,X),bot_bo8632255151191677115_nat_o) = X ).

% boolean_algebra.disj_zero_right
tff(fact_1609_prod__set__defs_I2_J,axiom,
    ! [X4: produc3658429121746597890et_nat] : basic_2698842327037663054et_nat(X4) = insert_set_nat(aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X4),bot_bot_set_set_nat) ).

% prod_set_defs(2)
tff(fact_1610_prod__set__defs_I2_J,axiom,
    ! [X4: product_prod_int_int] : basic_snds_int_int(X4) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,X4)),bot_bot_set_int) ).

% prod_set_defs(2)
tff(fact_1611_snds_Ocases,axiom,
    ! [A2: set_nat,P2: produc3658429121746597890et_nat] :
      ( member_set_nat(A2,basic_2698842327037663054et_nat(P2))
     => ( A2 = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2) ) ) ).

% snds.cases
tff(fact_1612_snds_Ocases,axiom,
    ! [A2: int,P2: product_prod_int_int] :
      ( aa_set_int_o(member_int(A2),basic_snds_int_int(P2))
     => ( A2 = aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) ) ) ).

% snds.cases
tff(fact_1613_snds_Osimps,axiom,
    ! [A2: set_nat,P2: produc3658429121746597890et_nat] :
      ( member_set_nat(A2,basic_2698842327037663054et_nat(P2))
    <=> ( A2 = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2) ) ) ).

% snds.simps
tff(fact_1614_snds_Osimps,axiom,
    ! [A2: int,P2: product_prod_int_int] :
      ( aa_set_int_o(member_int(A2),basic_snds_int_int(P2))
    <=> ( A2 = aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) ) ) ).

% snds.simps
tff(fact_1615_snds_Ointros,axiom,
    ! [P2: produc3658429121746597890et_nat] : member_set_nat(aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2),basic_2698842327037663054et_nat(P2)) ).

% snds.intros
tff(fact_1616_snds_Ointros,axiom,
    ! [P2: product_prod_int_int] : aa_set_int_o(member_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),basic_snds_int_int(P2)) ).

% snds.intros
tff(fact_1617_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(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( ( (Y)
                <=> ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
                    & ! [H4: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As6) = bot_bot_set_nat )
                          & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H4)
                          & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As2))
                          & aa_Pro653082453696054616_nat_o(X,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),As6)) )
                       => aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As6))) ) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_1618_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(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))))
               => ~ ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
                    & ! [H6: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As7) = bot_bot_set_nat )
                          & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H6)
                          & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H6),As2))
                          & aa_Pro653082453696054616_nat_o(X,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H6),As7)) )
                       => aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H6),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As7))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_1619_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(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(wand_raw_rel),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))))
               => ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
                  & ! [H5: heap_e7401611519738050253t_unit,As4: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As4) = bot_bot_set_nat )
                        & aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(relH(As2),H),H5)
                        & aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As2))
                        & aa_Pro653082453696054616_nat_o(X,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),As4)) )
                     => aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As4))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_1620_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_1621_le__numeral__extra_I4_J,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),one_one_Code_natural) ).

% le_numeral_extra(4)
tff(fact_1622_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_1623_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_1624_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_1625_less__eq__assn__def,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
    <=> ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B) ) ) ).

% less_eq_assn_def
tff(fact_1626_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(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))))
               => ? [As13: set_nat,As24: set_nat] :
                    ( ( As2 = 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,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As13))
                    & aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As24)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_1627_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(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))))
               => ~ ? [As12: set_nat,As23: set_nat] :
                      ( ( As2 = 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,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As12))
                      & aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_1628_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(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( ( (Y)
                <=> ? [As1: set_nat,As22: set_nat] :
                      ( ( As2 = 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,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As1))
                      & aa_Pro653082453696054616_nat_o(Xa,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As22)) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Xa),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_1629_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( order_7534304569044981540t_unit(field_Product_unit(R2),R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
          <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),A2),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_1630_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( order_preorder_on_o(field_o(R2),R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
          <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(A2)),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_1631_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( order_4859164337913492053on_int(field_int(R2),R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))),image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)))
          <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_1632_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( order_4728425272145416326_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))),image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)))
          <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,A2),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_1633_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( order_4861654808422542329on_nat(field_nat(R2),R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)))
          <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_1634_Field__insert,axiom,
    ! [A2: product_unit,B: product_unit,R2: set_Pr5094982260447487303t_unit] : field_Product_unit(insert4616891024742550839t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))),field_Product_unit(R2)) ).

% Field_insert
tff(fact_1635_Field__insert,axiom,
    ! [A2: $o,B: $o,R2: set_Product_prod_o_o] : field_o(insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))),field_o(R2)) ).

% Field_insert
tff(fact_1636_Field__insert,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] : field_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))),field_nat(R2)) ).

% Field_insert
tff(fact_1637_Field__insert,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] : field_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))),field_int(R2)) ).

% Field_insert
tff(fact_1638_Field__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr2890355626087285447_nat_o] : field_48612661033931452_nat_o(insert4451378821651018679_nat_o(produc4714251849521319255_nat_o(A2,B),R2)) = aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))),field_48612661033931452_nat_o(R2)) ).

% Field_insert
tff(fact_1639_dom__ran__disj__comp,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( ( inf_in4660618365625256667t_unit(domain4292024440186670924t_unit(R),range_6617724647562260917t_unit(R)) = bot_bo3957492148770167129t_unit )
     => ( relcom848357998638694028t_unit(R,R) = bot_bo1654022762305692595t_unit ) ) ).

% dom_ran_disj_comp
tff(fact_1640_dom__ran__disj__comp,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( inf_inf_set_o(domain_o_o(R),range_o_o(R)) = bot_bot_set_o )
     => ( relcomp_o_o_o(R,R) = bot_bo7073875226086086771od_o_o ) ) ).

% dom_ran_disj_comp
tff(fact_1641_dom__ran__disj__comp,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,domain_nat_nat(R)),range_nat_nat(R)) = bot_bot_set_nat )
     => ( relcomp_nat_nat_nat(R,R) = bot_bo2099793752762293965at_nat ) ) ).

% dom_ran_disj_comp
tff(fact_1642_dom__ran__disj__comp,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ( inf_inf_set_int(domain_int_int(R),range_int_int(R)) = bot_bot_set_int )
     => ( relcomp_int_int_int(R,R) = bot_bo1796632182523588997nt_int ) ) ).

% dom_ran_disj_comp
tff(fact_1643_dom__ran__disj__comp,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(domain5580505327626972364_nat_o(R),range_307841406276416053_nat_o(R)) = bot_bo8632255151191677115_nat_o )
     => ( relcom1922595779876297326_nat_o(R,R) = bot_bo6763312114214544691_nat_o ) ) ).

% dom_ran_disj_comp
tff(fact_1644_proj__iff,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit))),A)
       => ( ( aa_Pro234974305641502940t_unit(equiv_5763582929720877514t_unit(R2),X) = aa_Pro234974305641502940t_unit(equiv_5763582929720877514t_unit(R2),Y) )
        <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2) ) ) ) ).

% proj_iff
tff(fact_1645_proj__iff,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o))),A)
       => ( ( aa_o_set_o(equiv_proj_o_o(R2),(X)) = aa_o_set_o(equiv_proj_o_o(R2),(Y)) )
        <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2) ) ) ) ).

% proj_iff
tff(fact_1646_proj__iff,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int))),A)
       => ( ( aa_int_set_int(equiv_proj_int_int(R2),X) = aa_int_set_int(equiv_proj_int_int(R2),Y) )
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2) ) ) ) ).

% proj_iff
tff(fact_1647_proj__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o))),A)
       => ( ( equiv_5944703715125438794_nat_o(R2,X) = equiv_5944703715125438794_nat_o(R2,Y) )
        <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2) ) ) ) ).

% proj_iff
tff(fact_1648_proj__iff,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat))),A)
       => ( ( aa_nat_set_nat(equiv_proj_nat_nat(R2),X) = aa_nat_set_nat(equiv_proj_nat_nat(R2),Y) )
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2) ) ) ) ).

% proj_iff
tff(fact_1649_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( refl_on_Product_unit(field_Product_unit(R2),R2)
     => ( antisym_Product_unit(R2)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
         => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
           => ( ( image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) )
            <=> ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_1650_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( refl_on_o(field_o(R2),R2)
     => ( antisym_o(R2)
       => ( aa_set_o_o(member_o((A2)),field_o(R2))
         => ( aa_set_o_o(member_o((B)),field_o(R2))
           => ( ( image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) )
            <=> ( (A2)
              <=> (B) ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_1651_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( refl_on_nat(field_nat(R2),R2)
     => ( antisym_nat(R2)
       => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
         => ( aa_set_nat_o(member_nat(B),field_nat(R2))
           => ( ( aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) )
            <=> ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_1652_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( refl_on_int(field_int(R2),R2)
     => ( antisym_int(R2)
       => ( aa_set_int_o(member_int(A2),field_int(R2))
         => ( aa_set_int_o(member_int(B),field_int(R2))
           => ( ( image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) )
            <=> ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_1653_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( refl_o5094713388964525678_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( antisy7184066905350198161_nat_o(R2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
         => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
           => ( ( image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) )
            <=> ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_1654_equiv__class__nondisjoint,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,A2: product_unit,B: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),inf_in4660618365625256667t_unit(image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))))
       => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_1655_equiv__class__nondisjoint,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,A2: $o,B: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(member_o((X)),inf_inf_set_o(image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))))
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_1656_equiv__class__nondisjoint,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,A2: nat,B: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_1657_equiv__class__nondisjoint,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,A2: int,B: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(member_int(X),inf_inf_set_int(image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)),image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_1658_equiv__class__nondisjoint,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),inf_in3376754634429691197_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)),image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))))
       => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_1659_finite__reachable__advance,axiom,
    ! [E: set_Pr4811707699266497531nteger,V0: code_integer,V: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_4147621952751189391nteger(transi3984306202085910426nteger(E),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,V0),bot_bo3990330152332043303nteger)))
     => ( member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(V0),V),transi3984306202085910426nteger(E))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_4147621952751189391nteger(transi3984306202085910426nteger(E),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,V),bot_bo3990330152332043303nteger))) ) ) ).

% finite_reachable_advance
tff(fact_1660_finite__reachable__advance,axiom,
    ! [E: set_Pr5094982260447487303t_unit,V0: product_unit,V: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_8431784415008483091t_unit(transi6481887417741755664t_unit(E),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V0),bot_bo3957492148770167129t_unit)))
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(V0),V),transi6481887417741755664t_unit(E))
       => aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_8431784415008483091t_unit(transi6481887417741755664t_unit(E),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,V),bot_bo3957492148770167129t_unit))) ) ) ).

% finite_reachable_advance
tff(fact_1661_finite__reachable__advance,axiom,
    ! [E: set_Product_prod_o_o,V0: $o,V: $o] :
      ( aa_set_o_o(finite_finite_o,image_o_o(transitive_rtrancl_o(E),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V0)),bot_bot_set_o)))
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((V0)),(V)),transitive_rtrancl_o(E))
       => aa_set_o_o(finite_finite_o,image_o_o(transitive_rtrancl_o(E),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(V)),bot_bot_set_o))) ) ) ).

% finite_reachable_advance
tff(fact_1662_finite__reachable__advance,axiom,
    ! [E: set_Pr1261947904930325089at_nat,V0: nat,V: nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V0),bot_bot_set_nat)))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V0),V),transi2905341329935302413cl_nat(E))
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,V),bot_bot_set_nat))) ) ) ).

% finite_reachable_advance
tff(fact_1663_finite__reachable__advance,axiom,
    ! [E: set_Pr958786334691620121nt_int,V0: int,V: int] :
      ( aa_set_int_o(finite_finite_int,image_int_int(transi2902850859426252137cl_int(E),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V0),bot_bot_set_int)))
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V0),V),transi2902850859426252137cl_int(E))
       => aa_set_int_o(finite_finite_int,image_int_int(transi2902850859426252137cl_int(E),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,V),bot_bot_set_int))) ) ) ).

% finite_reachable_advance
tff(fact_1664_finite__reachable__advance,axiom,
    ! [E: set_Pr2890355626087285447_nat_o,V0: fun_Pr6460944699146269935_nat_o,V: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(E),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V0),bot_bo8632255151191677115_nat_o)))
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(V0,V),transi4955142523599900786_nat_o(E))
       => aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,image_2188787866847195539_nat_o(transi4955142523599900786_nat_o(E),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(V),bot_bo8632255151191677115_nat_o))) ) ) ).

% finite_reachable_advance
tff(fact_1665_finite__Field__eq__finite,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,field_nat(R))
    <=> finite6177210948735845034at_nat(R) ) ).

% finite_Field_eq_finite
tff(fact_1666_finite__Field__eq__finite,axiom,
    ! [R: set_Pr4811707699266497531nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,field_Code_integer(R))
    <=> finite2154694888287295812nteger(R) ) ).

% finite_Field_eq_finite
tff(fact_1667_Field__empty,axiom,
    field_Product_unit(bot_bo1654022762305692595t_unit) = bot_bo3957492148770167129t_unit ).

% Field_empty
tff(fact_1668_Field__empty,axiom,
    field_o(bot_bo7073875226086086771od_o_o) = bot_bot_set_o ).

% Field_empty
tff(fact_1669_Field__empty,axiom,
    field_nat(bot_bo2099793752762293965at_nat) = bot_bot_set_nat ).

% Field_empty
tff(fact_1670_Field__empty,axiom,
    field_int(bot_bo1796632182523588997nt_int) = bot_bot_set_int ).

% Field_empty
tff(fact_1671_Field__empty,axiom,
    field_48612661033931452_nat_o(bot_bo6763312114214544691_nat_o) = bot_bo8632255151191677115_nat_o ).

% Field_empty
tff(fact_1672_Field__Un,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] : field_nat(sup_su6327502436637775413at_nat(R2,S2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,field_nat(R2)),field_nat(S2)) ).

% Field_Un
tff(fact_1673_finite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( finite6177210948735845034at_nat(R2)
     => aa_set_nat_o(finite_finite_nat,field_nat(R2)) ) ).

% finite_Field
tff(fact_1674_finite__Field,axiom,
    ! [R2: set_Pr4811707699266497531nteger] :
      ( finite2154694888287295812nteger(R2)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,field_Code_integer(R2)) ) ).

% finite_Field
tff(fact_1675_relcompEpair,axiom,
    ! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),C),relcom4930750947919833671et_nat(R2,S2))
     => ~ ! [B2: heap_e7401611519738050253t_unit] :
            ( member5046970461208427408t_unit(produc8256076736985659095t_unit(A2,B2),R2)
           => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B2),C),S2) ) ) ).

% relcompEpair
tff(fact_1676_relcompEpair,axiom,
    ! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),C),relcom8881708762719574682et_nat(R2,S2))
     => ~ ! [B2: set_nat] :
            ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B2),R2)
           => ~ member8277197624267554838et_nat(produc4532415448927165861et_nat(B2,C),S2) ) ) ).

% relcompEpair
tff(fact_1677_relcompEpair,axiom,
    ! [A2: nat,C: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),relcomp_nat_nat_nat(R2,S2))
     => ~ ! [B2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),R2)
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),C),S2) ) ) ).

% relcompEpair
tff(fact_1678_relcompEpair,axiom,
    ! [A2: int,C: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),relcomp_int_int_int(R2,S2))
     => ~ ! [B2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),R2)
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),C),S2) ) ) ).

% relcompEpair
tff(fact_1679_relcompEpair,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),C),relcom4607742480028120643et_nat(R2,S2))
     => ~ ! [B2: fun_Pr6460944699146269935_nat_o] :
            ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B2),R2)
           => ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),C),S2) ) ) ).

% relcompEpair
tff(fact_1680_relcompEpair,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr719794911490849221et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),C),relcom8637434619782166068et_nat(R2,S2))
     => ~ ! [B2: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B2),R2)
           => ~ member6099555550032318734et_nat(produc8199053930788261021et_nat(B2,C),S2) ) ) ).

% relcompEpair
tff(fact_1681_relcompEpair,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr6455445072687854819et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),C),relcom8550657552108312530et_nat(R2,S2))
     => ~ ! [B2: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B2),R2)
           => ~ member7022071448640610604et_nat(produc3782269066236780987et_nat(B2,C),S2) ) ) ).

% relcompEpair
tff(fact_1682_relcompEpair,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C: produc8799533983703829792et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),C),relcom9162728224921490145et_nat(R2,S2))
     => ~ ! [B2: fun_Pr6460944699146269935_nat_o] :
            ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B2),R2)
           => ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B2),C),S2) ) ) ).

% relcompEpair
tff(fact_1683_relcompEpair,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C: produc8799533983703829792et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr375601930021782115et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),C),relcom3443509513556836946et_nat(R2,S2))
     => ~ ! [B2: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B2),R2)
           => ~ member9093683042263559852et_nat(produc7898493064540081211et_nat(B2,C),S2) ) ) ).

% relcompEpair
tff(fact_1684_relcompEpair,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,C: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr9066344408807579265et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),C),relcom8040964428772967664et_nat(R2,S2))
     => ~ ! [B2: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B2),R2)
           => ~ member2216856746449346506et_nat(produc8050396023069295193et_nat(B2,C),S2) ) ) ).

% relcompEpair
tff(fact_1685_relcompE,axiom,
    ! [Xz: produc3658429121746597890et_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(Xz,relcom4930750947919833671et_nat(R2,S2))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: heap_e7401611519738050253t_unit,Z3: set_nat] :
            ( ( Xz = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Z3) )
           => ( member5046970461208427408t_unit(produc8256076736985659095t_unit(X3,Y2),R2)
             => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y2),Z3),S2) ) ) ) ).

% relcompE
tff(fact_1686_relcompE,axiom,
    ! [Xz: produc3658429121746597890et_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
      ( member6260224972018164377et_nat(Xz,relcom8881708762719574682et_nat(R2,S2))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
            ( ( Xz = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Z3) )
           => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R2)
             => ~ member8277197624267554838et_nat(produc4532415448927165861et_nat(Y2,Z3),S2) ) ) ) ).

% relcompE
tff(fact_1687_relcompE,axiom,
    ! [Xz: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(Xz,relcomp_nat_nat_nat(R2,S2))
     => ~ ! [X3: nat,Y2: nat,Z3: nat] :
            ( ( Xz = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Z3) )
           => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3),S2) ) ) ) ).

% relcompE
tff(fact_1688_relcompE,axiom,
    ! [Xz: product_prod_int_int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(Xz,relcomp_int_int_int(R2,S2))
     => ~ ! [X3: int,Y2: int,Z3: int] :
            ( ( Xz = aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Z3) )
           => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Z3),S2) ) ) ) ).

% relcompE
tff(fact_1689_relcompE,axiom,
    ! [Xz: produc8799533983703829792et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(Xz,relcom4607742480028120643et_nat(R2,S2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o,Z3: produc3658429121746597890et_nat] :
            ( ( Xz = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Z3) )
           => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X3,Y2),R2)
             => ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y2),Z3),S2) ) ) ) ).

% relcompE
tff(fact_1690_relcompE,axiom,
    ! [Xz: produc8799533983703829792et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr719794911490849221et_nat] :
      ( member4513763875858039863et_nat(Xz,relcom8637434619782166068et_nat(R2,S2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
            ( ( Xz = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Z3) )
           => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),R2)
             => ~ member6099555550032318734et_nat(produc8199053930788261021et_nat(Y2,Z3),S2) ) ) ) ).

% relcompE
tff(fact_1691_relcompE,axiom,
    ! [Xz: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr6455445072687854819et_nat] :
      ( member4513763875858039863et_nat(Xz,relcom8550657552108312530et_nat(R2,S2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat,Z3: produc3658429121746597890et_nat] :
            ( ( Xz = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Z3) )
           => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2),R2)
             => ~ member7022071448640610604et_nat(produc3782269066236780987et_nat(Y2,Z3),S2) ) ) ) ).

% relcompE
tff(fact_1692_relcompE,axiom,
    ! [Xz: produc3112576180687729086et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(Xz,relcom9162728224921490145et_nat(R2,S2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o,Z3: produc8799533983703829792et_nat] :
            ( ( Xz = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Z3) )
           => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X3,Y2),R2)
             => ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y2),Z3),S2) ) ) ) ).

% relcompE
tff(fact_1693_relcompE,axiom,
    ! [Xz: produc3112576180687729086et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr375601930021782115et_nat] :
      ( member9174161442226468181et_nat(Xz,relcom3443509513556836946et_nat(R2,S2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat,Z3: produc8799533983703829792et_nat] :
            ( ( Xz = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Z3) )
           => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),R2)
             => ~ member9093683042263559852et_nat(produc7898493064540081211et_nat(Y2,Z3),S2) ) ) ) ).

% relcompE
tff(fact_1694_relcompE,axiom,
    ! [Xz: produc3112576180687729086et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr9066344408807579265et_nat] :
      ( member9174161442226468181et_nat(Xz,relcom8040964428772967664et_nat(R2,S2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat,Z3: produc8799533983703829792et_nat] :
            ( ( Xz = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Z3) )
           => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2),R2)
             => ~ member2216856746449346506et_nat(produc8050396023069295193et_nat(Y2,Z3),S2) ) ) ) ).

% relcompE
tff(fact_1695_relcomp_OrelcompI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: heap_e7401611519738050253t_unit,R2: set_Pr1776176089890104647t_unit,C: set_nat,S2: set_Pr3948176798113811640et_nat] :
      ( member5046970461208427408t_unit(produc8256076736985659095t_unit(A2,B),R2)
     => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B),C),S2)
       => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),C),relcom4930750947919833671et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1696_relcomp_OrelcompI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr2890355626087285447_nat_o,C: produc3658429121746597890et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
     => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B),C),S2)
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),C),relcom4607742480028120643et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1697_relcomp_OrelcompI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr2890355626087285447_nat_o,C: produc8799533983703829792et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
     => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B),C),S2)
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),C),relcom9162728224921490145et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1698_relcomp_OrelcompI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat,C: set_nat,S2: set_Pr5488025237498180813et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R2)
     => ( member8277197624267554838et_nat(produc4532415448927165861et_nat(B,C),S2)
       => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),C),relcom8881708762719574682et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1699_relcomp_OrelcompI,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C),S2)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C),relcomp_nat_nat_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1700_relcomp_OrelcompI,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int,S2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),C),S2)
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),C),relcomp_int_int_int(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1701_relcomp_OrelcompI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,C: produc3658429121746597890et_nat,S2: set_Pr719794911490849221et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)
     => ( member6099555550032318734et_nat(produc8199053930788261021et_nat(B,C),S2)
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),C),relcom8637434619782166068et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1702_relcomp_OrelcompI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,C: produc8799533983703829792et_nat,S2: set_Pr375601930021782115et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)
     => ( member9093683042263559852et_nat(produc7898493064540081211et_nat(B,C),S2)
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),C),relcom3443509513556836946et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1703_relcomp_OrelcompI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,C: produc3658429121746597890et_nat,S2: set_Pr6455445072687854819et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)
     => ( member7022071448640610604et_nat(produc3782269066236780987et_nat(B,C),S2)
       => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),C),relcom8550657552108312530et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1704_relcomp_OrelcompI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,C: produc8799533983703829792et_nat,S2: set_Pr9066344408807579265et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)
     => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(B,C),S2)
       => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),C),relcom8040964428772967664et_nat(R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_1705_relcomp_Osimps,axiom,
    ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),A22),relcom4930750947919833671et_nat(R2,S2))
    <=> ? [A5: heap_e7401611519738050253t_unit,B6: heap_e7401611519738050253t_unit,C4: set_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member5046970461208427408t_unit(produc8256076736985659095t_unit(A5,B6),R2)
          & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B6),C4),S2) ) ) ).

% relcomp.simps
tff(fact_1706_relcomp_Osimps,axiom,
    ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),A22),relcom8881708762719574682et_nat(R2,S2))
    <=> ? [A5: heap_e7401611519738050253t_unit,B6: set_nat,C4: set_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A5),B6),R2)
          & member8277197624267554838et_nat(produc4532415448927165861et_nat(B6,C4),S2) ) ) ).

% relcomp.simps
tff(fact_1707_relcomp_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),relcomp_nat_nat_nat(R2,S2))
    <=> ? [A5: nat,B6: nat,C4: nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),R2)
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B6),C4),S2) ) ) ).

% relcomp.simps
tff(fact_1708_relcomp_Osimps,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),relcomp_int_int_int(R2,S2))
    <=> ? [A5: int,B6: int,C4: int] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),R2)
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B6),C4),S2) ) ) ).

% relcomp.simps
tff(fact_1709_relcomp_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),relcom4607742480028120643et_nat(R2,S2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: fun_Pr6460944699146269935_nat_o,C4: produc3658429121746597890et_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A5,B6),R2)
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B6),C4),S2) ) ) ).

% relcomp.simps
tff(fact_1710_relcomp_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr719794911490849221et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),relcom8637434619782166068et_nat(R2,S2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc3658429121746597890et_nat,C4: produc3658429121746597890et_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B6),R2)
          & member6099555550032318734et_nat(produc8199053930788261021et_nat(B6,C4),S2) ) ) ).

% relcomp.simps
tff(fact_1711_relcomp_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr6455445072687854819et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),relcom8550657552108312530et_nat(R2,S2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc8799533983703829792et_nat,C4: produc3658429121746597890et_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B6),R2)
          & member7022071448640610604et_nat(produc3782269066236780987et_nat(B6,C4),S2) ) ) ).

% relcomp.simps
tff(fact_1712_relcomp_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),relcom9162728224921490145et_nat(R2,S2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: fun_Pr6460944699146269935_nat_o,C4: produc8799533983703829792et_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A5,B6),R2)
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B6),C4),S2) ) ) ).

% relcomp.simps
tff(fact_1713_relcomp_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr375601930021782115et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),relcom3443509513556836946et_nat(R2,S2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc3658429121746597890et_nat,C4: produc8799533983703829792et_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B6),R2)
          & member9093683042263559852et_nat(produc7898493064540081211et_nat(B6,C4),S2) ) ) ).

% relcomp.simps
tff(fact_1714_relcomp_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr9066344408807579265et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),relcom8040964428772967664et_nat(R2,S2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc8799533983703829792et_nat,C4: produc8799533983703829792et_nat] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B6),R2)
          & member2216856746449346506et_nat(produc8050396023069295193et_nat(B6,C4),S2) ) ) ).

% relcomp.simps
tff(fact_1715_relcomp_Ocases,axiom,
    ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),A22),relcom4930750947919833671et_nat(R2,S2))
     => ~ ! [B2: heap_e7401611519738050253t_unit] :
            ( member5046970461208427408t_unit(produc8256076736985659095t_unit(A1,B2),R2)
           => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B2),A22),S2) ) ) ).

% relcomp.cases
tff(fact_1716_relcomp_Ocases,axiom,
    ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),A22),relcom8881708762719574682et_nat(R2,S2))
     => ~ ! [B2: set_nat] :
            ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),B2),R2)
           => ~ member8277197624267554838et_nat(produc4532415448927165861et_nat(B2,A22),S2) ) ) ).

% relcomp.cases
tff(fact_1717_relcomp_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),relcomp_nat_nat_nat(R2,S2))
     => ~ ! [B2: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),B2),R2)
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),A22),S2) ) ) ).

% relcomp.cases
tff(fact_1718_relcomp_Ocases,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),relcomp_int_int_int(R2,S2))
     => ~ ! [B2: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),B2),R2)
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),A22),S2) ) ) ).

% relcomp.cases
tff(fact_1719_relcomp_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),relcom4607742480028120643et_nat(R2,S2))
     => ~ ! [B2: fun_Pr6460944699146269935_nat_o] :
            ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A1,B2),R2)
           => ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B2),A22),S2) ) ) ).

% relcomp.cases
tff(fact_1720_relcomp_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr719794911490849221et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),relcom8637434619782166068et_nat(R2,S2))
     => ~ ! [B2: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),B2),R2)
           => ~ member6099555550032318734et_nat(produc8199053930788261021et_nat(B2,A22),S2) ) ) ).

% relcomp.cases
tff(fact_1721_relcomp_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr6455445072687854819et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),relcom8550657552108312530et_nat(R2,S2))
     => ~ ! [B2: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),B2),R2)
           => ~ member7022071448640610604et_nat(produc3782269066236780987et_nat(B2,A22),S2) ) ) ).

% relcomp.cases
tff(fact_1722_relcomp_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),relcom9162728224921490145et_nat(R2,S2))
     => ~ ! [B2: fun_Pr6460944699146269935_nat_o] :
            ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A1,B2),R2)
           => ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B2),A22),S2) ) ) ).

% relcomp.cases
tff(fact_1723_relcomp_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr1312406793078187350et_nat,S2: set_Pr375601930021782115et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),relcom3443509513556836946et_nat(R2,S2))
     => ~ ! [B2: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),B2),R2)
           => ~ member9093683042263559852et_nat(produc7898493064540081211et_nat(B2,A22),S2) ) ) ).

% relcomp.cases
tff(fact_1724_relcomp_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,S2: set_Pr9066344408807579265et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),relcom8040964428772967664et_nat(R2,S2))
     => ~ ! [B2: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),B2),R2)
           => ~ member2216856746449346506et_nat(produc8050396023069295193et_nat(B2,A22),S2) ) ) ).

% relcomp.cases
tff(fact_1725_FieldI2,axiom,
    ! [I2: fun_Pr6460944699146269935_nat_o,J: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(I2,J),R)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(J),field_48612661033931452_nat_o(R)) ) ).

% FieldI2
tff(fact_1726_FieldI2,axiom,
    ! [I2: $o,J: $o,R: set_Product_prod_o_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((I2)),(J)),R)
     => aa_set_o_o(member_o((J)),field_o(R)) ) ).

% FieldI2
tff(fact_1727_FieldI2,axiom,
    ! [I2: product_unit,J: product_unit,R: set_Pr5094982260447487303t_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(I2),J),R)
     => aa_set8780210009671629607unit_o(member_Product_unit(J),field_Product_unit(R)) ) ).

% FieldI2
tff(fact_1728_FieldI2,axiom,
    ! [I2: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I2),J),R)
     => aa_set_nat_o(member_nat(J),field_nat(R)) ) ).

% FieldI2
tff(fact_1729_FieldI2,axiom,
    ! [I2: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J),R)
     => aa_set_int_o(member_int(J),field_int(R)) ) ).

% FieldI2
tff(fact_1730_FieldI1,axiom,
    ! [I2: fun_Pr6460944699146269935_nat_o,J: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(I2,J),R)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I2),field_48612661033931452_nat_o(R)) ) ).

% FieldI1
tff(fact_1731_FieldI1,axiom,
    ! [I2: $o,J: $o,R: set_Product_prod_o_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((I2)),(J)),R)
     => aa_set_o_o(member_o((I2)),field_o(R)) ) ).

% FieldI1
tff(fact_1732_FieldI1,axiom,
    ! [I2: product_unit,J: product_unit,R: set_Pr5094982260447487303t_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(I2),J),R)
     => aa_set8780210009671629607unit_o(member_Product_unit(I2),field_Product_unit(R)) ) ).

% FieldI1
tff(fact_1733_FieldI1,axiom,
    ! [I2: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I2),J),R)
     => aa_set_nat_o(member_nat(I2),field_nat(R)) ) ).

% FieldI1
tff(fact_1734_FieldI1,axiom,
    ! [I2: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J),R)
     => aa_set_int_o(member_int(I2),field_int(R)) ) ).

% FieldI1
tff(fact_1735_mono__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(R2,S2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,field_nat(R2)),field_nat(S2)) ) ).

% mono_Field
tff(fact_1736_equiv__class__self,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(A2),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) ) ) ).

% equiv_class_self
tff(fact_1737_equiv__class__self,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set_o_o(member_o((A2)),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) ) ) ).

% equiv_class_self
tff(fact_1738_equiv__class__self,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(member_nat(A2),A)
       => aa_set_nat_o(member_nat(A2),aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) ) ) ).

% equiv_class_self
tff(fact_1739_equiv__class__self,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(member_int(A2),A)
       => aa_set_int_o(member_int(A2),image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) ) ) ).

% equiv_class_self
tff(fact_1740_equiv__class__self,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) ) ) ).

% equiv_class_self
tff(fact_1741_Field__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : field_nat(R2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,domain_nat_nat(R2)),range_nat_nat(R2)) ).

% Field_def
tff(fact_1742_equiv__class__eq__iff,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2)
      <=> ( ( image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit)) )
          & aa_set8780210009671629607unit_o(member_Product_unit(X),A)
          & aa_set8780210009671629607unit_o(member_Product_unit(Y),A) ) ) ) ).

% equiv_class_eq_iff
tff(fact_1743_equiv__class__eq__iff,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2)
      <=> ( ( image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o)) )
          & aa_set_o_o(member_o((X)),A)
          & aa_set_o_o(member_o((Y)),A) ) ) ) ).

% equiv_class_eq_iff
tff(fact_1744_equiv__class__eq__iff,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
      <=> ( ( aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat)) )
          & aa_set_nat_o(member_nat(X),A)
          & aa_set_nat_o(member_nat(Y),A) ) ) ) ).

% equiv_class_eq_iff
tff(fact_1745_equiv__class__eq__iff,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
      <=> ( ( image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int)) )
          & aa_set_int_o(member_int(X),A)
          & aa_set_int_o(member_int(Y),A) ) ) ) ).

% equiv_class_eq_iff
tff(fact_1746_equiv__class__eq__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2)
      <=> ( ( image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o)) )
          & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
          & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A) ) ) ) ).

% equiv_class_eq_iff
tff(fact_1747_eq__equiv__class__iff,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),A)
         => ( ( image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit)) )
          <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_1748_eq__equiv__class__iff,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(member_o((Y)),A)
         => ( ( image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o)) )
          <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_1749_eq__equiv__class__iff,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_o(member_nat(Y),A)
         => ( ( aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat)) )
          <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_1750_eq__equiv__class__iff,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(member_int(X),A)
       => ( aa_set_int_o(member_int(Y),A)
         => ( ( image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int)) )
          <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_1751_eq__equiv__class__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
         => ( ( image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o)) )
          <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_1752_equiv__class__eq,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)
       => ( image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) ) ) ) ).

% equiv_class_eq
tff(fact_1753_equiv__class__eq,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
       => ( image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) ) ) ) ).

% equiv_class_eq
tff(fact_1754_equiv__class__eq,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => ( aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) ) ) ) ).

% equiv_class_eq
tff(fact_1755_equiv__class__eq,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( equiv_equiv_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => ( image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) ) ) ) ).

% equiv_class_eq
tff(fact_1756_equiv__class__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
       => ( image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) ) ) ) ).

% equiv_class_eq
tff(fact_1757_eq__equiv__class,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,A: set_Product_unit] :
      ( ( image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) )
     => ( equiv_4189356295871601603t_unit(A,R2)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),A)
         => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ) ) ).

% eq_equiv_class
tff(fact_1758_eq__equiv__class,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o,A: set_o] :
      ( ( image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) )
     => ( equiv_equiv_o(A,R2)
       => ( aa_set_o_o(member_o((B)),A)
         => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ) ) ).

% eq_equiv_class
tff(fact_1759_eq__equiv__class,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,A: set_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) )
     => ( equiv_equiv_nat(A,R2)
       => ( aa_set_nat_o(member_nat(B),A)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2) ) ) ) ).

% eq_equiv_class
tff(fact_1760_eq__equiv__class,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,A: set_int] :
      ( ( image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) )
     => ( equiv_equiv_int(A,R2)
       => ( aa_set_int_o(member_int(B),A)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2) ) ) ) ).

% eq_equiv_class
tff(fact_1761_eq__equiv__class,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) )
     => ( equiv_5581975019310583717_nat_o(A,R2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A)
         => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2) ) ) ) ).

% eq_equiv_class
tff(fact_1762_rtrancl__Image__in__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat,V2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(R)),V2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,field_nat(R)),V2)) ).

% rtrancl_Image_in_Field
tff(fact_1763_refines__equiv__class__eq2,axiom,
    ! [R: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
      ( ord_le930853871177014503t_unit(R,S)
     => ( equiv_4189356295871601603t_unit(A,R)
       => ( equiv_4189356295871601603t_unit(A,S)
         => ( image_8431784415008483091t_unit(S,image_8431784415008483091t_unit(R,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = image_8431784415008483091t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_1764_refines__equiv__class__eq2,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
      ( ord_le192195778054999463od_o_o(R,S)
     => ( equiv_equiv_o(A,R)
       => ( equiv_equiv_o(A,S)
         => ( image_o_o(S,image_o_o(R,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = image_o_o(S,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_1765_refines__equiv__class__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
      ( ord_le3146513528884898305at_nat(R,S)
     => ( equiv_equiv_nat(A,R)
       => ( equiv_equiv_nat(A,S)
         => ( aa_set_nat_set_nat(image_nat_nat(S),aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = aa_set_nat_set_nat(image_nat_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_1766_refines__equiv__class__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
      ( ord_le2843351958646193337nt_int(R,S)
     => ( equiv_equiv_int(A,R)
       => ( equiv_equiv_int(A,S)
         => ( image_int_int(S,image_int_int(R,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = image_int_int(S,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_1767_refines__equiv__class__eq2,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,S: set_Pr2890355626087285447_nat_o,A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ord_le2940462717903690343_nat_o(R,S)
     => ( equiv_5581975019310583717_nat_o(A,R)
       => ( equiv_5581975019310583717_nat_o(A,S)
         => ( image_2188787866847195539_nat_o(S,image_2188787866847195539_nat_o(R,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) = image_2188787866847195539_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_1768_refines__equiv__class__eq,axiom,
    ! [R: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
      ( ord_le930853871177014503t_unit(R,S)
     => ( equiv_4189356295871601603t_unit(A,R)
       => ( equiv_4189356295871601603t_unit(A,S)
         => ( image_8431784415008483091t_unit(R,image_8431784415008483091t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = image_8431784415008483091t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_1769_refines__equiv__class__eq,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
      ( ord_le192195778054999463od_o_o(R,S)
     => ( equiv_equiv_o(A,R)
       => ( equiv_equiv_o(A,S)
         => ( image_o_o(R,image_o_o(S,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = image_o_o(S,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_1770_refines__equiv__class__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
      ( ord_le3146513528884898305at_nat(R,S)
     => ( equiv_equiv_nat(A,R)
       => ( equiv_equiv_nat(A,S)
         => ( aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(image_nat_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = aa_set_nat_set_nat(image_nat_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_1771_refines__equiv__class__eq,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
      ( ord_le2843351958646193337nt_int(R,S)
     => ( equiv_equiv_int(A,R)
       => ( equiv_equiv_int(A,S)
         => ( image_int_int(R,image_int_int(S,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = image_int_int(S,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_1772_refines__equiv__class__eq,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,S: set_Pr2890355626087285447_nat_o,A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ord_le2940462717903690343_nat_o(R,S)
     => ( equiv_5581975019310583717_nat_o(A,R)
       => ( equiv_5581975019310583717_nat_o(A,S)
         => ( image_2188787866847195539_nat_o(R,image_2188787866847195539_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) = image_2188787866847195539_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_1773_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( order_2505591618835473410t_unit(field_Product_unit(R2),R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( ( image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)) )
          <=> ( A2 = B ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_1774_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( order_8538205470752950541r_on_o(field_o(R2),R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( ( image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)) )
          <=> ( (A2)
            <=> (B) ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_1775_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( order_5251275573222108571on_nat(field_nat(R2),R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( ( aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)) )
          <=> ( A2 = B ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_1776_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( order_5248785102713058295on_int(field_int(R2),R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( ( image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)) )
          <=> ( A2 = B ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_1777_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( order_3152091435976458852_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( ( image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)) )
          <=> ( A2 = B ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_1778_finite__reachable__restrictedI,axiom,
    ! [Q: set_Code_integer,I: set_Code_integer,E: set_Pr4811707699266497531nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Q)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(I),Q)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(range_1907724884897625393nteger(E)),Q)
         => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_4147621952751189391nteger(transi3984306202085910426nteger(E),I)) ) ) ) ).

% finite_reachable_restrictedI
tff(fact_1779_finite__reachable__restrictedI,axiom,
    ! [Q: set_nat,I: set_nat,E: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,Q)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),Q)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,range_nat_nat(E)),Q)
         => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(transi2905341329935302413cl_nat(E)),I)) ) ) ) ).

% finite_reachable_restrictedI
tff(fact_1780_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A: set_int,B4: set_int] :
      ( order_4859164337913492053on_int(field_int(R2),R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),field_int(R2))
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
         => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(R2,A)),image_int_int(R2,B4))
          <=> ! [X2: int] :
                ( aa_set_int_o(member_int(X2),A)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),B4)
                    & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa2),X2),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_1781_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
      ( order_4861654808422542329on_nat(field_nat(R2),R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),field_nat(R2))
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
         => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R2),A)),aa_set_nat_set_nat(image_nat_nat(R2),B4))
          <=> ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),B4)
                    & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),X2),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_1782_equiv__class__subset,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)
       => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))) ) ) ).

% equiv_class_subset
tff(fact_1783_equiv__class__subset,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
       => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))) ) ) ).

% equiv_class_subset
tff(fact_1784_equiv__class__subset,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( equiv_equiv_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))),image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))) ) ) ).

% equiv_class_subset
tff(fact_1785_equiv__class__subset,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
       => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))),image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))) ) ) ).

% equiv_class_subset
tff(fact_1786_equiv__class__subset,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))) ) ) ).

% equiv_class_subset
tff(fact_1787_subset__equiv__class,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,B: product_unit,A2: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),A)
         => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ) ) ).

% subset_equiv_class
tff(fact_1788_subset__equiv__class,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,B: $o,A2: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))
       => ( aa_set_o_o(member_o((B)),A)
         => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ) ) ).

% subset_equiv_class
tff(fact_1789_subset__equiv__class,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,B: int,A2: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))),image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)))
       => ( aa_set_int_o(member_int(B),A)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2) ) ) ) ).

% subset_equiv_class
tff(fact_1790_subset__equiv__class,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,B: fun_Pr6460944699146269935_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))),image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A)
         => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2) ) ) ) ).

% subset_equiv_class
tff(fact_1791_subset__equiv__class,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B: nat,A2: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)))
       => ( aa_set_nat_o(member_nat(B),A)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2) ) ) ) ).

% subset_equiv_class
tff(fact_1792_finite__finite__vimage__IntI,axiom,
    ! [F2: set_Code_integer,Ha: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( ! [Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),F2)
           => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage1318793397690208119nteger(Ha,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger)),A)) )
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage1318793397690208119nteger(Ha,F2),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1793_finite__finite__vimage__IntI,axiom,
    ! [F2: set_Code_integer,Ha: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( ! [Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),F2)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage6104394758324456170nteger(Ha,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger))),A)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage6104394758324456170nteger(Ha,F2)),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1794_finite__finite__vimage__IntI,axiom,
    ! [F2: set_Product_unit,Ha: fun_Co1365676809456677961t_unit,A: set_Code_integer] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F2)
     => ( ! [Y2: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),F2)
           => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5651688891684837235t_unit(Ha,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit)),A)) )
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5651688891684837235t_unit(Ha,F2),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1795_finite__finite__vimage__IntI,axiom,
    ! [F2: set_Product_unit,Ha: fun_nat_Product_unit,A: set_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F2)
     => ( ! [Y2: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),F2)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage4884490618288580032t_unit(Ha,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit))),A)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage4884490618288580032t_unit(Ha,F2)),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1796_finite__finite__vimage__IntI,axiom,
    ! [F2: set_o,Ha: fun_Code_integer_o,A: set_Code_integer] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( ! [Y2: $o] :
            ( aa_set_o_o(member_o((Y2)),F2)
           => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage3005800680117336574eger_o(Ha,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o)),A)) )
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage3005800680117336574eger_o(Ha,F2),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1797_finite__finite__vimage__IntI,axiom,
    ! [F2: set_o,Ha: fun_nat_o,A: set_nat] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( ! [Y2: $o] :
            ( aa_set_o_o(member_o((Y2)),F2)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_o(Ha,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o))),A)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_o(Ha,F2)),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1798_finite__finite__vimage__IntI,axiom,
    ! [F2: set_nat,Ha: fun_Code_integer_nat,A: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),F2)
           => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5839839309545413354er_nat(Ha,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat)),A)) )
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5839839309545413354er_nat(Ha,F2),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1799_finite__finite__vimage__IntI,axiom,
    ! [F2: set_nat,Ha: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),F2)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(Ha),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat))),A)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(Ha),F2)),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1800_finite__finite__vimage__IntI,axiom,
    ! [F2: set_int,Ha: fun_Code_integer_int,A: set_Code_integer] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),F2)
           => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5837348839036363078er_int(Ha,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int)),A)) )
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5837348839036363078er_int(Ha,F2),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1801_finite__finite__vimage__IntI,axiom,
    ! [F2: set_int,Ha: fun_nat_int,A: set_nat] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),F2)
           => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_int(Ha,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int))),A)) )
       => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_int(Ha,F2)),A)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_1802_finite__insert,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))
    <=> aa_set_int_o(finite_finite_int,A) ) ).

% finite_insert
tff(fact_1803_finite__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> aa_set_o_o(finite_finite_o,A) ) ).

% finite_insert
tff(fact_1804_finite__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A) ) ).

% finite_insert
tff(fact_1805_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))
    <=> aa_set_nat_o(finite_finite_nat,A) ) ).

% finite_insert
tff(fact_1806_finite__insert,axiom,
    ! [A2: code_integer,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),A))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ).

% finite_insert
tff(fact_1807_finite__subset__induct_H,axiom,
    ! [F2: set_Code_integer,A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F2),A)
       => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F3: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
                 => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F3),A)
                   => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F3)
                     => ( aa_set5643387332196575111eger_o(P,F3)
                       => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A3),F3)) ) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1808_finite__subset__induct_H,axiom,
    ! [F2: set_Product_unit,A: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,F2),A)
       => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
         => ( ! [A3: product_unit,F3: set_Product_unit] :
                ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F3)
               => ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
                 => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,F3),A)
                   => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A3),F3)
                     => ( aa_set8780210009671629607unit_o(P,F3)
                       => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A3),F3)) ) ) ) ) )
           => aa_set8780210009671629607unit_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1809_finite__subset__induct_H,axiom,
    ! [F2: set_o,A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F2),A)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A)
                 => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F3),A)
                   => ( ~ aa_set_o_o(member_o((A3)),F3)
                     => ( aa_set_o_o(P,F3)
                       => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A3)),F3)) ) ) ) ) )
           => aa_set_o_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1810_finite__subset__induct_H,axiom,
    ! [F2: set_int,A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F2),A)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( aa_set_int_o(member_int(A3),A)
                 => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F3),A)
                   => ( ~ aa_set_int_o(member_int(A3),F3)
                     => ( aa_set_int_o(P,F3)
                       => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A3),F3)) ) ) ) ) )
           => aa_set_int_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1811_finite__subset__induct_H,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F2),A)
       => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
         => ( ! [A3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A)
                 => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F3),A)
                   => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),F3)
                     => ( aa_set5091322549519198405at_o_o(P,F3)
                       => aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),F3)) ) ) ) ) )
           => aa_set5091322549519198405at_o_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1812_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat(A3),A)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A)
                   => ( ~ aa_set_nat_o(member_nat(A3),F3)
                     => ( aa_set_nat_o(P,F3)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A3),F3)) ) ) ) ) )
           => aa_set_nat_o(P,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1813_finite__subset__induct,axiom,
    ! [F2: set_Code_integer,A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F2),A)
       => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F3: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F3)
                   => ( aa_set5643387332196575111eger_o(P,F3)
                     => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A3),F3)) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1814_finite__subset__induct,axiom,
    ! [F2: set_Product_unit,A: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,F2),A)
       => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
         => ( ! [A3: product_unit,F3: set_Product_unit] :
                ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F3)
               => ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
                 => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A3),F3)
                   => ( aa_set8780210009671629607unit_o(P,F3)
                     => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A3),F3)) ) ) ) )
           => aa_set8780210009671629607unit_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1815_finite__subset__induct,axiom,
    ! [F2: set_o,A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,F2),A)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A)
                 => ( ~ aa_set_o_o(member_o((A3)),F3)
                   => ( aa_set_o_o(P,F3)
                     => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A3)),F3)) ) ) ) )
           => aa_set_o_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1816_finite__subset__induct,axiom,
    ! [F2: set_int,A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,F2),A)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( aa_set_int_o(member_int(A3),A)
                 => ( ~ aa_set_int_o(member_int(A3),F3)
                   => ( aa_set_int_o(P,F3)
                     => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A3),F3)) ) ) ) )
           => aa_set_int_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1817_finite__subset__induct,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(F2),A)
       => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
         => ( ! [A3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A)
                 => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),F3)
                   => ( aa_set5091322549519198405at_o_o(P,F3)
                     => aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),F3)) ) ) ) )
           => aa_set5091322549519198405at_o_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1818_finite__subset__induct,axiom,
    ! [F2: set_nat,A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat(A3),A)
                 => ( ~ aa_set_nat_o(member_nat(A3),F3)
                   => ( aa_set_nat_o(P,F3)
                     => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A3),F3)) ) ) ) )
           => aa_set_nat_o(P,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1819_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: fun_se7289222521248229150eger_o,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,Y4)),aa_Code_integer_rat(F,X3)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1820_finite__ranking__induct,axiom,
    ! [S: set_Product_unit,P: fun_se4342725320470837934unit_o,F: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [X3: product_unit,S5: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S5)
             => ( ! [Y4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Product_unit_rat(F,Y4)),aa_Product_unit_rat(F,X3)) )
               => ( aa_set8780210009671629607unit_o(P,S5)
                 => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),S5)) ) ) )
         => aa_set8780210009671629607unit_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1821_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,S5: set_o] :
              ( aa_set_o_o(finite_finite_o,S5)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Y4))),aa_o_rat(F,(X3))) )
               => ( aa_set_o_o(P,S5)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),S5)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1822_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X3)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),S5)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1823_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X3)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),S5)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1824_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: fun_se7289222521248229150eger_o,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,Y4)),aa_Code_integer_nat(F,X3)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1825_finite__ranking__induct,axiom,
    ! [S: set_Product_unit,P: fun_se4342725320470837934unit_o,F: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [X3: product_unit,S5: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S5)
             => ( ! [Y4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,Y4)),aa_Product_unit_nat(F,X3)) )
               => ( aa_set8780210009671629607unit_o(P,S5)
                 => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),S5)) ) ) )
         => aa_set8780210009671629607unit_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1826_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,S5: set_o] :
              ( aa_set_o_o(finite_finite_o,S5)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Y4))),aa_o_nat(F,(X3))) )
               => ( aa_set_o_o(P,S5)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),S5)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1827_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Y4)),aa_nat_nat(F,X3)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),S5)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1828_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Y4)),aa_int_nat(F,X3)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),S5)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_1829_infinite__finite__induct,axiom,
    ! [P: fun_se7289222521248229150eger_o,A: set_Code_integer] :
      ( ! [A8: set_Code_integer] :
          ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8)
         => aa_set5643387332196575111eger_o(P,A8) )
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,F3: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
             => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X3),F3)
               => ( aa_set5643387332196575111eger_o(P,F3)
                 => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),F3)) ) ) )
         => aa_set5643387332196575111eger_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_1830_infinite__finite__induct,axiom,
    ! [P: fun_se4342725320470837934unit_o,A: set_Product_unit] :
      ( ! [A8: set_Product_unit] :
          ( ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8)
         => aa_set8780210009671629607unit_o(P,A8) )
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [X3: product_unit,F3: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F3)
             => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),F3)
               => ( aa_set8780210009671629607unit_o(P,F3)
                 => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),F3)) ) ) )
         => aa_set8780210009671629607unit_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_1831_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A: set_o] :
      ( ! [A8: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A8)
         => aa_set_o_o(P,A8) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X3)),F3)
               => ( aa_set_o_o(P,F3)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),F3)) ) ) )
         => aa_set_o_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_1832_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A: set_nat] :
      ( ! [A8: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A8)
         => aa_set_nat_o(P,A8) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X3),F3)
               => ( aa_set_nat_o(P,F3)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),F3)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_1833_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A: set_int] :
      ( ! [A8: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A8)
         => aa_set_int_o(P,A8) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X3),F3)
               => ( aa_set_int_o(P,F3)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),F3)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_1834_infinite__finite__induct,axiom,
    ! [P: fun_se7459991829846211660at_o_o,A: set_fu3555266933746557007_nat_o] :
      ( ! [A8: set_fu3555266933746557007_nat_o] :
          ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A8)
         => aa_set5091322549519198405at_o_o(P,A8) )
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
             => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),F3)
               => ( aa_set5091322549519198405at_o_o(P,F3)
                 => aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),F3)) ) ) )
         => aa_set5091322549519198405at_o_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_1835_finite__ne__induct,axiom,
    ! [F2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( ( F2 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] : aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),bot_bo3990330152332043303nteger))
         => ( ! [X3: code_integer,F3: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
               => ( ( F3 != bot_bo3990330152332043303nteger )
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X3),F3)
                   => ( aa_set5643387332196575111eger_o(P,F3)
                     => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),F3)) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1836_finite__ne__induct,axiom,
    ! [F2: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F2)
     => ( ( F2 != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit] : aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit))
         => ( ! [X3: product_unit,F3: set_Product_unit] :
                ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F3)
               => ( ( F3 != bot_bo3957492148770167129t_unit )
                 => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),F3)
                   => ( aa_set8780210009671629607unit_o(P,F3)
                     => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),F3)) ) ) ) )
           => aa_set8780210009671629607unit_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1837_finite__ne__induct,axiom,
    ! [F2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X3: $o] : aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o))
         => ( ! [X3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X3)),F3)
                   => ( aa_set_o_o(P,F3)
                     => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),F3)) ) ) ) )
           => aa_set_o_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1838_finite__ne__induct,axiom,
    ! [F2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X3: nat] : aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat))
         => ( ! [X3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X3),F3)
                   => ( aa_set_nat_o(P,F3)
                     => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),F3)) ) ) ) )
           => aa_set_nat_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1839_finite__ne__induct,axiom,
    ! [F2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X3: int] : aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int))
         => ( ! [X3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X3),F3)
                   => ( aa_set_int_o(P,F3)
                     => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),F3)) ) ) ) )
           => aa_set_int_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1840_finite__ne__induct,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( ( F2 != bot_bo8632255151191677115_nat_o )
       => ( ! [X3: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),bot_bo8632255151191677115_nat_o))
         => ( ! [X3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
                ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
               => ( ( F3 != bot_bo8632255151191677115_nat_o )
                 => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),F3)
                   => ( aa_set5091322549519198405at_o_o(P,F3)
                     => aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),F3)) ) ) ) )
           => aa_set5091322549519198405at_o_o(P,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1841_finite__induct,axiom,
    ! [F2: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,F3: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
             => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X3),F3)
               => ( aa_set5643387332196575111eger_o(P,F3)
                 => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),F3)) ) ) )
         => aa_set5643387332196575111eger_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1842_finite__induct,axiom,
    ! [F2: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F2)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [X3: product_unit,F3: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,F3)
             => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),F3)
               => ( aa_set8780210009671629607unit_o(P,F3)
                 => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),F3)) ) ) )
         => aa_set8780210009671629607unit_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1843_finite__induct,axiom,
    ! [F2: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X3)),F3)
               => ( aa_set_o_o(P,F3)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),F3)) ) ) )
         => aa_set_o_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1844_finite__induct,axiom,
    ! [F2: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X3),F3)
               => ( aa_set_nat_o(P,F3)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),F3)) ) ) )
         => aa_set_nat_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1845_finite__induct,axiom,
    ! [F2: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X3),F3)
               => ( aa_set_int_o(P,F3)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),F3)) ) ) )
         => aa_set_int_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1846_finite__induct,axiom,
    ! [F2: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F2)
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,F3: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,F3)
             => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),F3)
               => ( aa_set5091322549519198405at_o_o(P,F3)
                 => aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),F3)) ) ) )
         => aa_set5091322549519198405at_o_o(P,F2) ) ) ) ).

% finite_induct
tff(fact_1847_finite_Osimps,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
    <=> ( ( A2 = bot_bo3990330152332043303nteger )
        | ? [A9: set_Code_integer,A5: code_integer] :
            ( ( A2 = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A5),A9) )
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A9) ) ) ) ).

% finite.simps
tff(fact_1848_finite_Osimps,axiom,
    ! [A2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A2)
    <=> ( ( A2 = bot_bo3957492148770167129t_unit )
        | ? [A9: set_Product_unit,A5: product_unit] :
            ( ( A2 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A5),A9) )
            & aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A9) ) ) ) ).

% finite.simps
tff(fact_1849_finite_Osimps,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
    <=> ( ( A2 = bot_bot_set_o )
        | ? [A9: set_o,A5: $o] :
            ( ( A2 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A5)),A9) )
            & aa_set_o_o(finite_finite_o,A9) ) ) ) ).

% finite.simps
tff(fact_1850_finite_Osimps,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
    <=> ( ( A2 = bot_bot_set_nat )
        | ? [A9: set_nat,A5: nat] :
            ( ( A2 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A5),A9) )
            & aa_set_nat_o(finite_finite_nat,A9) ) ) ) ).

% finite.simps
tff(fact_1851_finite_Osimps,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
    <=> ( ( A2 = bot_bot_set_int )
        | ? [A9: set_int,A5: int] :
            ( ( A2 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A5),A9) )
            & aa_set_int_o(finite_finite_int,A9) ) ) ) ).

% finite.simps
tff(fact_1852_finite_Osimps,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
    <=> ( ( A2 = bot_bo8632255151191677115_nat_o )
        | ? [A9: set_fu3555266933746557007_nat_o,A5: fun_Pr6460944699146269935_nat_o] :
            ( ( A2 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A5),A9) )
            & aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A9) ) ) ) ).

% finite.simps
tff(fact_1853_infinite__imp__nonempty,axiom,
    ! [S: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( S != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
tff(fact_1854_infinite__imp__nonempty,axiom,
    ! [S: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( S != bot_bo3957492148770167129t_unit ) ) ).

% infinite_imp_nonempty
tff(fact_1855_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_1856_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_1857_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_1858_infinite__imp__nonempty,axiom,
    ! [S: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( S != bot_bo8632255151191677115_nat_o ) ) ).

% infinite_imp_nonempty
tff(fact_1859_finite_OemptyI,axiom,
    aa_set5643387332196575111eger_o(finite6017078050557962740nteger,bot_bo3990330152332043303nteger) ).

% finite.emptyI
tff(fact_1860_finite_OemptyI,axiom,
    aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,bot_bo3957492148770167129t_unit) ).

% finite.emptyI
tff(fact_1861_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_1862_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_1863_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_1864_finite_OemptyI,axiom,
    aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,bot_bo8632255151191677115_nat_o) ).

% finite.emptyI
tff(fact_1865_finite_OinsertI,axiom,
    ! [A: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A)) ) ).

% finite.insertI
tff(fact_1866_finite_OinsertI,axiom,
    ! [A: set_o,A2: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => aa_set_o_o(finite_finite_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A)) ) ).

% finite.insertI
tff(fact_1867_finite_OinsertI,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) ) ).

% finite.insertI
tff(fact_1868_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)) ) ).

% finite.insertI
tff(fact_1869_finite_OinsertI,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),A)) ) ).

% finite.insertI
tff(fact_1870_finite__has__minimal,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1871_finite__has__minimal,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
            & ! [Xa3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Xa3),A)
               => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1872_finite__has__minimal,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
            & ! [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),A)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa3)),(X3))
                 => ( (X3)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1873_finite__has__minimal,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ? [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
            & ! [Xa3: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa3),A)
               => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1874_finite__has__minimal,axiom,
    ! [A: set_filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ? [X3: filter_nat] :
            ( member_filter_nat(X3,A)
            & ! [Xa3: filter_nat] :
                ( member_filter_nat(Xa3,A)
               => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1875_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A)
            & ! [Xa3: set_nat] :
                ( member_set_nat(Xa3,A)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1876_finite__has__minimal,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A)
            & ! [Xa3: rat] :
                ( aa_set_rat_o(member_rat(Xa3),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1877_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1878_finite__has__minimal,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa3),X3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1879_finite__has__maximal,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1880_finite__has__maximal,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
            & ! [Xa3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Xa3),A)
               => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1881_finite__has__maximal,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
            & ! [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),A)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(Xa3))
                 => ( (X3)
                  <=> (Xa3) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1882_finite__has__maximal,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ? [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
            & ! [Xa3: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa3),A)
               => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1883_finite__has__maximal,axiom,
    ! [A: set_filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ? [X3: filter_nat] :
            ( member_filter_nat(X3,A)
            & ! [Xa3: filter_nat] :
                ( member_filter_nat(Xa3,A)
               => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1884_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A)
            & ! [Xa3: set_nat] :
                ( member_set_nat(Xa3,A)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1885_finite__has__maximal,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A)
            & ! [Xa3: rat] :
                ( aa_set_rat_o(member_rat(Xa3),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1886_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1887_finite__has__maximal,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa3)
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1888_finite_Ocases,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ~ ! [A8: set_Code_integer] :
              ( ? [A3: code_integer] : A2 = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A3),A8)
             => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8) ) ) ) ).

% finite.cases
tff(fact_1889_finite_Ocases,axiom,
    ! [A2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A2)
     => ( ( A2 != bot_bo3957492148770167129t_unit )
       => ~ ! [A8: set_Product_unit] :
              ( ? [A3: product_unit] : A2 = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A3),A8)
             => ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8) ) ) ) ).

% finite.cases
tff(fact_1890_finite_Ocases,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ~ ! [A8: set_o] :
              ( ? [A3: $o] : A2 = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A3)),A8)
             => ~ aa_set_o_o(finite_finite_o,A8) ) ) ) ).

% finite.cases
tff(fact_1891_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A8: set_nat] :
              ( ? [A3: nat] : A2 = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A3),A8)
             => ~ aa_set_nat_o(finite_finite_nat,A8) ) ) ) ).

% finite.cases
tff(fact_1892_finite_Ocases,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ~ ! [A8: set_int] :
              ( ? [A3: int] : A2 = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A3),A8)
             => ~ aa_set_int_o(finite_finite_int,A8) ) ) ) ).

% finite.cases
tff(fact_1893_finite_Ocases,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A2)
     => ( ( A2 != bot_bo8632255151191677115_nat_o )
       => ~ ! [A8: set_fu3555266933746557007_nat_o] :
              ( ? [A3: fun_Pr6460944699146269935_nat_o] : A2 = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),A8)
             => ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A8) ) ) ) ).

% finite.cases
tff(fact_1894_min__ext__compat,axiom,
    ! [R: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
      ( ord_le930853871177014503t_unit(relcom848357998638694028t_unit(R,S),R)
     => ord_le8072594200992201959t_unit(relcom7108816970757792620t_unit(min_ext_Product_unit(R),sup_su3573767163274163995t_unit(min_ext_Product_unit(S),insert6793119152790066487t_unit(produc7269801343787132631t_unit(bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit),bot_bo5113747506523071411t_unit))),min_ext_Product_unit(R)) ) ).

% min_ext_compat
tff(fact_1895_min__ext__compat,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ord_le192195778054999463od_o_o(relcomp_o_o_o(R,S),R)
     => ord_le2885120122233735591_set_o(relcom7534720040753365047_set_o(min_ext_o(R),sup_su7766833992073560027_set_o(min_ext_o(S),insert1494450971503307511_set_o(produc5838405689764958487_set_o(bot_bot_set_o,bot_bot_set_o),bot_bo8143257439550059635_set_o))),min_ext_o(R)) ) ).

% min_ext_compat
tff(fact_1896_min__ext__compat,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(relcomp_nat_nat_nat(R,S),R)
     => ord_le4402255868550298733et_nat(relcom236561585145426837et_nat(min_ext_nat(R),sup_su4251893418135750817et_nat(min_ext_nat(S),insert3810226134351308605et_nat(produc4532415448927165861et_nat(bot_bot_set_nat,bot_bot_set_nat),bot_bo4952753636535259449et_nat))),min_ext_nat(R)) ) ).

% min_ext_compat
tff(fact_1897_min__ext__compat,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ord_le2843351958646193337nt_int(relcomp_int_int_int(R,S),R)
     => ord_le1436784781161120549et_int(relcom4034950705519254697et_int(min_ext_int(R),sup_su1286422330746572633et_int(min_ext_int(S),insert7328952640415156725et_int(produc6363374080413544029et_int(bot_bot_set_int,bot_bot_set_int),bot_bo1987282549146081265et_int))),min_ext_int(R)) ) ).

% min_ext_compat
tff(fact_1898_min__ext__compat,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,S: set_Pr2890355626087285447_nat_o] :
      ( ord_le2940462717903690343_nat_o(relcom1922595779876297326_nat_o(R,S),R)
     => ord_le9137474638858277479_nat_o(relcom7987510920137042382_nat_o(min_ex8544404781473672728_nat_o(R),sup_su4458343269897816219_nat_o(min_ex8544404781473672728_nat_o(S),insert1134728238107055031_nat_o(produc5434073240646464855_nat_o(bot_bo8632255151191677115_nat_o,bot_bo8632255151191677115_nat_o),bot_bo7269894807694359859_nat_o))),min_ex8544404781473672728_nat_o(R)) ) ).

% min_ext_compat
tff(fact_1899_arg__min__least,axiom,
    ! [S: set_Code_integer,Y: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,lattic5439806495466278992er_rat(F,S))),aa_Code_integer_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1900_arg__min__least,axiom,
    ! [S: set_Product_unit,Y: product_unit,F: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( ( S != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Product_unit_rat(F,lattic725702332050428812it_rat(F,S))),aa_Product_unit_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1901_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Y)),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,aa_set_o_o(lattic2140725968369957399_o_rat(F),S))),aa_o_rat(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_1902_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S))),aa_nat_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1903_arg__min__least,axiom,
    ! [S: set_int,Y: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S))),aa_int_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1904_arg__min__least,axiom,
    ! [S: set_Code_integer,Y: code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,lattic6074936555552774728er_nat(F,S))),aa_Code_integer_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1905_arg__min__least,axiom,
    ! [S: set_Product_unit,Y: product_unit,F: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( ( S != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,lattic1360832392136924548it_nat(F,S))),aa_Product_unit_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1906_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Y)),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S))),aa_o_nat(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_1907_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_set_nat_nat(lattic7446932960582359483at_nat(F),S))),aa_nat_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1908_arg__min__least,axiom,
    ! [S: set_int,Y: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,lattic8446286672483414039nt_nat(F,S))),aa_int_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1909_Total__subset__Id,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( total_on_nat(field_nat(R2),R2)
     => ( ord_le3146513528884898305at_nat(R2,id_nat)
       => ( ( R2 = bot_bo2099793752762293965at_nat )
          | ? [A3: nat] : R2 = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),A3),bot_bo2099793752762293965at_nat) ) ) ) ).

% Total_subset_Id
tff(fact_1910_Total__subset__Id,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( total_on_int(field_int(R2),R2)
     => ( ord_le2843351958646193337nt_int(R2,id_int)
       => ( ( R2 = bot_bo1796632182523588997nt_int )
          | ? [A3: int] : R2 = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),A3),bot_bo1796632182523588997nt_int) ) ) ) ).

% Total_subset_Id
tff(fact_1911_finite__transitivity__chain,axiom,
    ! [A: set_Code_integer,R: fun_Co2081214437782444048eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [X3: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X3),X3)
       => ( ! [X3: code_integer,Y2: code_integer,Z3: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X3),Y2)
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,Y2),Z3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X3),Z3) ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
               => ? [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),A)
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X3),Y4) ) )
           => ( A = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1912_finite__transitivity__chain,axiom,
    ! [A: set_Product_unit,R: fun_Pr1520749766685617582unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ! [X3: product_unit] : ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X3),X3)
       => ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
              ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X3),Y2)
             => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,Y2),Z3)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X3),Z3) ) )
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
               => ? [Y4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(Y4),A)
                    & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X3),Y4) ) )
           => ( A = bot_bo3957492148770167129t_unit ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1913_finite__transitivity__chain,axiom,
    ! [A: set_o,R: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ! [X3: $o] : ~ aa_o_o(aa_o_fun_o_o(R,(X3)),(X3))
       => ( ! [X3: $o,Y2: $o,Z3: $o] :
              ( aa_o_o(aa_o_fun_o_o(R,(X3)),(Y2))
             => ( aa_o_o(aa_o_fun_o_o(R,(Y2)),(Z3))
               => aa_o_o(aa_o_fun_o_o(R,(X3)),(Z3)) ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A)
               => ? [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),A)
                    & aa_o_o(aa_o_fun_o_o(R,(X3)),(Y4)) ) )
           => ( A = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1914_finite__transitivity__chain,axiom,
    ! [A: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y2),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z3) ) )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),A)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X3),Y4) ) )
           => ( A = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1915_finite__transitivity__chain,axiom,
    ! [A: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [X3: int] : ~ aa_int_o(aa_int_fun_int_o(R,X3),X3)
       => ( ! [X3: int,Y2: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X3),Y2)
             => ( aa_int_o(aa_int_fun_int_o(R,Y2),Z3)
               => aa_int_o(aa_int_fun_int_o(R,X3),Z3) ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),A)
               => ? [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),A)
                    & aa_int_o(aa_int_fun_int_o(R,X3),Y4) ) )
           => ( A = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1916_finite__transitivity__chain,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R: fun_fu4164567611050918574at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] : ~ aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X3),X3)
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o,Z3: fun_Pr6460944699146269935_nat_o] :
              ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X3),Y2)
             => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,Y2),Z3)
               => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X3),Z3) ) )
         => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
               => ? [Y4: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),A)
                    & aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X3),Y4) ) )
           => ( A = bot_bo8632255151191677115_nat_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1917_disjnt__equiv__class,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(disjnt_Product_unit(image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))),image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
      <=> ~ member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ) ).

% disjnt_equiv_class
tff(fact_1918_disjnt__equiv__class,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(disjnt_o(image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))),image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
      <=> ~ member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ) ).

% disjnt_equiv_class
tff(fact_1919_disjnt__equiv__class,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))),aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)))
      <=> ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2) ) ) ).

% disjnt_equiv_class
tff(fact_1920_disjnt__equiv__class,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(disjnt_int(image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))),image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)))
      <=> ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2) ) ) ).

% disjnt_equiv_class
tff(fact_1921_disjnt__equiv__class,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))),image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)))
      <=> ~ member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2) ) ) ).

% disjnt_equiv_class
tff(fact_1922_Sup__fin_Oinsert,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( lattic3835124923745554447et_nat(insert_set_nat(X,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1923_Sup__fin_Oinsert,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( ( A != bot_bot_set_assn )
       => ( lattic2150320897289308081n_assn(insert_assn(X,A)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1924_Sup__fin_Oinsert,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic5294303975357428420t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),lattic5294303975357428420t_unit(A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1925_Sup__fin_Oinsert,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
        <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1926_Sup__fin_Oinsert,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_set_nat_nat(lattic1093996805478795353in_nat,A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1927_Sup__fin_Oinsert,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1928_Sup__fin_Oinsert,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( lattic7388253182610234406_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(A)) ) ) ) ).

% Sup_fin.insert
tff(fact_1929_Inf__fin_Oinsert,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( lattic3014633134055518761et_nat(insert_set_nat(X,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1930_Inf__fin_Oinsert,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( ( A != bot_bot_set_assn )
       => ( lattic47131356835913163n_assn(insert_assn(X,A)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1931_Inf__fin_Oinsert,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic1263872656861969706t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),lattic1263872656861969706t_unit(A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1932_Inf__fin_Oinsert,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
        <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1933_Inf__fin_Oinsert,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1934_Inf__fin_Oinsert,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1935_Inf__fin_Oinsert,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( lattic6089491107740071820_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X),lattic6089491107740071820_nat_o(A)) ) ) ) ).

% Inf_fin.insert
tff(fact_1936_in__quotient__imp__in__rel,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X5: set_Product_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit))),X5)
         => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_1937_in__quotient__imp__in__rel,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X5: set_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member_set_o(X5,equiv_quotient_o(A,R2))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o))),X5)
         => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_1938_in__quotient__imp__in__rel,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X5: set_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( member_set_int(X5,equiv_quotient_int(A,R2))
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int))),X5)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_1939_in__quotient__imp__in__rel,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X5: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
       => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o))),X5)
         => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_1940_in__quotient__imp__in__rel,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X5: set_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member_set_nat(X5,equiv_quotient_nat(A,R2))
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat))),X5)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_1941_prod__set__simps_I1_J,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat] : basic_7833836874476830220et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = insert8563697753282147485t_unit(X,bot_bo3686245532977126937t_unit) ).

% prod_set_simps(1)
tff(fact_1942_prod__set__simps_I1_J,axiom,
    ! [X: nat,Y: nat] : basic_fsts_nat_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ).

% prod_set_simps(1)
tff(fact_1943_prod__set__simps_I1_J,axiom,
    ! [X: int,Y: int] : basic_fsts_int_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ).

% prod_set_simps(1)
tff(fact_1944_prod__set__simps_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : basic_6868354978976217258et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ).

% prod_set_simps(1)
tff(fact_1945_prod__set__simps_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : basic_7906824686833514952et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ).

% prod_set_simps(1)
tff(fact_1946_rel__restrict__tranclI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),transi8445014155877283344_nat_o(E))
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),R)
       => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),R)
         => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(E,R)),R)
           => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),transi8445014155877283344_nat_o(rel_re4406848170174997969_nat_o(E,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_1947_rel__restrict__tranclI,axiom,
    ! [X: $o,Y: $o,E: set_Product_prod_o_o,R: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),transitive_trancl_o(E))
     => ( ~ aa_set_o_o(member_o((X)),R)
       => ( ~ aa_set_o_o(member_o((Y)),R)
         => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(E,R)),R)
           => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),transitive_trancl_o(rel_restrict_o(E,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_1948_rel__restrict__tranclI,axiom,
    ! [X: product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit,R: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),transi430408012122777006t_unit(E))
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),R)
       => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(Y),R)
         => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(E,R)),R)
           => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),transi430408012122777006t_unit(rel_re9207996470947987439t_unit(E,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_1949_rel__restrict__tranclI,axiom,
    ! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(E))
     => ( ~ aa_set_int_o(member_int(X),R)
       => ( ~ aa_set_int_o(member_int(Y),R)
         => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(E,R)),R)
           => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(rel_restrict_int(E,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_1950_rel__restrict__tranclI,axiom,
    ! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(E))
     => ( ~ aa_set_nat_o(member_nat(X),R)
       => ( ~ aa_set_nat_o(member_nat(Y),R)
         => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(E),R)),R)
           => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(rel_restrict_nat(E,R))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_1951_pair__in__Id__conv,axiom,
    ! [A2: nat,B: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),id_nat)
    <=> ( A2 = B ) ) ).

% pair_in_Id_conv
tff(fact_1952_pair__in__Id__conv,axiom,
    ! [A2: int,B: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),id_int)
    <=> ( A2 = B ) ) ).

% pair_in_Id_conv
tff(fact_1953_IdI,axiom,
    ! [A2: nat] : member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),id_nat) ).

% IdI
tff(fact_1954_IdI,axiom,
    ! [A2: int] : member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),id_int) ).

% IdI
tff(fact_1955_quotient__empty,axiom,
    ! [R2: set_Pr5094982260447487303t_unit] : equiv_5172566880711946708t_unit(bot_bo3957492148770167129t_unit,R2) = bot_bo6284674557190688313t_unit ).

% quotient_empty
tff(fact_1956_quotient__empty,axiom,
    ! [R2: set_Product_prod_o_o] : equiv_quotient_o(bot_bot_set_o,R2) = bot_bot_set_set_o ).

% quotient_empty
tff(fact_1957_quotient__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : equiv_quotient_nat(bot_bot_set_nat,R2) = bot_bot_set_set_nat ).

% quotient_empty
tff(fact_1958_quotient__empty,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : equiv_quotient_int(bot_bot_set_int,R2) = bot_bot_set_set_int ).

% quotient_empty
tff(fact_1959_quotient__empty,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o] : equiv_6536735752891577142_nat_o(bot_bo8632255151191677115_nat_o,R2) = bot_bo4069457504818663451_nat_o ).

% quotient_empty
tff(fact_1960_quotient__is__empty,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( ( equiv_5172566880711946708t_unit(A,R2) = bot_bo6284674557190688313t_unit )
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% quotient_is_empty
tff(fact_1961_quotient__is__empty,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o] :
      ( ( equiv_quotient_o(A,R2) = bot_bot_set_set_o )
    <=> ( A = bot_bot_set_o ) ) ).

% quotient_is_empty
tff(fact_1962_quotient__is__empty,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( equiv_quotient_nat(A,R2) = bot_bot_set_set_nat )
    <=> ( A = bot_bot_set_nat ) ) ).

% quotient_is_empty
tff(fact_1963_quotient__is__empty,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( ( equiv_quotient_int(A,R2) = bot_bot_set_set_int )
    <=> ( A = bot_bot_set_int ) ) ).

% quotient_is_empty
tff(fact_1964_quotient__is__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o] :
      ( ( equiv_6536735752891577142_nat_o(A,R2) = bot_bo4069457504818663451_nat_o )
    <=> ( A = bot_bo8632255151191677115_nat_o ) ) ).

% quotient_is_empty
tff(fact_1965_quotient__is__empty2,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( ( bot_bo6284674557190688313t_unit = equiv_5172566880711946708t_unit(A,R2) )
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% quotient_is_empty2
tff(fact_1966_quotient__is__empty2,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o] :
      ( ( bot_bot_set_set_o = equiv_quotient_o(A,R2) )
    <=> ( A = bot_bot_set_o ) ) ).

% quotient_is_empty2
tff(fact_1967_quotient__is__empty2,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bot_set_set_nat = equiv_quotient_nat(A,R2) )
    <=> ( A = bot_bot_set_nat ) ) ).

% quotient_is_empty2
tff(fact_1968_quotient__is__empty2,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( ( bot_bot_set_set_int = equiv_quotient_int(A,R2) )
    <=> ( A = bot_bot_set_int ) ) ).

% quotient_is_empty2
tff(fact_1969_quotient__is__empty2,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o] :
      ( ( bot_bo4069457504818663451_nat_o = equiv_6536735752891577142_nat_o(A,R2) )
    <=> ( A = bot_bo8632255151191677115_nat_o ) ) ).

% quotient_is_empty2
tff(fact_1970_disjnt__self__iff__empty,axiom,
    ! [S: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(disjnt_Product_unit(S),S)
    <=> ( S = bot_bo3957492148770167129t_unit ) ) ).

% disjnt_self_iff_empty
tff(fact_1971_disjnt__self__iff__empty,axiom,
    ! [S: set_o] :
      ( aa_set_o_o(disjnt_o(S),S)
    <=> ( S = bot_bot_set_o ) ) ).

% disjnt_self_iff_empty
tff(fact_1972_disjnt__self__iff__empty,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,S),S)
    <=> ( S = bot_bot_set_nat ) ) ).

% disjnt_self_iff_empty
tff(fact_1973_disjnt__self__iff__empty,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(disjnt_int(S),S)
    <=> ( S = bot_bot_set_int ) ) ).

% disjnt_self_iff_empty
tff(fact_1974_disjnt__self__iff__empty,axiom,
    ! [S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(S),S)
    <=> ( S = bot_bo8632255151191677115_nat_o ) ) ).

% disjnt_self_iff_empty
tff(fact_1975_disjnt__insert1,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,X5: set_fu3555266933746557007_nat_o,Y5: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),X5)),Y5)
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),Y5)
        & aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(X5),Y5) ) ) ).

% disjnt_insert1
tff(fact_1976_disjnt__insert1,axiom,
    ! [A2: nat,X5: set_nat,Y5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),X5)),Y5)
    <=> ( ~ aa_set_nat_o(member_nat(A2),Y5)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,X5),Y5) ) ) ).

% disjnt_insert1
tff(fact_1977_disjnt__insert1,axiom,
    ! [A2: int,X5: set_int,Y5: set_int] :
      ( aa_set_int_o(disjnt_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),X5)),Y5)
    <=> ( ~ aa_set_int_o(member_int(A2),Y5)
        & aa_set_int_o(disjnt_int(X5),Y5) ) ) ).

% disjnt_insert1
tff(fact_1978_disjnt__insert1,axiom,
    ! [A2: $o,X5: set_o,Y5: set_o] :
      ( aa_set_o_o(disjnt_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),X5)),Y5)
    <=> ( ~ aa_set_o_o(member_o((A2)),Y5)
        & aa_set_o_o(disjnt_o(X5),Y5) ) ) ).

% disjnt_insert1
tff(fact_1979_disjnt__insert1,axiom,
    ! [A2: product_unit,X5: set_Product_unit,Y5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(disjnt_Product_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),X5)),Y5)
    <=> ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),Y5)
        & aa_set8780210009671629607unit_o(disjnt_Product_unit(X5),Y5) ) ) ).

% disjnt_insert1
tff(fact_1980_disjnt__insert2,axiom,
    ! [Y5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o,X5: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(Y5),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),X5))
    <=> ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),Y5)
        & aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(Y5),X5) ) ) ).

% disjnt_insert2
tff(fact_1981_disjnt__insert2,axiom,
    ! [Y5: set_nat,A2: nat,X5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,Y5),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),X5))
    <=> ( ~ aa_set_nat_o(member_nat(A2),Y5)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,Y5),X5) ) ) ).

% disjnt_insert2
tff(fact_1982_disjnt__insert2,axiom,
    ! [Y5: set_int,A2: int,X5: set_int] :
      ( aa_set_int_o(disjnt_int(Y5),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),X5))
    <=> ( ~ aa_set_int_o(member_int(A2),Y5)
        & aa_set_int_o(disjnt_int(Y5),X5) ) ) ).

% disjnt_insert2
tff(fact_1983_disjnt__insert2,axiom,
    ! [Y5: set_o,A2: $o,X5: set_o] :
      ( aa_set_o_o(disjnt_o(Y5),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),X5))
    <=> ( ~ aa_set_o_o(member_o((A2)),Y5)
        & aa_set_o_o(disjnt_o(Y5),X5) ) ) ).

% disjnt_insert2
tff(fact_1984_disjnt__insert2,axiom,
    ! [Y5: set_Product_unit,A2: product_unit,X5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(disjnt_Product_unit(Y5),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),X5))
    <=> ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),Y5)
        & aa_set8780210009671629607unit_o(disjnt_Product_unit(Y5),X5) ) ) ).

% disjnt_insert2
tff(fact_1985_rel__restrict__empty,axiom,
    ! [R: set_Pr5094982260447487303t_unit] : rel_re9207996470947987439t_unit(R,bot_bo3957492148770167129t_unit) = R ).

% rel_restrict_empty
tff(fact_1986_rel__restrict__empty,axiom,
    ! [R: set_Product_prod_o_o] : rel_restrict_o(R,bot_bot_set_o) = R ).

% rel_restrict_empty
tff(fact_1987_rel__restrict__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : rel_restrict_nat(R,bot_bot_set_nat) = R ).

% rel_restrict_empty
tff(fact_1988_rel__restrict__empty,axiom,
    ! [R: set_Pr958786334691620121nt_int] : rel_restrict_int(R,bot_bot_set_int) = R ).

% rel_restrict_empty
tff(fact_1989_rel__restrict__empty,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] : rel_re4406848170174997969_nat_o(R,bot_bo8632255151191677115_nat_o) = R ).

% rel_restrict_empty
tff(fact_1990_disjnt__Un1,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)),C5)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),C5)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,B4),C5) ) ) ).

% disjnt_Un1
tff(fact_1991_disjnt__Un2,axiom,
    ! [C5: set_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,C5),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,C5),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,C5),B4) ) ) ).

% disjnt_Un2
tff(fact_1992_Sup__fin_Osingleton,axiom,
    ! [X: product_unit] : lattic5294303975357428420t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% Sup_fin.singleton
tff(fact_1993_Sup__fin_Osingleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% Sup_fin.singleton
tff(fact_1994_Sup__fin_Osingleton,axiom,
    ! [X: nat] : aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% Sup_fin.singleton
tff(fact_1995_Sup__fin_Osingleton,axiom,
    ! [X: int] : lattic1091506334969745077in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% Sup_fin.singleton
tff(fact_1996_Sup__fin_Osingleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : lattic7388253182610234406_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X ).

% Sup_fin.singleton
tff(fact_1997_Inf__fin_Osingleton,axiom,
    ! [X: product_unit] : lattic1263872656861969706t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% Inf_fin.singleton
tff(fact_1998_Inf__fin_Osingleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% Inf_fin.singleton
tff(fact_1999_Inf__fin_Osingleton,axiom,
    ! [X: nat] : aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% Inf_fin.singleton
tff(fact_2000_Inf__fin_Osingleton,axiom,
    ! [X: int] : lattic5235898064620869839in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% Inf_fin.singleton
tff(fact_2001_Inf__fin_Osingleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : lattic6089491107740071820_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X ).

% Inf_fin.singleton
tff(fact_2002_disjnt__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(A),B4)
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ~ ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
            & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),B4) ) ) ).

% disjnt_iff
tff(fact_2003_disjnt__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4)
    <=> ! [X2: nat] :
          ~ ( aa_set_nat_o(member_nat(X2),A)
            & aa_set_nat_o(member_nat(X2),B4) ) ) ).

% disjnt_iff
tff(fact_2004_disjnt__iff,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(disjnt_int(A),B4)
    <=> ! [X2: int] :
          ~ ( aa_set_int_o(member_int(X2),A)
            & aa_set_int_o(member_int(X2),B4) ) ) ).

% disjnt_iff
tff(fact_2005_disjnt__iff,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(disjnt_o(A),B4)
    <=> ! [X2: $o] :
          ~ ( aa_set_o_o(member_o((X2)),A)
            & aa_set_o_o(member_o((X2)),B4) ) ) ).

% disjnt_iff
tff(fact_2006_disjnt__iff,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(disjnt_Product_unit(A),B4)
    <=> ! [X2: product_unit] :
          ~ ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
            & aa_set8780210009671629607unit_o(member_Product_unit(X2),B4) ) ) ).

% disjnt_iff
tff(fact_2007_IdE,axiom,
    ! [P2: product_prod_nat_nat] :
      ( member8440522571783428010at_nat(P2,id_nat)
     => ~ ! [X3: nat] : P2 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3) ) ).

% IdE
tff(fact_2008_IdE,axiom,
    ! [P2: product_prod_int_int] :
      ( member5262025264175285858nt_int(P2,id_int)
     => ~ ! [X3: int] : P2 != aa_int7462565939943104719nt_int(product_Pair_int_int(X3),X3) ) ).

% IdE
tff(fact_2009_disjnt__empty1,axiom,
    ! [A: set_Product_unit] : aa_set8780210009671629607unit_o(disjnt_Product_unit(bot_bo3957492148770167129t_unit),A) ).

% disjnt_empty1
tff(fact_2010_disjnt__empty1,axiom,
    ! [A: set_o] : aa_set_o_o(disjnt_o(bot_bot_set_o),A) ).

% disjnt_empty1
tff(fact_2011_disjnt__empty1,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,bot_bot_set_nat),A) ).

% disjnt_empty1
tff(fact_2012_disjnt__empty1,axiom,
    ! [A: set_int] : aa_set_int_o(disjnt_int(bot_bot_set_int),A) ).

% disjnt_empty1
tff(fact_2013_disjnt__empty1,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(bot_bo8632255151191677115_nat_o),A) ).

% disjnt_empty1
tff(fact_2014_disjnt__empty2,axiom,
    ! [A: set_Product_unit] : aa_set8780210009671629607unit_o(disjnt_Product_unit(A),bot_bo3957492148770167129t_unit) ).

% disjnt_empty2
tff(fact_2015_disjnt__empty2,axiom,
    ! [A: set_o] : aa_set_o_o(disjnt_o(A),bot_bot_set_o) ).

% disjnt_empty2
tff(fact_2016_disjnt__empty2,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),bot_bot_set_nat) ).

% disjnt_empty2
tff(fact_2017_disjnt__empty2,axiom,
    ! [A: set_int] : aa_set_int_o(disjnt_int(A),bot_bot_set_int) ).

% disjnt_empty2
tff(fact_2018_disjnt__empty2,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(A),bot_bo8632255151191677115_nat_o) ).

% disjnt_empty2
tff(fact_2019_disjnt__subset1,axiom,
    ! [X5: set_nat,Y5: set_nat,Z4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,X5),Y5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),X5)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,Z4),Y5) ) ) ).

% disjnt_subset1
tff(fact_2020_disjnt__subset2,axiom,
    ! [X5: set_nat,Y5: set_nat,Z4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,X5),Y5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z4),Y5)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,X5),Z4) ) ) ).

% disjnt_subset2
tff(fact_2021_disjnt__insert,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,N: set_fu3555266933746557007_nat_o,M: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),N)
     => ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(M),N)
       => aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),M)),N) ) ) ).

% disjnt_insert
tff(fact_2022_disjnt__insert,axiom,
    ! [X: nat,N: set_nat,M: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),N)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,M),N)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),M)),N) ) ) ).

% disjnt_insert
tff(fact_2023_disjnt__insert,axiom,
    ! [X: int,N: set_int,M: set_int] :
      ( ~ aa_set_int_o(member_int(X),N)
     => ( aa_set_int_o(disjnt_int(M),N)
       => aa_set_int_o(disjnt_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),M)),N) ) ) ).

% disjnt_insert
tff(fact_2024_disjnt__insert,axiom,
    ! [X: $o,N: set_o,M: set_o] :
      ( ~ aa_set_o_o(member_o((X)),N)
     => ( aa_set_o_o(disjnt_o(M),N)
       => aa_set_o_o(disjnt_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),M)),N) ) ) ).

% disjnt_insert
tff(fact_2025_disjnt__insert,axiom,
    ! [X: product_unit,N: set_Product_unit,M: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),N)
     => ( aa_set8780210009671629607unit_o(disjnt_Product_unit(M),N)
       => aa_set8780210009671629607unit_o(disjnt_Product_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),M)),N) ) ) ).

% disjnt_insert
tff(fact_2026_rel__restrict__notR_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),rel_re4406848170174997969_nat_o(A,R))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_2027_rel__restrict__notR_I2_J,axiom,
    ! [X: $o,Y: $o,A: set_Product_prod_o_o,R: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),rel_restrict_o(A,R))
     => ~ aa_set_o_o(member_o((Y)),R) ) ).

% rel_restrict_notR(2)
tff(fact_2028_rel__restrict__notR_I2_J,axiom,
    ! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),rel_re9207996470947987439t_unit(A,R))
     => ~ aa_set8780210009671629607unit_o(member_Product_unit(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_2029_rel__restrict__notR_I2_J,axiom,
    ! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),rel_restrict_nat(A,R))
     => ~ aa_set_nat_o(member_nat(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_2030_rel__restrict__notR_I2_J,axiom,
    ! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),rel_restrict_int(A,R))
     => ~ aa_set_int_o(member_int(Y),R) ) ).

% rel_restrict_notR(2)
tff(fact_2031_rel__restrict__notR_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),rel_re4406848170174997969_nat_o(A,R))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_2032_rel__restrict__notR_I1_J,axiom,
    ! [X: $o,Y: $o,A: set_Product_prod_o_o,R: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),rel_restrict_o(A,R))
     => ~ aa_set_o_o(member_o((X)),R) ) ).

% rel_restrict_notR(1)
tff(fact_2033_rel__restrict__notR_I1_J,axiom,
    ! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),rel_re9207996470947987439t_unit(A,R))
     => ~ aa_set8780210009671629607unit_o(member_Product_unit(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_2034_rel__restrict__notR_I1_J,axiom,
    ! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),rel_restrict_nat(A,R))
     => ~ aa_set_nat_o(member_nat(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_2035_rel__restrict__notR_I1_J,axiom,
    ! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),rel_restrict_int(A,R))
     => ~ aa_set_int_o(member_int(X),R) ) ).

% rel_restrict_notR(1)
tff(fact_2036_rel__restrictI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,R: set_fu3555266933746557007_nat_o,Y: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),R)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),R)
       => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),E)
         => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),rel_re4406848170174997969_nat_o(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_2037_rel__restrictI,axiom,
    ! [X: $o,R: set_o,Y: $o,E: set_Product_prod_o_o] :
      ( ~ aa_set_o_o(member_o((X)),R)
     => ( ~ aa_set_o_o(member_o((Y)),R)
       => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),E)
         => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),rel_restrict_o(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_2038_rel__restrictI,axiom,
    ! [X: product_unit,R: set_Product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),R)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(Y),R)
       => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),E)
         => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),rel_re9207996470947987439t_unit(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_2039_rel__restrictI,axiom,
    ! [X: nat,R: set_nat,Y: nat,E: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set_nat_o(member_nat(X),R)
     => ( ~ aa_set_nat_o(member_nat(Y),R)
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),E)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),rel_restrict_nat(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_2040_rel__restrictI,axiom,
    ! [X: int,R: set_int,Y: int,E: set_Pr958786334691620121nt_int] :
      ( ~ aa_set_int_o(member_int(X),R)
     => ( ~ aa_set_int_o(member_int(Y),R)
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),E)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),rel_restrict_int(E,R)) ) ) ) ).

% rel_restrictI
tff(fact_2041_rel__restrict__lift,axiom,
    ! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),rel_restrict_nat(E,R))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),E) ) ).

% rel_restrict_lift
tff(fact_2042_rel__restrict__lift,axiom,
    ! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),rel_restrict_int(E,R))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),E) ) ).

% rel_restrict_lift
tff(fact_2043_rel__restrict__union,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] : rel_restrict_nat(R,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = rel_restrict_nat(rel_restrict_nat(R,A),B4) ).

% rel_restrict_union
tff(fact_2044_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1263872656861969706t_unit(A)),lattic5294303975357428420t_unit(A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2045_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,A)),aa_set_o_o(lattic1508158080041050831_fin_o,A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2046_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic6089491107740071820_nat_o(A)),lattic7388253182610234406_nat_o(A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2047_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,lattic9128708996678779395er_nat(A)),lattic5930898082463196905er_nat(A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2048_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(A)),lattic3835124923745554447et_nat(A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2049_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic4603258475043424379in_rat(A)),lattic458866745392299617in_rat(A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2050_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,A)),aa_set_nat_nat(lattic1093996805478795353in_nat,A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2051_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(A)),lattic1091506334969745077in_int(A)) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_2052_disjnt__Sigma__iff,axiom,
    ! [A: set_nat,C5: fun_na896991816813928310t_unit,B4: set_nat] :
      ( disjnt595863415355979947t_unit(produc5506539147108004339t_unit(A,C5),produc5506539147108004339t_unit(B4,C5))
    <=> ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
           => ( aa_nat2714212862977473631t_unit(C5,X2) = bot_bo3957492148770167129t_unit ) )
        | aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4) ) ) ).

% disjnt_Sigma_iff
tff(fact_2053_disjnt__Sigma__iff,axiom,
    ! [A: set_nat,C5: fun_nat_set_o,B4: set_nat] :
      ( disjnt5652777605541633334_nat_o(product_Sigma_nat_o(A,C5),product_Sigma_nat_o(B4,C5))
    <=> ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
           => ( aa_nat_set_o(C5,X2) = bot_bot_set_o ) )
        | aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4) ) ) ).

% disjnt_Sigma_iff
tff(fact_2054_disjnt__Sigma__iff,axiom,
    ! [A: set_nat,C5: fun_nat_set_nat,B4: set_nat] :
      ( disjnt2713546992526078024at_nat(produc457027306803732586at_nat(A,C5),produc457027306803732586at_nat(B4,C5))
    <=> ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
           => ( aa_nat_set_nat(C5,X2) = bot_bot_set_nat ) )
        | aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4) ) ) ).

% disjnt_Sigma_iff
tff(fact_2055_disjnt__Sigma__iff,axiom,
    ! [A: set_nat,C5: fun_nat_set_int,B4: set_nat] :
      ( disjnt7759068009871657124at_int(produc454536836294682310at_int(A,C5),produc454536836294682310at_int(B4,C5))
    <=> ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
           => ( aa_nat_set_int(C5,X2) = bot_bot_set_int ) )
        | aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4) ) ) ).

% disjnt_Sigma_iff
tff(fact_2056_disjnt__Sigma__iff,axiom,
    ! [A: set_nat,C5: fun_na5933469815046955224_nat_o,B4: set_nat] :
      ( disjnt8382938109279558413_nat_o(produc3779179696562253269_nat_o(A,C5),produc3779179696562253269_nat_o(B4,C5))
    <=> ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))
           => ( aa_nat7189845332094958785_nat_o(C5,X2) = bot_bo8632255151191677115_nat_o ) )
        | aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4) ) ) ).

% disjnt_Sigma_iff
tff(fact_2057_disjnt__def,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(disjnt_Product_unit(A),B4)
    <=> ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit ) ) ).

% disjnt_def
tff(fact_2058_disjnt__def,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(disjnt_o(A),B4)
    <=> ( inf_inf_set_o(A,B4) = bot_bot_set_o ) ) ).

% disjnt_def
tff(fact_2059_disjnt__def,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(disjnt_nat,A),B4)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat ) ) ).

% disjnt_def
tff(fact_2060_disjnt__def,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(disjnt_int(A),B4)
    <=> ( inf_inf_set_int(A,B4) = bot_bot_set_int ) ) ).

% disjnt_def
tff(fact_2061_disjnt__def,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(disjnt3987968171041746742_nat_o(A),B4)
    <=> ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o ) ) ).

% disjnt_def
tff(fact_2062_quotient__eqI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X5: set_fu3555266933746557007_nat_o,Y5: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
       => ( member361801372707117048_nat_o(Y5,equiv_6536735752891577142_nat_o(A,R2))
         => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
           => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),Y5)
             => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2)
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_2063_quotient__eqI,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X5: set_o,Y5: set_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member_set_o(X5,equiv_quotient_o(A,R2))
       => ( member_set_o(Y5,equiv_quotient_o(A,R2))
         => ( aa_set_o_o(member_o((X)),X5)
           => ( aa_set_o_o(member_o((Y)),Y5)
             => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2)
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_2064_quotient__eqI,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X5: set_Product_unit,Y5: set_Product_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
       => ( member5877623283571906838t_unit(Y5,equiv_5172566880711946708t_unit(A,R2))
         => ( aa_set8780210009671629607unit_o(member_Product_unit(X),X5)
           => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),Y5)
             => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2)
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_2065_quotient__eqI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X5: set_nat,Y5: set_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member_set_nat(X5,equiv_quotient_nat(A,R2))
       => ( member_set_nat(Y5,equiv_quotient_nat(A,R2))
         => ( aa_set_nat_o(member_nat(X),X5)
           => ( aa_set_nat_o(member_nat(Y),Y5)
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_2066_quotient__eqI,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X5: set_int,Y5: set_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( member_set_int(X5,equiv_quotient_int(A,R2))
       => ( member_set_int(Y5,equiv_quotient_int(A,R2))
         => ( aa_set_int_o(member_int(X),X5)
           => ( aa_set_int_o(member_int(Y),Y5)
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_2067_quotient__eq__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X5: set_fu3555266933746557007_nat_o,Y5: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
       => ( member361801372707117048_nat_o(Y5,equiv_6536735752891577142_nat_o(A,R2))
         => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
           => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),Y5)
             => ( ( X5 = Y5 )
              <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_2068_quotient__eq__iff,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X5: set_o,Y5: set_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member_set_o(X5,equiv_quotient_o(A,R2))
       => ( member_set_o(Y5,equiv_quotient_o(A,R2))
         => ( aa_set_o_o(member_o((X)),X5)
           => ( aa_set_o_o(member_o((Y)),Y5)
             => ( ( X5 = Y5 )
              <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_2069_quotient__eq__iff,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X5: set_Product_unit,Y5: set_Product_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
       => ( member5877623283571906838t_unit(Y5,equiv_5172566880711946708t_unit(A,R2))
         => ( aa_set8780210009671629607unit_o(member_Product_unit(X),X5)
           => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),Y5)
             => ( ( X5 = Y5 )
              <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_2070_quotient__eq__iff,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X5: set_nat,Y5: set_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member_set_nat(X5,equiv_quotient_nat(A,R2))
       => ( member_set_nat(Y5,equiv_quotient_nat(A,R2))
         => ( aa_set_nat_o(member_nat(X),X5)
           => ( aa_set_nat_o(member_nat(Y),Y5)
             => ( ( X5 = Y5 )
              <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_2071_quotient__eq__iff,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X5: set_int,Y5: set_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( member_set_int(X5,equiv_quotient_int(A,R2))
       => ( member_set_int(Y5,equiv_quotient_int(A,R2))
         => ( aa_set_int_o(member_int(X),X5)
           => ( aa_set_int_o(member_int(Y),Y5)
             => ( ( X5 = Y5 )
              <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_2072_in__quotient__imp__closed,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X5: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
         => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2)
           => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_2073_in__quotient__imp__closed,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X5: set_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( member_set_o(X5,equiv_quotient_o(A,R2))
       => ( aa_set_o_o(member_o((X)),X5)
         => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2)
           => aa_set_o_o(member_o((Y)),X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_2074_in__quotient__imp__closed,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X5: set_Product_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(X),X5)
         => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2)
           => aa_set8780210009671629607unit_o(member_Product_unit(Y),X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_2075_in__quotient__imp__closed,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X5: set_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member_set_nat(X5,equiv_quotient_nat(A,R2))
       => ( aa_set_nat_o(member_nat(X),X5)
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
           => aa_set_nat_o(member_nat(Y),X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_2076_in__quotient__imp__closed,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X5: set_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( member_set_int(X5,equiv_quotient_int(A,R2))
       => ( aa_set_int_o(member_int(X),X5)
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
           => aa_set_int_o(member_int(Y),X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_2077_in__quotient__imp__non__empty,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X5: set_Product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
       => ( X5 != bot_bo3957492148770167129t_unit ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_2078_in__quotient__imp__non__empty,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X5: set_o] :
      ( equiv_equiv_o(A,R2)
     => ( member_set_o(X5,equiv_quotient_o(A,R2))
       => ( X5 != bot_bot_set_o ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_2079_in__quotient__imp__non__empty,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X5: set_nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member_set_nat(X5,equiv_quotient_nat(A,R2))
       => ( X5 != bot_bot_set_nat ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_2080_in__quotient__imp__non__empty,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X5: set_int] :
      ( equiv_equiv_int(A,R2)
     => ( member_set_int(X5,equiv_quotient_int(A,R2))
       => ( X5 != bot_bot_set_int ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_2081_in__quotient__imp__non__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X5: set_fu3555266933746557007_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
       => ( X5 != bot_bo8632255151191677115_nat_o ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_2082_rel__restrict__trancl__notR_I2_J,axiom,
    ! [V: fun_Pr6460944699146269935_nat_o,W: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(V,W),transi8445014155877283344_nat_o(rel_re4406848170174997969_nat_o(E,R)))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(W),R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_2083_rel__restrict__trancl__notR_I2_J,axiom,
    ! [V: $o,W: $o,E: set_Product_prod_o_o,R: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((V)),(W)),transitive_trancl_o(rel_restrict_o(E,R)))
     => ~ aa_set_o_o(member_o((W)),R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_2084_rel__restrict__trancl__notR_I2_J,axiom,
    ! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(V),W),transi430408012122777006t_unit(rel_re9207996470947987439t_unit(E,R)))
     => ~ aa_set8780210009671629607unit_o(member_Product_unit(W),R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_2085_rel__restrict__trancl__notR_I2_J,axiom,
    ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V),W),transi6264000038957366511cl_nat(rel_restrict_nat(E,R)))
     => ~ aa_set_nat_o(member_nat(W),R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_2086_rel__restrict__trancl__notR_I2_J,axiom,
    ! [V: int,W: int,E: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V),W),transi6261509568448316235cl_int(rel_restrict_int(E,R)))
     => ~ aa_set_int_o(member_int(W),R) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_2087_rel__restrict__trancl__notR_I1_J,axiom,
    ! [V: fun_Pr6460944699146269935_nat_o,W: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(V,W),transi8445014155877283344_nat_o(rel_re4406848170174997969_nat_o(E,R)))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(V),R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_2088_rel__restrict__trancl__notR_I1_J,axiom,
    ! [V: $o,W: $o,E: set_Product_prod_o_o,R: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((V)),(W)),transitive_trancl_o(rel_restrict_o(E,R)))
     => ~ aa_set_o_o(member_o((V)),R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_2089_rel__restrict__trancl__notR_I1_J,axiom,
    ! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(V),W),transi430408012122777006t_unit(rel_re9207996470947987439t_unit(E,R)))
     => ~ aa_set8780210009671629607unit_o(member_Product_unit(V),R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_2090_rel__restrict__trancl__notR_I1_J,axiom,
    ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(V),W),transi6264000038957366511cl_nat(rel_restrict_nat(E,R)))
     => ~ aa_set_nat_o(member_nat(V),R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_2091_rel__restrict__trancl__notR_I1_J,axiom,
    ! [V: int,W: int,E: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(V),W),transi6261509568448316235cl_int(rel_restrict_int(E,R)))
     => ~ aa_set_int_o(member_int(V),R) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_2092_rel__restrict__trancl__mem,axiom,
    ! [A2: nat,B: nat,A: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi6264000038957366511cl_nat(rel_restrict_nat(A,R)))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),rel_restrict_nat(transi6264000038957366511cl_nat(A),R)) ) ).

% rel_restrict_trancl_mem
tff(fact_2093_rel__restrict__trancl__mem,axiom,
    ! [A2: int,B: int,A: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi6261509568448316235cl_int(rel_restrict_int(A,R)))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),rel_restrict_int(transi6261509568448316235cl_int(A),R)) ) ).

% rel_restrict_trancl_mem
tff(fact_2094_rel__restrict__mono2,axiom,
    ! [R: set_nat,S: set_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,R),S)
     => ord_le3146513528884898305at_nat(rel_restrict_nat(A,S),rel_restrict_nat(A,R)) ) ).

% rel_restrict_mono2
tff(fact_2095_singleton__quotient,axiom,
    ! [X: product_unit,R2: set_Pr5094982260447487303t_unit] : equiv_5172566880711946708t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit),R2) = insert8986028188087627965t_unit(image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)),bot_bo6284674557190688313t_unit) ).

% singleton_quotient
tff(fact_2096_singleton__quotient,axiom,
    ! [X: $o,R2: set_Product_prod_o_o] : equiv_quotient_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o),R2) = insert_set_o(image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)),bot_bot_set_set_o) ).

% singleton_quotient
tff(fact_2097_singleton__quotient,axiom,
    ! [X: nat,R2: set_Pr1261947904930325089at_nat] : equiv_quotient_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat),R2) = insert_set_nat(aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)),bot_bot_set_set_nat) ).

% singleton_quotient
tff(fact_2098_singleton__quotient,axiom,
    ! [X: int,R2: set_Pr958786334691620121nt_int] : equiv_quotient_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int),R2) = insert_set_int(image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)),bot_bot_set_set_int) ).

% singleton_quotient
tff(fact_2099_singleton__quotient,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,R2: set_Pr2890355626087285447_nat_o] : equiv_6536735752891577142_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o),R2) = insert755438361227728415_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)),bot_bo4069457504818663451_nat_o) ).

% singleton_quotient
tff(fact_2100_quotientI,axiom,
    ! [X: product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => member5877623283571906838t_unit(image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)),equiv_5172566880711946708t_unit(A,R2)) ) ).

% quotientI
tff(fact_2101_quotientI,axiom,
    ! [X: $o,A: set_o,R2: set_Product_prod_o_o] :
      ( aa_set_o_o(member_o((X)),A)
     => member_set_o(image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)),equiv_quotient_o(A,R2)) ) ).

% quotientI
tff(fact_2102_quotientI,axiom,
    ! [X: nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => member_set_nat(aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)),equiv_quotient_nat(A,R2)) ) ).

% quotientI
tff(fact_2103_quotientI,axiom,
    ! [X: int,A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(X),A)
     => member_set_int(image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)),equiv_quotient_int(A,R2)) ) ).

% quotientI
tff(fact_2104_quotientI,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => member361801372707117048_nat_o(image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)),equiv_6536735752891577142_nat_o(A,R2)) ) ).

% quotientI
tff(fact_2105_quotientE,axiom,
    ! [X5: set_Product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
     => ~ ! [X3: product_unit] :
            ( ( X5 = image_8431784415008483091t_unit(R2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit)) )
           => ~ aa_set8780210009671629607unit_o(member_Product_unit(X3),A) ) ) ).

% quotientE
tff(fact_2106_quotientE,axiom,
    ! [X5: set_o,A: set_o,R2: set_Product_prod_o_o] :
      ( member_set_o(X5,equiv_quotient_o(A,R2))
     => ~ ! [X3: $o] :
            ( ( X5 = image_o_o(R2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o)) )
           => ~ aa_set_o_o(member_o((X3)),A) ) ) ).

% quotientE
tff(fact_2107_quotientE,axiom,
    ! [X5: set_nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member_set_nat(X5,equiv_quotient_nat(A,R2))
     => ~ ! [X3: nat] :
            ( ( X5 = aa_set_nat_set_nat(image_nat_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat)) )
           => ~ aa_set_nat_o(member_nat(X3),A) ) ) ).

% quotientE
tff(fact_2108_quotientE,axiom,
    ! [X5: set_int,A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( member_set_int(X5,equiv_quotient_int(A,R2))
     => ~ ! [X3: int] :
            ( ( X5 = image_int_int(R2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int)) )
           => ~ aa_set_int_o(member_int(X3),A) ) ) ).

% quotientE
tff(fact_2109_quotientE,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o] :
      ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( ( X5 = image_2188787866847195539_nat_o(R2,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),bot_bo8632255151191677115_nat_o)) )
           => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A) ) ) ).

% quotientE
tff(fact_2110_Inf__fin_OboundedE,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic1263872656861969706t_unit(A))
         => ! [A10: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A10),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2111_Inf__fin_OboundedE,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A))
         => ! [A10: $o] :
              ( aa_set_o_o(member_o((A10)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A10)) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2112_Inf__fin_OboundedE,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(A))
         => ! [A10: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A10),A)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2113_Inf__fin_OboundedE,axiom,
    ! [A: set_filter_nat,X: filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),lattic9128708996678779395er_nat(A))
         => ! [A10: filter_nat] :
              ( member_filter_nat(A10,A)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2114_Inf__fin_OboundedE,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),lattic3014633134055518761et_nat(A))
         => ! [A10: set_nat] :
              ( member_set_nat(A10,A)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2115_Inf__fin_OboundedE,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic4603258475043424379in_rat(A))
         => ! [A10: rat] :
              ( aa_set_rat_o(member_rat(A10),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2116_Inf__fin_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A))
         => ! [A10: nat] :
              ( aa_set_nat_o(member_nat(A10),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2117_Inf__fin_OboundedE,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(A))
         => ! [A10: int] :
              ( aa_set_int_o(member_int(A10),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A10) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_2118_Inf__fin_OboundedI,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A3) )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic1263872656861969706t_unit(A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2119_Inf__fin_OboundedI,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A3)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2120_Inf__fin_OboundedI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),A3) )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2121_Inf__fin_OboundedI,axiom,
    ! [A: set_filter_nat,X: filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( ! [A3: filter_nat] :
              ( member_filter_nat(A3,A)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),A3) )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),lattic9128708996678779395er_nat(A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2122_Inf__fin_OboundedI,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( member_set_nat(A3,A)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A3) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),lattic3014633134055518761et_nat(A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2123_Inf__fin_OboundedI,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A3) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic4603258475043424379in_rat(A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2124_Inf__fin_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A3) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2125_Inf__fin_OboundedI,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A3) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(A)) ) ) ) ).

% Inf_fin.boundedI
tff(fact_2126_Sup__fin_OboundedE,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic5294303975357428420t_unit(A)),X)
         => ! [A10: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A10),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2127_Sup__fin_OboundedE,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A)),(X))
         => ! [A10: $o] :
              ( aa_set_o_o(member_o((A10)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A10)),(X)) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2128_Sup__fin_OboundedE,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(A)),X)
         => ! [A10: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A10),A)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2129_Sup__fin_OboundedE,axiom,
    ! [A: set_filter_nat,X: filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,lattic5930898082463196905er_nat(A)),X)
         => ! [A10: filter_nat] :
              ( member_filter_nat(A10,A)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2130_Sup__fin_OboundedE,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A)),X)
         => ! [A10: set_nat] :
              ( member_set_nat(A10,A)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2131_Sup__fin_OboundedE,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A)),X)
         => ! [A10: rat] :
              ( aa_set_rat_o(member_rat(A10),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2132_Sup__fin_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic1093996805478795353in_nat,A)),X)
         => ! [A10: nat] :
              ( aa_set_nat_o(member_nat(A10),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2133_Sup__fin_OboundedE,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A)),X)
         => ! [A10: int] :
              ( aa_set_int_o(member_int(A10),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A10),X) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_2134_Sup__fin_OboundedI,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A3),X) )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic5294303975357428420t_unit(A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2135_Sup__fin_OboundedI,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(X)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A)),(X)) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2136_Sup__fin_OboundedI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A3),X) )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2137_Sup__fin_OboundedI,axiom,
    ! [A: set_filter_nat,X: filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( ! [A3: filter_nat] :
              ( member_filter_nat(A3,A)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A3),X) )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,lattic5930898082463196905er_nat(A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2138_Sup__fin_OboundedI,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [A3: set_nat] :
              ( member_set_nat(A3,A)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),X) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2139_Sup__fin_OboundedI,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),X) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2140_Sup__fin_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),X) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic1093996805478795353in_nat,A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2141_Sup__fin_OboundedI,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),X) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A)),X) ) ) ) ).

% Sup_fin.boundedI
tff(fact_2142_Inf__fin_Obounded__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic1263872656861969706t_unit(A))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2143_Inf__fin_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X2)) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2144_Inf__fin_Obounded__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),lattic6089491107740071820_nat_o(A))
        <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2145_Inf__fin_Obounded__iff,axiom,
    ! [A: set_filter_nat,X: filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),lattic9128708996678779395er_nat(A))
        <=> ! [X2: filter_nat] :
              ( member_filter_nat(X2,A)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2146_Inf__fin_Obounded__iff,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),lattic3014633134055518761et_nat(A))
        <=> ! [X2: set_nat] :
              ( member_set_nat(X2,A)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2147_Inf__fin_Obounded__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic4603258475043424379in_rat(A))
        <=> ! [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2148_Inf__fin_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2149_Inf__fin_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic5235898064620869839in_int(A))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X2) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_2150_Sup__fin_Obounded__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic5294303975357428420t_unit(A)),X)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2151_Sup__fin_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A)),(X))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(X)) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2152_Sup__fin_Obounded__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(A)),X)
        <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2153_Sup__fin_Obounded__iff,axiom,
    ! [A: set_filter_nat,X: filter_nat] :
      ( finite2119507909894593271er_nat(A)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,lattic5930898082463196905er_nat(A)),X)
        <=> ! [X2: filter_nat] :
              ( member_filter_nat(X2,A)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2154_Sup__fin_Obounded__iff,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A)),X)
        <=> ! [X2: set_nat] :
              ( member_set_nat(X2,A)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2155_Sup__fin_Obounded__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A)),X)
        <=> ! [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2156_Sup__fin_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic1093996805478795353in_nat,A)),X)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2157_Sup__fin_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A)),X)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),X) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_2158_quotient__disj,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X5: set_Product_unit,Y5: set_Product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( member5877623283571906838t_unit(X5,equiv_5172566880711946708t_unit(A,R2))
       => ( member5877623283571906838t_unit(Y5,equiv_5172566880711946708t_unit(A,R2))
         => ( ( X5 = Y5 )
            | ( inf_in4660618365625256667t_unit(X5,Y5) = bot_bo3957492148770167129t_unit ) ) ) ) ) ).

% quotient_disj
tff(fact_2159_quotient__disj,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X5: set_o,Y5: set_o] :
      ( equiv_equiv_o(A,R2)
     => ( member_set_o(X5,equiv_quotient_o(A,R2))
       => ( member_set_o(Y5,equiv_quotient_o(A,R2))
         => ( ( X5 = Y5 )
            | ( inf_inf_set_o(X5,Y5) = bot_bot_set_o ) ) ) ) ) ).

% quotient_disj
tff(fact_2160_quotient__disj,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X5: set_nat,Y5: set_nat] :
      ( equiv_equiv_nat(A,R2)
     => ( member_set_nat(X5,equiv_quotient_nat(A,R2))
       => ( member_set_nat(Y5,equiv_quotient_nat(A,R2))
         => ( ( X5 = Y5 )
            | ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X5),Y5) = bot_bot_set_nat ) ) ) ) ) ).

% quotient_disj
tff(fact_2161_quotient__disj,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X5: set_int,Y5: set_int] :
      ( equiv_equiv_int(A,R2)
     => ( member_set_int(X5,equiv_quotient_int(A,R2))
       => ( member_set_int(Y5,equiv_quotient_int(A,R2))
         => ( ( X5 = Y5 )
            | ( inf_inf_set_int(X5,Y5) = bot_bot_set_int ) ) ) ) ) ).

% quotient_disj
tff(fact_2162_quotient__disj,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X5: set_fu3555266933746557007_nat_o,Y5: set_fu3555266933746557007_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( member361801372707117048_nat_o(X5,equiv_6536735752891577142_nat_o(A,R2))
       => ( member361801372707117048_nat_o(Y5,equiv_6536735752891577142_nat_o(A,R2))
         => ( ( X5 = Y5 )
            | ( inf_in3376754634429691197_nat_o(X5,Y5) = bot_bo8632255151191677115_nat_o ) ) ) ) ) ).

% quotient_disj
tff(fact_2163_rel__restrict__Int__empty,axiom,
    ! [A: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
      ( ( inf_in4660618365625256667t_unit(A,field_Product_unit(R)) = bot_bo3957492148770167129t_unit )
     => ( rel_re9207996470947987439t_unit(R,A) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_2164_rel__restrict__Int__empty,axiom,
    ! [A: set_o,R: set_Product_prod_o_o] :
      ( ( inf_inf_set_o(A,field_o(R)) = bot_bot_set_o )
     => ( rel_restrict_o(R,A) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_2165_rel__restrict__Int__empty,axiom,
    ! [A: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),field_nat(R)) = bot_bot_set_nat )
     => ( rel_restrict_nat(R,A) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_2166_rel__restrict__Int__empty,axiom,
    ! [A: set_int,R: set_Pr958786334691620121nt_int] :
      ( ( inf_inf_set_int(A,field_int(R)) = bot_bot_set_int )
     => ( rel_restrict_int(R,A) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_2167_rel__restrict__Int__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A,field_48612661033931452_nat_o(R)) = bot_bo8632255151191677115_nat_o )
     => ( rel_re4406848170174997969_nat_o(R,A) = R ) ) ).

% rel_restrict_Int_empty
tff(fact_2168_eq__equiv__class__iff2,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
      ( equiv_4189356295871601603t_unit(A,R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),A)
         => ( ( equiv_5172566880711946708t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit),R2) = equiv_5172566880711946708t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit),R2) )
          <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_2169_eq__equiv__class__iff2,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
      ( equiv_equiv_o(A,R2)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(member_o((Y)),A)
         => ( ( equiv_quotient_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o),R2) = equiv_quotient_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o),R2) )
          <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_2170_eq__equiv__class__iff2,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( equiv_equiv_nat(A,R2)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_o(member_nat(Y),A)
         => ( ( equiv_quotient_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat),R2) = equiv_quotient_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat),R2) )
          <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_2171_eq__equiv__class__iff2,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( equiv_equiv_int(A,R2)
     => ( aa_set_int_o(member_int(X),A)
       => ( aa_set_int_o(member_int(Y),A)
         => ( ( equiv_quotient_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int),R2) = equiv_quotient_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int),R2) )
          <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_2172_eq__equiv__class__iff2,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( equiv_5581975019310583717_nat_o(A,R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
         => ( ( equiv_6536735752891577142_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o),R2) = equiv_6536735752891577142_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o),R2) )
          <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X,Y),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_2173_Inf__fin_Osubset__imp,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1263872656861969706t_unit(B4)),lattic1263872656861969706t_unit(A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2174_Inf__fin_Osubset__imp,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic4107685809792843317_fin_o,B4)),aa_set_o_o(lattic4107685809792843317_fin_o,A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2175_Inf__fin_Osubset__imp,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic6089491107740071820_nat_o(B4)),lattic6089491107740071820_nat_o(A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2176_Inf__fin_Osubset__imp,axiom,
    ! [A: set_filter_nat,B4: set_filter_nat] :
      ( ord_le2426478655948331894er_nat(A,B4)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( finite2119507909894593271er_nat(B4)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,lattic9128708996678779395er_nat(B4)),lattic9128708996678779395er_nat(A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2177_Inf__fin_Osubset__imp,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A,B4)
     => ( ( A != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B4)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3014633134055518761et_nat(B4)),lattic3014633134055518761et_nat(A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2178_Inf__fin_Osubset__imp,axiom,
    ! [A: set_rat,B4: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A),B4)
     => ( ( A != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic4603258475043424379in_rat(B4)),lattic4603258475043424379in_rat(A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2179_Inf__fin_Osubset__imp,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic5235898064620869839in_int(B4)),lattic5235898064620869839in_int(A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2180_Inf__fin_Osubset__imp,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,B4)),aa_set_nat_nat(lattic5238388535129920115in_nat,A)) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_2181_Sup__fin_Osubset__imp,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic5294303975357428420t_unit(A)),lattic5294303975357428420t_unit(B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2182_Sup__fin_Osubset__imp,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1508158080041050831_fin_o,A)),aa_set_o_o(lattic1508158080041050831_fin_o,B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2183_Sup__fin_Osubset__imp,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,lattic7388253182610234406_nat_o(A)),lattic7388253182610234406_nat_o(B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2184_Sup__fin_Osubset__imp,axiom,
    ! [A: set_filter_nat,B4: set_filter_nat] :
      ( ord_le2426478655948331894er_nat(A,B4)
     => ( ( A != bot_bo498966703094740906er_nat )
       => ( finite2119507909894593271er_nat(B4)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,lattic5930898082463196905er_nat(A)),lattic5930898082463196905er_nat(B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2185_Sup__fin_Osubset__imp,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A,B4)
     => ( ( A != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B4)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,lattic3835124923745554447et_nat(A)),lattic3835124923745554447et_nat(B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2186_Sup__fin_Osubset__imp,axiom,
    ! [A: set_rat,B4: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A),B4)
     => ( ( A != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic458866745392299617in_rat(A)),lattic458866745392299617in_rat(B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2187_Sup__fin_Osubset__imp,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic1091506334969745077in_int(A)),lattic1091506334969745077in_int(B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2188_Sup__fin_Osubset__imp,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic1093996805478795353in_nat,A)),aa_set_nat_nat(lattic1093996805478795353in_nat,B4)) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_2189_Inf__fin_Osubset,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( B4 != bot_bot_set_set_nat )
       => ( ord_le6893508408891458716et_nat(B4,A)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,lattic3014633134055518761et_nat(B4)),lattic3014633134055518761et_nat(A)) = lattic3014633134055518761et_nat(A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2190_Inf__fin_Osubset,axiom,
    ! [A: set_assn,B4: set_assn] :
      ( finite_finite_assn(A)
     => ( ( B4 != bot_bot_set_assn )
       => ( ord_less_eq_set_assn(B4,A)
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,lattic47131356835913163n_assn(B4)),lattic47131356835913163n_assn(A)) = lattic47131356835913163n_assn(A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2191_Inf__fin_Osubset,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( B4 != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),A)
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,lattic1263872656861969706t_unit(B4)),lattic1263872656861969706t_unit(A)) = lattic1263872656861969706t_unit(A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2192_Inf__fin_Osubset,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( B4 != bot_bot_set_o )
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),A)
         => ( aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,B4)),aa_set_o_o(lattic4107685809792843317_fin_o,A))
          <=> aa_set_o_o(lattic4107685809792843317_fin_o,A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2193_Inf__fin_Osubset,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( B4 != bot_bot_set_int )
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A)
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,lattic5235898064620869839in_int(B4)),lattic5235898064620869839in_int(A)) = lattic5235898064620869839in_int(A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2194_Inf__fin_Osubset,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( B4 != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),A)
         => ( aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(lattic6089491107740071820_nat_o(B4)),lattic6089491107740071820_nat_o(A)) = lattic6089491107740071820_nat_o(A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2195_Inf__fin_Osubset,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( B4 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,B4)),aa_set_nat_nat(lattic5238388535129920115in_nat,A)) = aa_set_nat_nat(lattic5238388535129920115in_nat,A) ) ) ) ) ).

% Inf_fin.subset
tff(fact_2196_Sup__fin_Osubset,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( B4 != bot_bot_set_set_nat )
       => ( ord_le6893508408891458716et_nat(B4,A)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(B4)),lattic3835124923745554447et_nat(A)) = lattic3835124923745554447et_nat(A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2197_Sup__fin_Osubset,axiom,
    ! [A: set_assn,B4: set_assn] :
      ( finite_finite_assn(A)
     => ( ( B4 != bot_bot_set_assn )
       => ( ord_less_eq_set_assn(B4,A)
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic2150320897289308081n_assn(B4)),lattic2150320897289308081n_assn(A)) = lattic2150320897289308081n_assn(A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2198_Sup__fin_Osubset,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( B4 != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),A)
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,lattic5294303975357428420t_unit(B4)),lattic5294303975357428420t_unit(A)) = lattic5294303975357428420t_unit(A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2199_Sup__fin_Osubset,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( B4 != bot_bot_set_o )
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),A)
         => ( aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,B4)),aa_set_o_o(lattic1508158080041050831_fin_o,A))
          <=> aa_set_o_o(lattic1508158080041050831_fin_o,A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2200_Sup__fin_Osubset,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( B4 != bot_bot_set_int )
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A)
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic1091506334969745077in_int(B4)),lattic1091506334969745077in_int(A)) = lattic1091506334969745077in_int(A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2201_Sup__fin_Osubset,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( B4 != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),A)
         => ( aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(lattic7388253182610234406_nat_o(B4)),lattic7388253182610234406_nat_o(A)) = lattic7388253182610234406_nat_o(A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2202_Sup__fin_Osubset,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( B4 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(lattic1093996805478795353in_nat,B4)),aa_set_nat_nat(lattic1093996805478795353in_nat,A)) = aa_set_nat_nat(lattic1093996805478795353in_nat,A) ) ) ) ) ).

% Sup_fin.subset
tff(fact_2203_Inf__fin_Oclosed,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y2: set_nat] : member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y2),insert_set_nat(X3,insert_set_nat(Y2,bot_bot_set_set_nat)))
         => member_set_nat(lattic3014633134055518761et_nat(A),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2204_Inf__fin_Oclosed,axiom,
    ! [A: set_assn] :
      ( finite_finite_assn(A)
     => ( ( A != bot_bot_set_assn )
       => ( ! [X3: assn,Y2: assn] : member_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y2),insert_assn(X3,insert_assn(Y2,bot_bot_set_assn)))
         => member_assn(lattic47131356835913163n_assn(A),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2205_Inf__fin_Oclosed,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit,Y2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X3),Y2)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit)))
         => aa_set8780210009671629607unit_o(member_Product_unit(lattic1263872656861969706t_unit(A)),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2206_Inf__fin_Oclosed,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(inf_inf_o,(X3)),(Y2))),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic4107685809792843317_fin_o,A)),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2207_Inf__fin_Oclosed,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [X3: nat,Y2: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y2)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat)))
         => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic5238388535129920115in_nat,A)),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2208_Inf__fin_Oclosed,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X3: int,Y2: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y2)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int)))
         => aa_set_int_o(member_int(lattic5235898064620869839in_int(A)),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2209_Inf__fin_Oclosed,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X3),Y2)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y2),bot_bo8632255151191677115_nat_o)))
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic6089491107740071820_nat_o(A)),A) ) ) ) ).

% Inf_fin.closed
tff(fact_2210_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ~ member_set_nat(X,A)
       => ( ( A != bot_bot_set_set_nat )
         => ( lattic3014633134055518761et_nat(insert_set_nat(X,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2211_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( ~ member_assn(X,A)
       => ( ( A != bot_bot_set_assn )
         => ( lattic47131356835913163n_assn(insert_assn(X,A)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2212_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( ( A != bot_bo3957492148770167129t_unit )
         => ( lattic1263872656861969706t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),lattic1263872656861969706t_unit(A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2213_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( ( A != bot_bot_set_o )
         => ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
          <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2214_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( ( A != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2215_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( ( A != bot_bot_set_int )
         => ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2216_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( ( A != bot_bo8632255151191677115_nat_o )
         => ( lattic6089491107740071820_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X),lattic6089491107740071820_nat_o(A)) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_2217_Sup__fin_Oclosed,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y2: set_nat] : member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y2),insert_set_nat(X3,insert_set_nat(Y2,bot_bot_set_set_nat)))
         => member_set_nat(lattic3835124923745554447et_nat(A),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2218_Sup__fin_Oclosed,axiom,
    ! [A: set_assn] :
      ( finite_finite_assn(A)
     => ( ( A != bot_bot_set_assn )
       => ( ! [X3: assn,Y2: assn] : member_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y2),insert_assn(X3,insert_assn(Y2,bot_bot_set_assn)))
         => member_assn(lattic2150320897289308081n_assn(A),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2219_Sup__fin_Oclosed,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit,Y2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X3),Y2)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit)))
         => aa_set8780210009671629607unit_o(member_Product_unit(lattic5294303975357428420t_unit(A)),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2220_Sup__fin_Oclosed,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(sup_sup_o,(X3)),(Y2))),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1508158080041050831_fin_o,A)),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2221_Sup__fin_Oclosed,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [X3: nat,Y2: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y2)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat)))
         => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic1093996805478795353in_nat,A)),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2222_Sup__fin_Oclosed,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X3: int,Y2: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y2)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int)))
         => aa_set_int_o(member_int(lattic1091506334969745077in_int(A)),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2223_Sup__fin_Oclosed,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X3),Y2)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y2),bot_bo8632255151191677115_nat_o)))
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(lattic7388253182610234406_nat_o(A)),A) ) ) ) ).

% Sup_fin.closed
tff(fact_2224_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ~ member_set_nat(X,A)
       => ( ( A != bot_bot_set_set_nat )
         => ( lattic3835124923745554447et_nat(insert_set_nat(X,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2225_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( ~ member_assn(X,A)
       => ( ( A != bot_bot_set_assn )
         => ( lattic2150320897289308081n_assn(insert_assn(X,A)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2226_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( ( A != bot_bo3957492148770167129t_unit )
         => ( lattic5294303975357428420t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),lattic5294303975357428420t_unit(A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2227_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( ( A != bot_bot_set_o )
         => ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
          <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2228_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( ( A != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_set_nat_nat(lattic1093996805478795353in_nat,A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2229_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( ( A != bot_bot_set_int )
         => ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2230_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( ( A != bot_bo8632255151191677115_nat_o )
         => ( lattic7388253182610234406_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(A)) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_2231_Inf__fin_Ounion,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B4)
         => ( ( B4 != bot_bot_set_set_nat )
           => ( lattic3014633134055518761et_nat(sup_sup_set_set_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,lattic3014633134055518761et_nat(A)),lattic3014633134055518761et_nat(B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2232_Inf__fin_Ounion,axiom,
    ! [A: set_assn,B4: set_assn] :
      ( finite_finite_assn(A)
     => ( ( A != bot_bot_set_assn )
       => ( finite_finite_assn(B4)
         => ( ( B4 != bot_bot_set_assn )
           => ( lattic47131356835913163n_assn(sup_sup_set_assn(A,B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,lattic47131356835913163n_assn(A)),lattic47131356835913163n_assn(B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2233_Inf__fin_Ounion,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => ( ( B4 != bot_bo3957492148770167129t_unit )
           => ( lattic1263872656861969706t_unit(aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,lattic1263872656861969706t_unit(A)),lattic1263872656861969706t_unit(B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2234_Inf__fin_Ounion,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => ( ( B4 != bot_bot_set_o )
           => ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
            <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(lattic4107685809792843317_fin_o,A)),aa_set_o_o(lattic4107685809792843317_fin_o,B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2235_Inf__fin_Ounion,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( B4 != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(lattic5238388535129920115in_nat,A)),aa_set_nat_nat(lattic5238388535129920115in_nat,B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2236_Inf__fin_Ounion,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => ( ( B4 != bot_bot_set_int )
           => ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,lattic5235898064620869839in_int(A)),lattic5235898064620869839in_int(B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2237_Inf__fin_Ounion,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
         => ( ( B4 != bot_bo8632255151191677115_nat_o )
           => ( lattic6089491107740071820_nat_o(aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(lattic6089491107740071820_nat_o(A)),lattic6089491107740071820_nat_o(B4)) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_2238_Sup__fin_Ounion,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( finite1152437895449049373et_nat(B4)
         => ( ( B4 != bot_bot_set_set_nat )
           => ( lattic3835124923745554447et_nat(sup_sup_set_set_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,lattic3835124923745554447et_nat(A)),lattic3835124923745554447et_nat(B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2239_Sup__fin_Ounion,axiom,
    ! [A: set_assn,B4: set_assn] :
      ( finite_finite_assn(A)
     => ( ( A != bot_bot_set_assn )
       => ( finite_finite_assn(B4)
         => ( ( B4 != bot_bot_set_assn )
           => ( lattic2150320897289308081n_assn(sup_sup_set_assn(A,B4)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,lattic2150320897289308081n_assn(A)),lattic2150320897289308081n_assn(B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2240_Sup__fin_Ounion,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => ( ( B4 != bot_bo3957492148770167129t_unit )
           => ( lattic5294303975357428420t_unit(aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,lattic5294303975357428420t_unit(A)),lattic5294303975357428420t_unit(B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2241_Sup__fin_Ounion,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => ( ( B4 != bot_bot_set_o )
           => ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
            <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(lattic1508158080041050831_fin_o,A)),aa_set_o_o(lattic1508158080041050831_fin_o,B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2242_Sup__fin_Ounion,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( B4 != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(lattic1093996805478795353in_nat,A)),aa_set_nat_nat(lattic1093996805478795353in_nat,B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2243_Sup__fin_Ounion,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => ( ( B4 != bot_bot_set_int )
           => ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,lattic1091506334969745077in_int(A)),lattic1091506334969745077in_int(B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2244_Sup__fin_Ounion,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
         => ( ( B4 != bot_bo8632255151191677115_nat_o )
           => ( lattic7388253182610234406_nat_o(aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(lattic7388253182610234406_nat_o(A)),lattic7388253182610234406_nat_o(B4)) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_2245_refl__on__reflcl__Image,axiom,
    ! [B4: set_nat,A: set_Pr1261947904930325089at_nat,C5: set_nat] :
      ( refl_on_nat(B4,A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),B4)
       => ( aa_set_nat_set_nat(image_nat_nat(sup_su6327502436637775413at_nat(A,id_nat)),C5) = aa_set_nat_set_nat(image_nat_nat(A),C5) ) ) ) ).

% refl_on_reflcl_Image
tff(fact_2246_E__closed__restr__reach__cases,axiom,
    ! [U: fun_Pr6460944699146269935_nat_o,V: fun_Pr6460944699146269935_nat_o,E: set_Pr2890355626087285447_nat_o,R: set_fu3555266933746557007_nat_o] :
      ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(U,V),transi4955142523599900786_nat_o(E))
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_2188787866847195539_nat_o(E,R)),R)
       => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(V),R)
         => ~ ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(U),R)
             => ~ member3012145478770660624_nat_o(produc4714251849521319255_nat_o(U,V),transi4955142523599900786_nat_o(rel_re4406848170174997969_nat_o(E,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_2247_E__closed__restr__reach__cases,axiom,
    ! [U: $o,V: $o,E: set_Product_prod_o_o,R: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((U)),(V)),transitive_rtrancl_o(E))
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o(E,R)),R)
       => ( ~ aa_set_o_o(member_o((V)),R)
         => ~ ( ~ aa_set_o_o(member_o((U)),R)
             => ~ member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((U)),(V)),transitive_rtrancl_o(rel_restrict_o(E,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_2248_E__closed__restr__reach__cases,axiom,
    ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit,R: set_Product_unit] :
      ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(U),V),transi6481887417741755664t_unit(E))
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8431784415008483091t_unit(E,R)),R)
       => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(V),R)
         => ~ ( ~ aa_set8780210009671629607unit_o(member_Product_unit(U),R)
             => ~ member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(U),V),transi6481887417741755664t_unit(rel_re9207996470947987439t_unit(E,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_2249_E__closed__restr__reach__cases,axiom,
    ! [U: int,V: int,E: set_Pr958786334691620121nt_int,R: set_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),V),transi2902850859426252137cl_int(E))
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(E,R)),R)
       => ( ~ aa_set_int_o(member_int(V),R)
         => ~ ( ~ aa_set_int_o(member_int(U),R)
             => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(U),V),transi2902850859426252137cl_int(rel_restrict_int(E,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_2250_E__closed__restr__reach__cases,axiom,
    ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,R: set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V),transi2905341329935302413cl_nat(E))
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(E),R)),R)
       => ( ~ aa_set_nat_o(member_nat(V),R)
         => ~ ( ~ aa_set_nat_o(member_nat(U),R)
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V),transi2905341329935302413cl_nat(rel_restrict_nat(E,R))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_2251_max__ext__compat,axiom,
    ! [R: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
      ( ord_le930853871177014503t_unit(relcom848357998638694028t_unit(R,S),R)
     => ord_le8072594200992201959t_unit(relcom7108816970757792620t_unit(max_ext_Product_unit(R),sup_su3573767163274163995t_unit(max_ext_Product_unit(S),insert6793119152790066487t_unit(produc7269801343787132631t_unit(bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit),bot_bo5113747506523071411t_unit))),max_ext_Product_unit(R)) ) ).

% max_ext_compat
tff(fact_2252_max__ext__compat,axiom,
    ! [R: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ord_le192195778054999463od_o_o(relcomp_o_o_o(R,S),R)
     => ord_le2885120122233735591_set_o(relcom7534720040753365047_set_o(max_ext_o(R),sup_su7766833992073560027_set_o(max_ext_o(S),insert1494450971503307511_set_o(produc5838405689764958487_set_o(bot_bot_set_o,bot_bot_set_o),bot_bo8143257439550059635_set_o))),max_ext_o(R)) ) ).

% max_ext_compat
tff(fact_2253_max__ext__compat,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(relcomp_nat_nat_nat(R,S),R)
     => ord_le4402255868550298733et_nat(relcom236561585145426837et_nat(max_ext_nat(R),sup_su4251893418135750817et_nat(max_ext_nat(S),insert3810226134351308605et_nat(produc4532415448927165861et_nat(bot_bot_set_nat,bot_bot_set_nat),bot_bo4952753636535259449et_nat))),max_ext_nat(R)) ) ).

% max_ext_compat
tff(fact_2254_max__ext__compat,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ord_le2843351958646193337nt_int(relcomp_int_int_int(R,S),R)
     => ord_le1436784781161120549et_int(relcom4034950705519254697et_int(max_ext_int(R),sup_su1286422330746572633et_int(max_ext_int(S),insert7328952640415156725et_int(produc6363374080413544029et_int(bot_bot_set_int,bot_bot_set_int),bot_bo1987282549146081265et_int))),max_ext_int(R)) ) ).

% max_ext_compat
tff(fact_2255_max__ext__compat,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,S: set_Pr2890355626087285447_nat_o] :
      ( ord_le2940462717903690343_nat_o(relcom1922595779876297326_nat_o(R,S),R)
     => ord_le9137474638858277479_nat_o(relcom7987510920137042382_nat_o(max_ex8279603704637224490_nat_o(R),sup_su4458343269897816219_nat_o(max_ex8279603704637224490_nat_o(S),insert1134728238107055031_nat_o(produc5434073240646464855_nat_o(bot_bo8632255151191677115_nat_o,bot_bo8632255151191677115_nat_o),bot_bo7269894807694359859_nat_o))),max_ex8279603704637224490_nat_o(R)) ) ).

% max_ext_compat
tff(fact_2256_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A2: nat,B: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),id_nat)
     => ( A2 = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_2257_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A2: int,B: int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),id_int)
     => ( A2 = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_2258_Sup__fin_Oremove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( member_set_nat(X,A)
       => ( lattic3835124923745554447et_nat(A) = $ite(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2259_Sup__fin_Oremove,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( member_assn(X,A)
       => ( lattic2150320897289308081n_assn(A) = $ite(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2260_Sup__fin_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( lattic5294303975357428420t_unit(A) = $ite(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),lattic5294303975357428420t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2261_Sup__fin_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,A)
        <=> $ite(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2262_Sup__fin_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( lattic1091506334969745077in_int(A) = $ite(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2263_Sup__fin_Oremove,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( lattic7388253182610234406_nat_o(A) = $ite(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2264_Sup__fin_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_nat(lattic1093996805478795353in_nat,A) = $ite(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ) ) ) ).

% Sup_fin.remove
tff(fact_2265_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( lattic3835124923745554447et_nat(insert_set_nat(X,A)) = $ite(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),lattic3835124923745554447et_nat(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2266_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( lattic2150320897289308081n_assn(insert_assn(X,A)) = $ite(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),lattic2150320897289308081n_assn(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2267_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( lattic5294303975357428420t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),lattic5294303975357428420t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2268_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
      <=> $ite(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(sup_sup_o,(X)),aa_set_o_o(lattic1508158080041050831_fin_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2269_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),lattic1091506334969745077in_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2270_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( lattic7388253182610234406_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = $ite(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X),lattic7388253182610234406_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2271_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ) ) ).

% Sup_fin.insert_remove
tff(fact_2272_Inf__fin_Oremove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( member_set_nat(X,A)
       => ( lattic3014633134055518761et_nat(A) = $ite(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2273_Inf__fin_Oremove,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( member_assn(X,A)
       => ( lattic47131356835913163n_assn(A) = $ite(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2274_Inf__fin_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( lattic1263872656861969706t_unit(A) = $ite(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),lattic1263872656861969706t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2275_Inf__fin_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,A)
        <=> $ite(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2276_Inf__fin_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( lattic5235898064620869839in_int(A) = $ite(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2277_Inf__fin_Oremove,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( lattic6089491107740071820_nat_o(A) = $ite(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X),lattic6089491107740071820_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2278_Inf__fin_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_nat(lattic5238388535129920115in_nat,A) = $ite(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ) ) ) ).

% Inf_fin.remove
tff(fact_2279_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( lattic3014633134055518761et_nat(insert_set_nat(X,A)) = $ite(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat)) = bot_bot_set_set_nat,X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),lattic3014633134055518761et_nat(minus_2163939370556025621et_nat(A,insert_set_nat(X,bot_bot_set_set_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2280_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( lattic47131356835913163n_assn(insert_assn(X,A)) = $ite(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn)) = bot_bot_set_assn,X,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),lattic47131356835913163n_assn(minus_minus_set_assn(A,insert_assn(X,bot_bot_set_assn))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2281_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( lattic1263872656861969706t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),lattic1263872656861969706t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2282_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
      <=> $ite(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(inf_inf_o,(X)),aa_set_o_o(lattic4107685809792843317_fin_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2283_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),lattic5235898064620869839in_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2284_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( lattic6089491107740071820_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = $ite(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = bot_bo8632255151191677115_nat_o,X,aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X),lattic6089491107740071820_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2285_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ) ) ).

% Inf_fin.insert_remove
tff(fact_2286_acyclic__insert,axiom,
    ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
      ( transi3139058337892597825ic_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X),R2))
    <=> ( transi3139058337892597825ic_nat(R2)
        & ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R2)) ) ) ).

% acyclic_insert
tff(fact_2287_acyclic__insert,axiom,
    ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
      ( transi3136567867383547549ic_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X),R2))
    <=> ( transi3136567867383547549ic_int(R2)
        & ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R2)) ) ) ).

% acyclic_insert
tff(fact_2288_irrefl__tranclI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( inf_in2572325071724192079at_nat(converse_nat_nat(R2),transi2905341329935302413cl_nat(R2)) = bot_bo2099793752762293965at_nat )
     => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),transi6264000038957366511cl_nat(R2)) ) ).

% irrefl_tranclI
tff(fact_2289_irrefl__tranclI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int] :
      ( ( inf_in2269163501485487111nt_int(converse_int_int(R2),transi2902850859426252137cl_int(R2)) = bot_bo1796632182523588997nt_int )
     => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),X),transi6261509568448316235cl_int(R2)) ) ).

% irrefl_tranclI
tff(fact_2290_inf__img__fin__dom_H,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage1318793397690208119nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),bot_bo3990330152332043303nteger)),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2291_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage6104394758324456170nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),bot_bo3990330152332043303nteger))),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2292_inf__img__fin__dom_H,axiom,
    ! [F: fun_Co1365676809456677961t_unit,A: set_Code_integer] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_7182656606676592809t_unit(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),image_7182656606676592809t_unit(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5651688891684837235t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit)),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2293_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_8730104196221521654t_unit(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),image_8730104196221521654t_unit(F,A))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit))),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2294_inf__img__fin__dom_H,axiom,
    ! [F: fun_Code_integer_o,A: set_Code_integer] :
      ( aa_set_o_o(finite_finite_o,image_Code_integer_o(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),image_Code_integer_o(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage3005800680117336574eger_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o)),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2295_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_o,A: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),image_nat_o2(F,A))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o))),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2296_inf__img__fin__dom_H,axiom,
    ! [F: fun_Code_integer_nat,A: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),image_951025933927791156er_nat(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5839839309545413354er_nat(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat)),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2297_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(image_nat_nat2(F),A))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat))),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2298_inf__img__fin__dom_H,axiom,
    ! [F: fun_Code_integer_int,A: set_Code_integer] :
      ( aa_set_int_o(finite_finite_int,image_948535463418740880er_int(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),image_948535463418740880er_int(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5837348839036363078er_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int)),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2299_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_int,A: set_nat] :
      ( aa_set_int_o(finite_finite_int,image_nat_int2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),image_nat_int2(F,A))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int))),A)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_2300_inf__img__fin__domE_H,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage1318793397690208119nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger)),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2301_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage6104394758324456170nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger))),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2302_inf__img__fin__domE_H,axiom,
    ! [F: fun_Co1365676809456677961t_unit,A: set_Code_integer] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_7182656606676592809t_unit(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),image_7182656606676592809t_unit(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5651688891684837235t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit)),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2303_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_8730104196221521654t_unit(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),image_8730104196221521654t_unit(F,A))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit))),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2304_inf__img__fin__domE_H,axiom,
    ! [F: fun_Code_integer_o,A: set_Code_integer] :
      ( aa_set_o_o(finite_finite_o,image_Code_integer_o(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: $o] :
              ( aa_set_o_o(member_o((Y2)),image_Code_integer_o(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage3005800680117336574eger_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o)),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2305_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_o,A: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: $o] :
              ( aa_set_o_o(member_o((Y2)),image_nat_o2(F,A))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o))),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2306_inf__img__fin__domE_H,axiom,
    ! [F: fun_Code_integer_nat,A: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: nat] :
              ( aa_set_nat_o(member_nat(Y2),image_951025933927791156er_nat(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5839839309545413354er_nat(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat)),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2307_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: nat] :
              ( aa_set_nat_o(member_nat(Y2),aa_set_nat_set_nat(image_nat_nat2(F),A))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat))),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2308_inf__img__fin__domE_H,axiom,
    ! [F: fun_Code_integer_int,A: set_Code_integer] :
      ( aa_set_int_o(finite_finite_int,image_948535463418740880er_int(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: int] :
              ( aa_set_int_o(member_int(Y2),image_948535463418740880er_int(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,inf_in1364745209274528805nteger(vimage5837348839036363078er_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int)),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2309_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_int,A: set_nat] :
      ( aa_set_int_o(finite_finite_int,image_nat_int2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: int] :
              ( aa_set_int_o(member_int(Y2),image_nat_int2(F,A))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,vimage_nat_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int))),A)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_2310_image__eqI,axiom,
    ! [B: code_integer,F: fun_Co8499238673658658103nteger,X: code_integer,A: set_Code_integer] :
      ( ( B = aa_Cod1783974652533396728nteger(F,X) )
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => aa_set5643387332196575111eger_o(member_Code_integer(B),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) ) ) ).

% image_eqI
tff(fact_2311_image__eqI,axiom,
    ! [B: code_integer,F: fun_nat_Code_integer,X: nat,A: set_nat] :
      ( ( B = aa_nat_Code_integer(F,X) )
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_set5643387332196575111eger_o(member_Code_integer(B),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)) ) ) ).

% image_eqI
tff(fact_2312_image__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,X: nat,A: set_nat] :
      ( ( B = aa_nat_nat(F,X) )
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat2(F),A)) ) ) ).

% image_eqI
tff(fact_2313_image__eqI,axiom,
    ! [B: int,F: fun_nat_int,X: nat,A: set_nat] :
      ( ( B = aa_nat_int(F,X) )
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_set_int_o(member_int(B),image_nat_int2(F,A)) ) ) ).

% image_eqI
tff(fact_2314_image__eqI,axiom,
    ! [B: $o,F: fun_nat_o,X: nat,A: set_nat] :
      ( ( (B)
      <=> aa_nat_o(F,X) )
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_set_o_o(member_o((B)),image_nat_o2(F,A)) ) ) ).

% image_eqI
tff(fact_2315_image__eqI,axiom,
    ! [B: product_unit,F: fun_nat_Product_unit,X: nat,A: set_nat] :
      ( ( B = aa_nat_Product_unit(F,X) )
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_8730104196221521654t_unit(F,A)) ) ) ).

% image_eqI
tff(fact_2316_image__eqI,axiom,
    ! [B: nat,F: fun_int_nat,X: int,A: set_int] :
      ( ( B = aa_int_nat(F,X) )
     => ( aa_set_int_o(member_int(X),A)
       => aa_set_nat_o(member_nat(B),image_int_nat2(F,A)) ) ) ).

% image_eqI
tff(fact_2317_image__eqI,axiom,
    ! [B: int,F: fun_int_int,X: int,A: set_int] :
      ( ( B = aa_int_int(F,X) )
     => ( aa_set_int_o(member_int(X),A)
       => aa_set_int_o(member_int(B),image_int_int2(F,A)) ) ) ).

% image_eqI
tff(fact_2318_image__eqI,axiom,
    ! [B: $o,F: fun_int_o,X: int,A: set_int] :
      ( ( (B)
      <=> aa_int_o(F,X) )
     => ( aa_set_int_o(member_int(X),A)
       => aa_set_o_o(member_o((B)),image_int_o2(F,A)) ) ) ).

% image_eqI
tff(fact_2319_image__eqI,axiom,
    ! [B: product_unit,F: fun_int_Product_unit,X: int,A: set_int] :
      ( ( B = aa_int_Product_unit(F,X) )
     => ( aa_set_int_o(member_int(X),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_5846526143423701914t_unit(F,A)) ) ) ).

% image_eqI
tff(fact_2320_Diff__idemp,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),B4) = aa_set_nat_set_nat(minus_minus_set_nat(A),B4) ).

% Diff_idemp
tff(fact_2321_Diff__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),minus_6456595905716665654_nat_o(A,B4))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% Diff_iff
tff(fact_2322_Diff__iff,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A,B4))
    <=> ( aa_set_int_o(member_int(C),A)
        & ~ aa_set_int_o(member_int(C),B4) ) ) ).

% Diff_iff
tff(fact_2323_Diff__iff,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A,B4))
    <=> ( aa_set_o_o(member_o((C)),A)
        & ~ aa_set_o_o(member_o((C)),B4) ) ) ).

% Diff_iff
tff(fact_2324_Diff__iff,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),minus_6452836326544984404t_unit(A,B4))
    <=> ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
        & ~ aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% Diff_iff
tff(fact_2325_Diff__iff,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))
    <=> ( aa_set_nat_o(member_nat(C),A)
        & ~ aa_set_nat_o(member_nat(C),B4) ) ) ).

% Diff_iff
tff(fact_2326_DiffI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),minus_6456595905716665654_nat_o(A,B4)) ) ) ).

% DiffI
tff(fact_2327_DiffI,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),A)
     => ( ~ aa_set_int_o(member_int(C),B4)
       => aa_set_int_o(member_int(C),minus_minus_set_int(A,B4)) ) ) ).

% DiffI
tff(fact_2328_DiffI,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),A)
     => ( ~ aa_set_o_o(member_o((C)),B4)
       => aa_set_o_o(member_o((C)),minus_minus_set_o(A,B4)) ) ) ).

% DiffI
tff(fact_2329_DiffI,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(C),B4)
       => aa_set8780210009671629607unit_o(member_Product_unit(C),minus_6452836326544984404t_unit(A,B4)) ) ) ).

% DiffI
tff(fact_2330_DiffI,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A)
     => ( ~ aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) ) ) ).

% DiffI
tff(fact_2331_image__is__empty,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A) = bot_bo3990330152332043303nteger )
    <=> ( A = bot_bo3990330152332043303nteger ) ) ).

% image_is_empty
tff(fact_2332_image__is__empty,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat] :
      ( ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A) = bot_bo3990330152332043303nteger )
    <=> ( A = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2333_image__is__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A: set_Product_unit] :
      ( ( image_405062704495631173t_unit(F,A) = bot_bo3957492148770167129t_unit )
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% image_is_empty
tff(fact_2334_image__is__empty,axiom,
    ! [F: fun_o_Product_unit,A: set_o] :
      ( ( image_o_Product_unit2(F,A) = bot_bo3957492148770167129t_unit )
    <=> ( A = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_2335_image__is__empty,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat] :
      ( ( image_8730104196221521654t_unit(F,A) = bot_bo3957492148770167129t_unit )
    <=> ( A = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2336_image__is__empty,axiom,
    ! [F: fun_int_Product_unit,A: set_int] :
      ( ( image_5846526143423701914t_unit(F,A) = bot_bo3957492148770167129t_unit )
    <=> ( A = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_2337_image__is__empty,axiom,
    ! [F: fun_Product_unit_o,A: set_Product_unit] :
      ( ( image_Product_unit_o2(F,A) = bot_bot_set_o )
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% image_is_empty
tff(fact_2338_image__is__empty,axiom,
    ! [F: fun_o_o,A: set_o] :
      ( ( image_o_o2(F,A) = bot_bot_set_o )
    <=> ( A = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_2339_image__is__empty,axiom,
    ! [F: fun_nat_o,A: set_nat] :
      ( ( image_nat_o2(F,A) = bot_bot_set_o )
    <=> ( A = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2340_image__is__empty,axiom,
    ! [F: fun_int_o,A: set_int] :
      ( ( image_int_o2(F,A) = bot_bot_set_o )
    <=> ( A = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_2341_empty__is__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( ( bot_bo3990330152332043303nteger = aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A) )
    <=> ( A = bot_bo3990330152332043303nteger ) ) ).

% empty_is_image
tff(fact_2342_empty__is__image,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat] :
      ( ( bot_bo3990330152332043303nteger = aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A) )
    <=> ( A = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2343_empty__is__image,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = image_405062704495631173t_unit(F,A) )
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% empty_is_image
tff(fact_2344_empty__is__image,axiom,
    ! [F: fun_o_Product_unit,A: set_o] :
      ( ( bot_bo3957492148770167129t_unit = image_o_Product_unit2(F,A) )
    <=> ( A = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_2345_empty__is__image,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat] :
      ( ( bot_bo3957492148770167129t_unit = image_8730104196221521654t_unit(F,A) )
    <=> ( A = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2346_empty__is__image,axiom,
    ! [F: fun_int_Product_unit,A: set_int] :
      ( ( bot_bo3957492148770167129t_unit = image_5846526143423701914t_unit(F,A) )
    <=> ( A = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_2347_empty__is__image,axiom,
    ! [F: fun_Product_unit_o,A: set_Product_unit] :
      ( ( bot_bot_set_o = image_Product_unit_o2(F,A) )
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% empty_is_image
tff(fact_2348_empty__is__image,axiom,
    ! [F: fun_o_o,A: set_o] :
      ( ( bot_bot_set_o = image_o_o2(F,A) )
    <=> ( A = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_2349_empty__is__image,axiom,
    ! [F: fun_nat_o,A: set_nat] :
      ( ( bot_bot_set_o = image_nat_o2(F,A) )
    <=> ( A = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2350_empty__is__image,axiom,
    ! [F: fun_int_o,A: set_int] :
      ( ( bot_bot_set_o = image_int_o2(F,A) )
    <=> ( A = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_2351_image__empty,axiom,
    ! [F: fun_Co8499238673658658103nteger] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),bot_bo3990330152332043303nteger) = bot_bo3990330152332043303nteger ).

% image_empty
tff(fact_2352_image__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit] : image_405062704495631173t_unit(F,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% image_empty
tff(fact_2353_image__empty,axiom,
    ! [F: fun_Product_unit_o] : image_Product_unit_o2(F,bot_bo3957492148770167129t_unit) = bot_bot_set_o ).

% image_empty
tff(fact_2354_image__empty,axiom,
    ! [F: fun_Product_unit_nat] : image_875570014554754200it_nat(F,bot_bo3957492148770167129t_unit) = bot_bot_set_nat ).

% image_empty
tff(fact_2355_image__empty,axiom,
    ! [F: fun_Product_unit_int] : image_873079544045703924it_int(F,bot_bo3957492148770167129t_unit) = bot_bot_set_int ).

% image_empty
tff(fact_2356_image__empty,axiom,
    ! [F: fun_o_Product_unit] : image_o_Product_unit2(F,bot_bot_set_o) = bot_bo3957492148770167129t_unit ).

% image_empty
tff(fact_2357_image__empty,axiom,
    ! [F: fun_o_o] : image_o_o2(F,bot_bot_set_o) = bot_bot_set_o ).

% image_empty
tff(fact_2358_image__empty,axiom,
    ! [F: fun_o_nat] : image_o_nat2(F,bot_bot_set_o) = bot_bot_set_nat ).

% image_empty
tff(fact_2359_image__empty,axiom,
    ! [F: fun_o_int] : image_o_int2(F,bot_bot_set_o) = bot_bot_set_int ).

% image_empty
tff(fact_2360_image__empty,axiom,
    ! [F: fun_nat_Code_integer] : aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),bot_bot_set_nat) = bot_bo3990330152332043303nteger ).

% image_empty
tff(fact_2361_insert__image,axiom,
    ! [X: code_integer,A: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
     => ( aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,aa_Cod1783974652533396728nteger(F,X)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) = aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A) ) ) ).

% insert_image
tff(fact_2362_insert__image,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,aa_nat_Code_integer(F,X)),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)) = aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A) ) ) ).

% insert_image
tff(fact_2363_insert__image,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_nat_nat(F,X)),aa_set_nat_set_nat(image_nat_nat2(F),A)) = aa_set_nat_set_nat(image_nat_nat2(F),A) ) ) ).

% insert_image
tff(fact_2364_insert__image,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_nat_int(F,X)),image_nat_int2(F,A)) = image_nat_int2(F,A) ) ) ).

% insert_image
tff(fact_2365_insert__image,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,aa_nat_o(F,X)),image_nat_o2(F,A)) = image_nat_o2(F,A) ) ) ).

% insert_image
tff(fact_2366_insert__image,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,aa_nat_Product_unit(F,X)),image_8730104196221521654t_unit(F,A)) = image_8730104196221521654t_unit(F,A) ) ) ).

% insert_image
tff(fact_2367_insert__image,axiom,
    ! [X: int,A: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(X),A)
     => ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_int_nat(F,X)),image_int_nat2(F,A)) = image_int_nat2(F,A) ) ) ).

% insert_image
tff(fact_2368_insert__image,axiom,
    ! [X: int,A: set_int,F: fun_int_int] :
      ( aa_set_int_o(member_int(X),A)
     => ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(F,X)),image_int_int2(F,A)) = image_int_int2(F,A) ) ) ).

% insert_image
tff(fact_2369_insert__image,axiom,
    ! [X: int,A: set_int,F: fun_int_o] :
      ( aa_set_int_o(member_int(X),A)
     => ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,aa_int_o(F,X)),image_int_o2(F,A)) = image_int_o2(F,A) ) ) ).

% insert_image
tff(fact_2370_insert__image,axiom,
    ! [X: int,A: set_int,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(X),A)
     => ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,aa_int_Product_unit(F,X)),image_5846526143423701914t_unit(F,A)) = image_5846526143423701914t_unit(F,A) ) ) ).

% insert_image
tff(fact_2371_image__insert,axiom,
    ! [F: fun_Co8499238673658658103nteger,A2: code_integer,B4: set_Code_integer] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),B4)) = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,aa_Cod1783974652533396728nteger(F,A2)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),B4)) ).

% image_insert
tff(fact_2372_image__insert,axiom,
    ! [F: fun_nat_Code_integer,A2: nat,B4: set_nat] : aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,aa_nat_Code_integer(F,A2)),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),B4)) ).

% image_insert
tff(fact_2373_image__insert,axiom,
    ! [F: fun_nat_nat,A2: nat,B4: set_nat] : aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_nat_nat(F,A2)),aa_set_nat_set_nat(image_nat_nat2(F),B4)) ).

% image_insert
tff(fact_2374_image__insert,axiom,
    ! [F: fun_nat_int,A2: nat,B4: set_nat] : image_nat_int2(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_nat_int(F,A2)),image_nat_int2(F,B4)) ).

% image_insert
tff(fact_2375_image__insert,axiom,
    ! [F: fun_nat_o,A2: nat,B4: set_nat] : image_nat_o2(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,aa_nat_o(F,A2)),image_nat_o2(F,B4)) ).

% image_insert
tff(fact_2376_image__insert,axiom,
    ! [F: fun_nat_Product_unit,A2: nat,B4: set_nat] : image_8730104196221521654t_unit(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,aa_nat_Product_unit(F,A2)),image_8730104196221521654t_unit(F,B4)) ).

% image_insert
tff(fact_2377_image__insert,axiom,
    ! [F: fun_int_nat,A2: int,B4: set_int] : image_int_nat2(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_int_nat(F,A2)),image_int_nat2(F,B4)) ).

% image_insert
tff(fact_2378_image__insert,axiom,
    ! [F: fun_int_int,A2: int,B4: set_int] : image_int_int2(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(F,A2)),image_int_int2(F,B4)) ).

% image_insert
tff(fact_2379_image__insert,axiom,
    ! [F: fun_int_o,A2: int,B4: set_int] : image_int_o2(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,aa_int_o(F,A2)),image_int_o2(F,B4)) ).

% image_insert
tff(fact_2380_image__insert,axiom,
    ! [F: fun_int_Product_unit,A2: int,B4: set_int] : image_5846526143423701914t_unit(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,aa_int_Product_unit(F,A2)),image_5846526143423701914t_unit(F,B4)) ).

% image_insert
tff(fact_2381_img__snd,axiom,
    ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),S)
     => aa_set_nat_o(member_nat(B),image_2486076414777270412at_nat(product_snd_nat_nat,S)) ) ).

% img_snd
tff(fact_2382_img__snd,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),S)
     => member6260224972018164377et_nat(B,image_5643293374885392937et_nat(produc4960056325182568940et_nat,S)) ) ).

% img_snd
tff(fact_2383_img__snd,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),S)
     => member4513763875858039863et_nat(B,image_1425188809386248805et_nat(produc9081944055658670602et_nat,S)) ) ).

% img_snd
tff(fact_2384_img__snd,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),S)
     => member_set_nat(B,image_6088997261112373647et_nat(produc8586169260539613262et_nat,S)) ) ).

% img_snd
tff(fact_2385_img__snd,axiom,
    ! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),S)
     => aa_set_int_o(member_int(B),image_5042161079198086560nt_int(product_snd_int_int,S)) ) ).

% img_snd
tff(fact_2386_Diff__cancel,axiom,
    ! [A: set_Product_unit] : minus_6452836326544984404t_unit(A,A) = bot_bo3957492148770167129t_unit ).

% Diff_cancel
tff(fact_2387_Diff__cancel,axiom,
    ! [A: set_o] : minus_minus_set_o(A,A) = bot_bot_set_o ).

% Diff_cancel
tff(fact_2388_Diff__cancel,axiom,
    ! [A: set_int] : minus_minus_set_int(A,A) = bot_bot_set_int ).

% Diff_cancel
tff(fact_2389_Diff__cancel,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : minus_6456595905716665654_nat_o(A,A) = bot_bo8632255151191677115_nat_o ).

% Diff_cancel
tff(fact_2390_Diff__cancel,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),A) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_2391_empty__Diff,axiom,
    ! [A: set_Product_unit] : minus_6452836326544984404t_unit(bot_bo3957492148770167129t_unit,A) = bot_bo3957492148770167129t_unit ).

% empty_Diff
tff(fact_2392_empty__Diff,axiom,
    ! [A: set_o] : minus_minus_set_o(bot_bot_set_o,A) = bot_bot_set_o ).

% empty_Diff
tff(fact_2393_empty__Diff,axiom,
    ! [A: set_int] : minus_minus_set_int(bot_bot_set_int,A) = bot_bot_set_int ).

% empty_Diff
tff(fact_2394_empty__Diff,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : minus_6456595905716665654_nat_o(bot_bo8632255151191677115_nat_o,A) = bot_bo8632255151191677115_nat_o ).

% empty_Diff
tff(fact_2395_empty__Diff,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(bot_bot_set_nat),A) = bot_bot_set_nat ).

% empty_Diff
tff(fact_2396_Diff__empty,axiom,
    ! [A: set_Product_unit] : minus_6452836326544984404t_unit(A,bot_bo3957492148770167129t_unit) = A ).

% Diff_empty
tff(fact_2397_Diff__empty,axiom,
    ! [A: set_o] : minus_minus_set_o(A,bot_bot_set_o) = A ).

% Diff_empty
tff(fact_2398_Diff__empty,axiom,
    ! [A: set_int] : minus_minus_set_int(A,bot_bot_set_int) = A ).

% Diff_empty
tff(fact_2399_Diff__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : minus_6456595905716665654_nat_o(A,bot_bo8632255151191677115_nat_o) = A ).

% Diff_empty
tff(fact_2400_Diff__empty,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),bot_bot_set_nat) = A ).

% Diff_empty
tff(fact_2401_insert__Diff1,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4)
     => ( minus_6456595905716665654_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A),B4) = minus_6456595905716665654_nat_o(A,B4) ) ) ).

% insert_Diff1
tff(fact_2402_insert__Diff1,axiom,
    ! [X: int,B4: set_int,A: set_int] :
      ( aa_set_int_o(member_int(X),B4)
     => ( minus_minus_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A),B4) = minus_minus_set_int(A,B4) ) ) ).

% insert_Diff1
tff(fact_2403_insert__Diff1,axiom,
    ! [X: $o,B4: set_o,A: set_o] :
      ( aa_set_o_o(member_o((X)),B4)
     => ( minus_minus_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A),B4) = minus_minus_set_o(A,B4) ) ) ).

% insert_Diff1
tff(fact_2404_insert__Diff1,axiom,
    ! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),B4)
     => ( minus_6452836326544984404t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A),B4) = minus_6452836326544984404t_unit(A,B4) ) ) ).

% insert_Diff1
tff(fact_2405_insert__Diff1,axiom,
    ! [X: nat,B4: set_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(X),B4)
     => ( aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)),B4) = aa_set_nat_set_nat(minus_minus_set_nat(A),B4) ) ) ).

% insert_Diff1
tff(fact_2406_Diff__insert0,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ( minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B4)) = minus_6456595905716665654_nat_o(A,B4) ) ) ).

% Diff_insert0
tff(fact_2407_Diff__insert0,axiom,
    ! [X: int,A: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(X),A)
     => ( minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4)) = minus_minus_set_int(A,B4) ) ) ).

% Diff_insert0
tff(fact_2408_Diff__insert0,axiom,
    ! [X: $o,A: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A)
     => ( minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4)) = minus_minus_set_o(A,B4) ) ) ).

% Diff_insert0
tff(fact_2409_Diff__insert0,axiom,
    ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ( minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4)) = minus_6452836326544984404t_unit(A,B4) ) ) ).

% Diff_insert0
tff(fact_2410_Diff__insert0,axiom,
    ! [X: nat,A: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4)) = aa_set_nat_set_nat(minus_minus_set_nat(A),B4) ) ) ).

% Diff_insert0
tff(fact_2411_Un__Diff__cancel2,axiom,
    ! [B4: set_nat,A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(B4),A)),A) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),A) ).

% Un_Diff_cancel2
tff(fact_2412_Un__Diff__cancel,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) ).

% Un_Diff_cancel
tff(fact_2413_converse__iff,axiom,
    ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member727419437372278867t_unit(produc2471974422597015918t_unit(A2,B),conver7272282955447398104et_nat(R2))
    <=> member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B),A2),R2) ) ).

% converse_iff
tff(fact_2414_converse__iff,axiom,
    ! [A2: produc3658429121746597890et_nat,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A2,B),conver3104641359166825334et_nat(R2))
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B),A2),R2) ) ).

% converse_iff
tff(fact_2415_converse__iff,axiom,
    ! [A2: produc8799533983703829792et_nat,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A2,B),conver953664324188626580et_nat(R2))
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B),A2),R2) ) ).

% converse_iff
tff(fact_2416_converse__iff,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),conver2236330673913229642t_unit(R2))
    <=> member727419437372278867t_unit(produc2471974422597015918t_unit(B,A2),R2) ) ).

% converse_iff
tff(fact_2417_converse__iff,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),converse_nat_nat(R2))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),R2) ) ).

% converse_iff
tff(fact_2418_converse__iff,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),converse_int_int(R2))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),R2) ) ).

% converse_iff
tff(fact_2419_converse__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr3013394129895136920_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),conver3426625790779277168_nat_o(R2))
    <=> member7677167966932745849_nat_o(produc6339119935501530516_nat_o(B,A2),R2) ) ).

% converse_iff
tff(fact_2420_converse__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr7905283282039288826_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),conver9036929979925900498_nat_o(R2))
    <=> member3696715088396795611_nat_o(produc6151749222514847990_nat_o(B,A2),R2) ) ).

% converse_iff
tff(fact_2421_Diff__eq__empty__iff,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( minus_6452836326544984404t_unit(A,B4) = bot_bo3957492148770167129t_unit )
    <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2422_Diff__eq__empty__iff,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A,B4) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2423_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A,B4) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2424_Diff__eq__empty__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( minus_6456595905716665654_nat_o(A,B4) = bot_bo8632255151191677115_nat_o )
    <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2425_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(minus_minus_set_nat(A),B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2426_insert__Diff__single,axiom,
    ! [A2: product_unit,A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A) ).

% insert_Diff_single
tff(fact_2427_insert__Diff__single,axiom,
    ! [A2: $o,A: set_o] : aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A) ).

% insert_Diff_single
tff(fact_2428_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] : aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A) ).

% insert_Diff_single
tff(fact_2429_insert__Diff__single,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] : aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),A) ).

% insert_Diff_single
tff(fact_2430_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A) ).

% insert_Diff_single
tff(fact_2431_finite__Diff__insert,axiom,
    ! [A: set_int,A2: int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)))
    <=> aa_set_int_o(finite_finite_int,minus_minus_set_int(A,B4)) ) ).

% finite_Diff_insert
tff(fact_2432_finite__Diff__insert,axiom,
    ! [A: set_o,A2: $o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)))
    <=> aa_set_o_o(finite_finite_o,minus_minus_set_o(A,B4)) ) ).

% finite_Diff_insert
tff(fact_2433_finite__Diff__insert,axiom,
    ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)))
    <=> aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,minus_6452836326544984404t_unit(A,B4)) ) ).

% finite_Diff_insert
tff(fact_2434_finite__Diff__insert,axiom,
    ! [A: set_Code_integer,A2: code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),B4)))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B4)) ) ).

% finite_Diff_insert
tff(fact_2435_finite__Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)))
    <=> aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) ) ).

% finite_Diff_insert
tff(fact_2436_Diff__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] : inf_in4660618365625256667t_unit(A,minus_6452836326544984404t_unit(B4,A)) = bot_bo3957492148770167129t_unit ).

% Diff_disjoint
tff(fact_2437_Diff__disjoint,axiom,
    ! [A: set_o,B4: set_o] : inf_inf_set_o(A,minus_minus_set_o(B4,A)) = bot_bot_set_o ).

% Diff_disjoint
tff(fact_2438_Diff__disjoint,axiom,
    ! [A: set_int,B4: set_int] : inf_inf_set_int(A,minus_minus_set_int(B4,A)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_2439_Diff__disjoint,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(A,minus_6456595905716665654_nat_o(B4,A)) = bot_bo8632255151191677115_nat_o ).

% Diff_disjoint
tff(fact_2440_Diff__disjoint,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_2441_image__diff__subset,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] : aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(minus_2355218937544613996nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),B4))),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),minus_2355218937544613996nteger(A,B4))) ).

% image_diff_subset
tff(fact_2442_image__diff__subset,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_nat] : ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4)),aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ).

% image_diff_subset
tff(fact_2443_image__diff__subset,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_nat] : aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(minus_2355218937544613996nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),B4))),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ).

% image_diff_subset
tff(fact_2444_image__diff__subset,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(image_nat_nat2(F),A)),aa_set_nat_set_nat(image_nat_nat2(F),B4))),aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ).

% image_diff_subset
tff(fact_2445_rev__image__eqI,axiom,
    ! [X: code_integer,A: set_Code_integer,B: code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
     => ( ( B = aa_Cod1783974652533396728nteger(F,X) )
       => aa_set5643387332196575111eger_o(member_Code_integer(B),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) ) ) ).

% rev_image_eqI
tff(fact_2446_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: code_integer,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( ( B = aa_nat_Code_integer(F,X) )
       => aa_set5643387332196575111eger_o(member_Code_integer(B),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)) ) ) ).

% rev_image_eqI
tff(fact_2447_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( ( B = aa_nat_nat(F,X) )
       => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat2(F),A)) ) ) ).

% rev_image_eqI
tff(fact_2448_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: int,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( ( B = aa_nat_int(F,X) )
       => aa_set_int_o(member_int(B),image_nat_int2(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2449_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: $o,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( ( (B)
        <=> aa_nat_o(F,X) )
       => aa_set_o_o(member_o((B)),image_nat_o2(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2450_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B: product_unit,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( ( B = aa_nat_Product_unit(F,X) )
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_8730104196221521654t_unit(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2451_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: nat,F: fun_int_nat] :
      ( aa_set_int_o(member_int(X),A)
     => ( ( B = aa_int_nat(F,X) )
       => aa_set_nat_o(member_nat(B),image_int_nat2(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2452_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: int,F: fun_int_int] :
      ( aa_set_int_o(member_int(X),A)
     => ( ( B = aa_int_int(F,X) )
       => aa_set_int_o(member_int(B),image_int_int2(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2453_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: $o,F: fun_int_o] :
      ( aa_set_int_o(member_int(X),A)
     => ( ( (B)
        <=> aa_int_o(F,X) )
       => aa_set_o_o(member_o((B)),image_int_o2(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2454_rev__image__eqI,axiom,
    ! [X: int,A: set_int,B: product_unit,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(X),A)
     => ( ( B = aa_int_Product_unit(F,X) )
       => aa_set8780210009671629607unit_o(member_Product_unit(B),image_5846526143423701914t_unit(F,A)) ) ) ).

% rev_image_eqI
tff(fact_2455_ball__imageD,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,P: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
         => aa_set_nat_o(P,X3) )
     => ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A)
         => aa_set_nat_o(P,aa_nat_set_nat(F,X4)) ) ) ).

% ball_imageD
tff(fact_2456_ball__imageD,axiom,
    ! [F: fun_nat_nat,A: set_nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(image_nat_nat2(F),A))
         => aa_nat_o(P,X3) )
     => ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A)
         => aa_nat_o(P,aa_nat_nat(F,X4)) ) ) ).

% ball_imageD
tff(fact_2457_ball__imageD,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,P: fun_Code_integer_o] :
      ( ! [X3: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X3),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
         => aa_Code_integer_o(P,X3) )
     => ! [X4: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A)
         => aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(F,X4)) ) ) ).

% ball_imageD
tff(fact_2458_ball__imageD,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,P: fun_Code_integer_o] :
      ( ! [X3: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X3),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
         => aa_Code_integer_o(P,X3) )
     => ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A)
         => aa_Code_integer_o(P,aa_nat_Code_integer(F,X4)) ) ) ).

% ball_imageD
tff(fact_2459_image__cong,axiom,
    ! [M: set_Code_integer,N: set_Code_integer,F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( ( M = N )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),N)
           => ( aa_Cod1783974652533396728nteger(F,X3) = aa_Cod1783974652533396728nteger(G,X3) ) )
       => ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),M) = aa_set8622560600277679972nteger(image_4470545334726330049nteger(G),N) ) ) ) ).

% image_cong
tff(fact_2460_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( ( M = N )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),N)
           => ( aa_nat_set_nat(F,X3) = aa_nat_set_nat(G,X3) ) )
       => ( aa_set7246812951530217472et_nat(image_nat_set_nat(F),M) = aa_set7246812951530217472et_nat(image_nat_set_nat(G),N) ) ) ) ).

% image_cong
tff(fact_2461_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( M = N )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),N)
           => ( aa_nat_nat(F,X3) = aa_nat_nat(G,X3) ) )
       => ( aa_set_nat_set_nat(image_nat_nat2(F),M) = aa_set_nat_set_nat(image_nat_nat2(G),N) ) ) ) ).

% image_cong
tff(fact_2462_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( ( M = N )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),N)
           => ( aa_nat_Code_integer(F,X3) = aa_nat_Code_integer(G,X3) ) )
       => ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),M) = aa_set7438092518518805847nteger(image_1215581382706833972nteger(G),N) ) ) ) ).

% image_cong
tff(fact_2463_bex__imageD,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,P: fun_set_nat_o] :
      ( ? [X4: set_nat] :
          ( member_set_nat(X4,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
          & aa_set_nat_o(P,X4) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
          & aa_set_nat_o(P,aa_nat_set_nat(F,X3)) ) ) ).

% bex_imageD
tff(fact_2464_bex__imageD,axiom,
    ! [F: fun_nat_nat,A: set_nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(image_nat_nat2(F),A))
          & aa_nat_o(P,X4) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
          & aa_nat_o(P,aa_nat_nat(F,X3)) ) ) ).

% bex_imageD
tff(fact_2465_bex__imageD,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,P: fun_Code_integer_o] :
      ( ? [X4: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X4),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
          & aa_Code_integer_o(P,X4) )
     => ? [X3: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
          & aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(F,X3)) ) ) ).

% bex_imageD
tff(fact_2466_bex__imageD,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,P: fun_Code_integer_o] :
      ( ? [X4: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X4),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
          & aa_Code_integer_o(P,X4) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
          & aa_Code_integer_o(P,aa_nat_Code_integer(F,X3)) ) ) ).

% bex_imageD
tff(fact_2467_image__iff,axiom,
    ! [Z: set_nat,F: fun_nat_set_nat,A: set_nat] :
      ( member_set_nat(Z,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
    <=> ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
          & ( Z = aa_nat_set_nat(F,X2) ) ) ) ).

% image_iff
tff(fact_2468_image__iff,axiom,
    ! [Z: code_integer,F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(Z),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
    <=> ? [X2: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
          & ( Z = aa_Cod1783974652533396728nteger(F,X2) ) ) ) ).

% image_iff
tff(fact_2469_image__iff,axiom,
    ! [Z: code_integer,F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(Z),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
    <=> ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
          & ( Z = aa_nat_Code_integer(F,X2) ) ) ) ).

% image_iff
tff(fact_2470_image__iff,axiom,
    ! [Z: nat,F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(Z),aa_set_nat_set_nat(image_nat_nat2(F),A))
    <=> ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
          & ( Z = aa_nat_nat(F,X2) ) ) ) ).

% image_iff
tff(fact_2471_imageI,axiom,
    ! [X: code_integer,A: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
     => aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(F,X)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) ) ).

% imageI
tff(fact_2472_imageI,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(member_nat(X),A)
     => aa_set5643387332196575111eger_o(member_Code_integer(aa_nat_Code_integer(F,X)),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)) ) ).

% imageI
tff(fact_2473_imageI,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => aa_set_nat_o(member_nat(aa_nat_nat(F,X)),aa_set_nat_set_nat(image_nat_nat2(F),A)) ) ).

% imageI
tff(fact_2474_imageI,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(X),A)
     => aa_set_int_o(member_int(aa_nat_int(F,X)),image_nat_int2(F,A)) ) ).

% imageI
tff(fact_2475_imageI,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(X),A)
     => aa_set_o_o(member_o(aa_nat_o(F,X)),image_nat_o2(F,A)) ) ).

% imageI
tff(fact_2476_imageI,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(X),A)
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(F,X)),image_8730104196221521654t_unit(F,A)) ) ).

% imageI
tff(fact_2477_imageI,axiom,
    ! [X: int,A: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(X),A)
     => aa_set_nat_o(member_nat(aa_int_nat(F,X)),image_int_nat2(F,A)) ) ).

% imageI
tff(fact_2478_imageI,axiom,
    ! [X: int,A: set_int,F: fun_int_int] :
      ( aa_set_int_o(member_int(X),A)
     => aa_set_int_o(member_int(aa_int_int(F,X)),image_int_int2(F,A)) ) ).

% imageI
tff(fact_2479_imageI,axiom,
    ! [X: int,A: set_int,F: fun_int_o] :
      ( aa_set_int_o(member_int(X),A)
     => aa_set_o_o(member_o(aa_int_o(F,X)),image_int_o2(F,A)) ) ).

% imageI
tff(fact_2480_imageI,axiom,
    ! [X: int,A: set_int,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(X),A)
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(F,X)),image_5846526143423701914t_unit(F,A)) ) ).

% imageI
tff(fact_2481_DiffD2,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),minus_6456595905716665654_nat_o(A,B4))
     => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ).

% DiffD2
tff(fact_2482_DiffD2,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A,B4))
     => ~ aa_set_int_o(member_int(C),B4) ) ).

% DiffD2
tff(fact_2483_DiffD2,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A,B4))
     => ~ aa_set_o_o(member_o((C)),B4) ) ).

% DiffD2
tff(fact_2484_DiffD2,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),minus_6452836326544984404t_unit(A,B4))
     => ~ aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ).

% DiffD2
tff(fact_2485_DiffD2,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))
     => ~ aa_set_nat_o(member_nat(C),B4) ) ).

% DiffD2
tff(fact_2486_DiffD1,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),minus_6456595905716665654_nat_o(A,B4))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A) ) ).

% DiffD1
tff(fact_2487_DiffD1,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A,B4))
     => aa_set_int_o(member_int(C),A) ) ).

% DiffD1
tff(fact_2488_DiffD1,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A,B4))
     => aa_set_o_o(member_o((C)),A) ) ).

% DiffD1
tff(fact_2489_DiffD1,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),minus_6452836326544984404t_unit(A,B4))
     => aa_set8780210009671629607unit_o(member_Product_unit(C),A) ) ).

% DiffD1
tff(fact_2490_DiffD1,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))
     => aa_set_nat_o(member_nat(C),A) ) ).

% DiffD1
tff(fact_2491_DiffE,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),minus_6456595905716665654_nat_o(A,B4))
     => ~ ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% DiffE
tff(fact_2492_DiffE,axiom,
    ! [C: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C),minus_minus_set_int(A,B4))
     => ~ ( aa_set_int_o(member_int(C),A)
         => aa_set_int_o(member_int(C),B4) ) ) ).

% DiffE
tff(fact_2493_DiffE,axiom,
    ! [C: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((C)),minus_minus_set_o(A,B4))
     => ~ ( aa_set_o_o(member_o((C)),A)
         => aa_set_o_o(member_o((C)),B4) ) ) ).

% DiffE
tff(fact_2494_DiffE,axiom,
    ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(C),minus_6452836326544984404t_unit(A,B4))
     => ~ ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% DiffE
tff(fact_2495_DiffE,axiom,
    ! [C: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))
     => ~ ( aa_set_nat_o(member_nat(C),A)
         => aa_set_nat_o(member_nat(C),B4) ) ) ).

% DiffE
tff(fact_2496_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2497_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C: nat,B: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(A2),C)),B) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(A2),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2498_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),C)),B) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2499_diff__eq__diff__eq,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(minus_minus_rat(C),D2) )
     => ( ( A2 = B )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2500_diff__eq__diff__eq,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(minus_minus_int(C),D2) )
     => ( ( A2 = B )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2501_in__image__insert__iff,axiom,
    ! [B4: set_set_Product_unit,X: product_unit,A: set_Product_unit] :
      ( ! [C6: set_Product_unit] :
          ( member5877623283571906838t_unit(C6,B4)
         => ~ aa_set8780210009671629607unit_o(member_Product_unit(X),C6) )
     => ( member5877623283571906838t_unit(A,image_4019945301284682565t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4))
      <=> ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
          & member5877623283571906838t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_2502_in__image__insert__iff,axiom,
    ! [B4: set_set_o,X: $o,A: set_o] :
      ( ! [C6: set_o] :
          ( member_set_o(C6,B4)
         => ~ aa_set_o_o(member_o((X)),C6) )
     => ( member_set_o(A,image_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4))
      <=> ( aa_set_o_o(member_o((X)),A)
          & member_set_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_2503_in__image__insert__iff,axiom,
    ! [B4: set_set_int,X: int,A: set_int] :
      ( ! [C6: set_int] :
          ( member_set_int(C6,B4)
         => ~ aa_set_int_o(member_int(X),C6) )
     => ( member_set_int(A,image_524474410958335435et_int(aa_int7955932875426912299et_int(insert_int,X),B4))
      <=> ( aa_set_int_o(member_int(X),A)
          & member_set_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_2504_in__image__insert__iff,axiom,
    ! [B4: set_se6690607124073039791_nat_o,X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ! [C6: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(C6,B4)
         => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),C6) )
     => ( member361801372707117048_nat_o(A,image_7577113955783394245_nat_o(insert4923026906511862463_nat_o(X),B4))
      <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
          & member361801372707117048_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_2505_in__image__insert__iff,axiom,
    ! [B4: set_set_nat,X: nat,A: set_nat] :
      ( ! [C6: set_nat] :
          ( member_set_nat(C6,B4)
         => ~ aa_set_nat_o(member_nat(X),C6) )
     => ( member_set_nat(A,image_7916887816326733075et_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4))
      <=> ( aa_set_nat_o(member_nat(X),A)
          & member_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_2506_converse_Ocases,axiom,
    ! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member727419437372278867t_unit(produc2471974422597015918t_unit(A1,A22),conver7272282955447398104et_nat(R2))
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A22),A1),R2) ) ).

% converse.cases
tff(fact_2507_converse_Ocases,axiom,
    ! [A1: produc3658429121746597890et_nat,A22: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A1,A22),conver3104641359166825334et_nat(R2))
     => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A22),A1),R2) ) ).

% converse.cases
tff(fact_2508_converse_Ocases,axiom,
    ! [A1: produc8799533983703829792et_nat,A22: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A1,A22),conver953664324188626580et_nat(R2))
     => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A22),A1),R2) ) ).

% converse.cases
tff(fact_2509_converse_Ocases,axiom,
    ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),A22),conver2236330673913229642t_unit(R2))
     => member727419437372278867t_unit(produc2471974422597015918t_unit(A22,A1),R2) ) ).

% converse.cases
tff(fact_2510_converse_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),converse_nat_nat(R2))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A22),A1),R2) ) ).

% converse.cases
tff(fact_2511_converse_Ocases,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),converse_int_int(R2))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A22),A1),R2) ) ).

% converse.cases
tff(fact_2512_converse_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr3013394129895136920_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),conver3426625790779277168_nat_o(R2))
     => member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A22,A1),R2) ) ).

% converse.cases
tff(fact_2513_converse_Ocases,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr7905283282039288826_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),conver9036929979925900498_nat_o(R2))
     => member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A22,A1),R2) ) ).

% converse.cases
tff(fact_2514_converse_Osimps,axiom,
    ! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member727419437372278867t_unit(produc2471974422597015918t_unit(A1,A22),conver7272282955447398104et_nat(R2))
    <=> ? [A5: heap_e7401611519738050253t_unit,B6: set_nat] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A5),B6),R2) ) ) ).

% converse.simps
tff(fact_2515_converse_Osimps,axiom,
    ! [A1: produc3658429121746597890et_nat,A22: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A1,A22),conver3104641359166825334et_nat(R2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc3658429121746597890et_nat] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A5),B6),R2) ) ) ).

% converse.simps
tff(fact_2516_converse_Osimps,axiom,
    ! [A1: produc8799533983703829792et_nat,A22: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A1,A22),conver953664324188626580et_nat(R2))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B6: produc8799533983703829792et_nat] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A5),B6),R2) ) ) ).

% converse.simps
tff(fact_2517_converse_Osimps,axiom,
    ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A1),A22),conver2236330673913229642t_unit(R2))
    <=> ? [A5: set_nat,B6: heap_e7401611519738050253t_unit] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member727419437372278867t_unit(produc2471974422597015918t_unit(A5,B6),R2) ) ) ).

% converse.simps
tff(fact_2518_converse_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),converse_nat_nat(R2))
    <=> ? [A5: nat,B6: nat] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B6),R2) ) ) ).

% converse.simps
tff(fact_2519_converse_Osimps,axiom,
    ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),converse_int_int(R2))
    <=> ? [A5: int,B6: int] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A5),B6),R2) ) ) ).

% converse.simps
tff(fact_2520_converse_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc3658429121746597890et_nat,R2: set_Pr3013394129895136920_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A1),A22),conver3426625790779277168_nat_o(R2))
    <=> ? [A5: produc3658429121746597890et_nat,B6: fun_Pr6460944699146269935_nat_o] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A5,B6),R2) ) ) ).

% converse.simps
tff(fact_2521_converse_Osimps,axiom,
    ! [A1: fun_Pr6460944699146269935_nat_o,A22: produc8799533983703829792et_nat,R2: set_Pr7905283282039288826_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A1),A22),conver9036929979925900498_nat_o(R2))
    <=> ? [A5: produc8799533983703829792et_nat,B6: fun_Pr6460944699146269935_nat_o] :
          ( ( A1 = B6 )
          & ( A22 = A5 )
          & member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A5,B6),R2) ) ) ).

% converse.simps
tff(fact_2522_converseD,axiom,
    ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member727419437372278867t_unit(produc2471974422597015918t_unit(A2,B),conver7272282955447398104et_nat(R2))
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B),A2),R2) ) ).

% converseD
tff(fact_2523_converseD,axiom,
    ! [A2: produc3658429121746597890et_nat,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A2,B),conver3104641359166825334et_nat(R2))
     => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B),A2),R2) ) ).

% converseD
tff(fact_2524_converseD,axiom,
    ! [A2: produc8799533983703829792et_nat,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A2,B),conver953664324188626580et_nat(R2))
     => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B),A2),R2) ) ).

% converseD
tff(fact_2525_converseD,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),conver2236330673913229642t_unit(R2))
     => member727419437372278867t_unit(produc2471974422597015918t_unit(B,A2),R2) ) ).

% converseD
tff(fact_2526_converseD,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),converse_nat_nat(R2))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),R2) ) ).

% converseD
tff(fact_2527_converseD,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),converse_int_int(R2))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),R2) ) ).

% converseD
tff(fact_2528_converseD,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr3013394129895136920_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),conver3426625790779277168_nat_o(R2))
     => member7677167966932745849_nat_o(produc6339119935501530516_nat_o(B,A2),R2) ) ).

% converseD
tff(fact_2529_converseD,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr7905283282039288826_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),conver9036929979925900498_nat_o(R2))
     => member3696715088396795611_nat_o(produc6151749222514847990_nat_o(B,A2),R2) ) ).

% converseD
tff(fact_2530_converseE,axiom,
    ! [Yx: produc7348995623955488188t_unit,R2: set_Pr3948176798113811640et_nat] :
      ( member727419437372278867t_unit(Yx,conver7272282955447398104et_nat(R2))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
            ( ( Yx = produc2471974422597015918t_unit(Y2,X3) )
           => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R2) ) ) ).

% converseE
tff(fact_2531_converseE,axiom,
    ! [Yx: produc2739566037923759970_nat_o,R2: set_Pr1312406793078187350et_nat] :
      ( member7677167966932745849_nat_o(Yx,conver3104641359166825334et_nat(R2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( Yx = produc6339119935501530516_nat_o(Y2,X3) )
           => ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),R2) ) ) ).

% converseE
tff(fact_2532_converseE,axiom,
    ! [Yx: produc6858501863712832324_nat_o,R2: set_Pr4040394263813967220et_nat] :
      ( member3696715088396795611_nat_o(Yx,conver953664324188626580et_nat(R2))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
            ( ( Yx = produc6151749222514847990_nat_o(Y2,X3) )
           => ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2),R2) ) ) ).

% converseE
tff(fact_2533_converseE,axiom,
    ! [Yx: produc3658429121746597890et_nat,R2: set_Pr4071100604818519410t_unit] :
      ( member6260224972018164377et_nat(Yx,conver2236330673913229642t_unit(R2))
     => ~ ! [X3: set_nat,Y2: heap_e7401611519738050253t_unit] :
            ( ( Yx = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y2),X3) )
           => ~ member727419437372278867t_unit(produc2471974422597015918t_unit(X3,Y2),R2) ) ) ).

% converseE
tff(fact_2534_converseE,axiom,
    ! [Yx: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(Yx,converse_nat_nat(R2))
     => ~ ! [X3: nat,Y2: nat] :
            ( ( Yx = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),X3) )
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2) ) ) ).

% converseE
tff(fact_2535_converseE,axiom,
    ! [Yx: product_prod_int_int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(Yx,converse_int_int(R2))
     => ~ ! [X3: int,Y2: int] :
            ( ( Yx = aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),X3) )
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2) ) ) ).

% converseE
tff(fact_2536_converseE,axiom,
    ! [Yx: produc8799533983703829792et_nat,R2: set_Pr3013394129895136920_nat_o] :
      ( member4513763875858039863et_nat(Yx,conver3426625790779277168_nat_o(R2))
     => ~ ! [X3: produc3658429121746597890et_nat,Y2: fun_Pr6460944699146269935_nat_o] :
            ( ( Yx = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y2),X3) )
           => ~ member7677167966932745849_nat_o(produc6339119935501530516_nat_o(X3,Y2),R2) ) ) ).

% converseE
tff(fact_2537_converseE,axiom,
    ! [Yx: produc3112576180687729086et_nat,R2: set_Pr7905283282039288826_nat_o] :
      ( member9174161442226468181et_nat(Yx,conver9036929979925900498_nat_o(R2))
     => ~ ! [X3: produc8799533983703829792et_nat,Y2: fun_Pr6460944699146269935_nat_o] :
            ( ( Yx = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y2),X3) )
           => ~ member3696715088396795611_nat_o(produc6151749222514847990_nat_o(X3,Y2),R2) ) ) ).

% converseE
tff(fact_2538_converseI,axiom,
    ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr4071100604818519410t_unit] :
      ( member727419437372278867t_unit(produc2471974422597015918t_unit(A2,B),R2)
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(B),A2),conver2236330673913229642t_unit(R2)) ) ).

% converseI
tff(fact_2539_converseI,axiom,
    ! [A2: produc3658429121746597890et_nat,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr3013394129895136920_nat_o] :
      ( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(A2,B),R2)
     => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(B),A2),conver3426625790779277168_nat_o(R2)) ) ).

% converseI
tff(fact_2540_converseI,axiom,
    ! [A2: produc8799533983703829792et_nat,B: fun_Pr6460944699146269935_nat_o,R2: set_Pr7905283282039288826_nat_o] :
      ( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(A2,B),R2)
     => member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(B),A2),conver9036929979925900498_nat_o(R2)) ) ).

% converseI
tff(fact_2541_converseI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R2)
     => member727419437372278867t_unit(produc2471974422597015918t_unit(B,A2),conver7272282955447398104et_nat(R2)) ) ).

% converseI
tff(fact_2542_converseI,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),converse_nat_nat(R2)) ) ).

% converseI
tff(fact_2543_converseI,axiom,
    ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),converse_int_int(R2)) ) ).

% converseI
tff(fact_2544_converseI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),R2)
     => member7677167966932745849_nat_o(produc6339119935501530516_nat_o(B,A2),conver3104641359166825334et_nat(R2)) ) ).

% converseI
tff(fact_2545_converseI,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),R2)
     => member3696715088396795611_nat_o(produc6151749222514847990_nat_o(B,A2),conver953664324188626580et_nat(R2)) ) ).

% converseI
tff(fact_2546_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(minus_minus_rat(C),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2547_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(minus_minus_int(C),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2548_diff__right__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_2549_diff__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),C)),aa_int_int(minus_minus_int(B),C)) ) ).

% diff_right_mono
tff(fact_2550_diff__left__mono,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A2)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_2551_diff__left__mono,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(C),A2)),aa_int_int(minus_minus_int(C),B)) ) ).

% diff_left_mono
tff(fact_2552_diff__mono,axiom,
    ! [A2: rat,B: rat,D2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),C)),aa_rat_rat(minus_minus_rat(B),D2)) ) ) ).

% diff_mono
tff(fact_2553_diff__mono,axiom,
    ! [A2: int,B: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),C)),aa_int_int(minus_minus_int(B),D2)) ) ) ).

% diff_mono
tff(fact_2554_subset__image__iff,axiom,
    ! [B4: set_Code_integer,F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
    <=> ? [AA: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(AA),A)
          & ( B4 = aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2555_subset__image__iff,axiom,
    ! [B4: set_set_nat,F: fun_nat_set_nat,A: set_nat] :
      ( ord_le6893508408891458716et_nat(B4,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A)
          & ( B4 = aa_set7246812951530217472et_nat(image_nat_set_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2556_subset__image__iff,axiom,
    ! [B4: set_Code_integer,F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A)
          & ( B4 = aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2557_subset__image__iff,axiom,
    ! [B4: set_nat,F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat2(F),A))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A)
          & ( B4 = aa_set_nat_set_nat(image_nat_nat2(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2558_image__subset__iff,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),B4)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => member_set_nat(aa_nat_set_nat(F,X2),B4) ) ) ).

% image_subset_iff
tff(fact_2559_image__subset__iff,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)),B4)
    <=> ! [X2: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
         => aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(F,X2)),B4) ) ) ).

% image_subset_iff
tff(fact_2560_image__subset__iff,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)),B4)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set5643387332196575111eger_o(member_Code_integer(aa_nat_Code_integer(F,X2)),B4) ) ) ).

% image_subset_iff
tff(fact_2561_image__subset__iff,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)),B4)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X2)),B4) ) ) ).

% image_subset_iff
tff(fact_2562_subset__imageE,axiom,
    ! [B4: set_Code_integer,F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => ~ ! [C6: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C6),A)
           => ( B4 != aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),C6) ) ) ) ).

% subset_imageE
tff(fact_2563_subset__imageE,axiom,
    ! [B4: set_set_nat,F: fun_nat_set_nat,A: set_nat] :
      ( ord_le6893508408891458716et_nat(B4,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
     => ~ ! [C6: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C6),A)
           => ( B4 != aa_set7246812951530217472et_nat(image_nat_set_nat(F),C6) ) ) ) ).

% subset_imageE
tff(fact_2564_subset__imageE,axiom,
    ! [B4: set_Code_integer,F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => ~ ! [C6: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C6),A)
           => ( B4 != aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),C6) ) ) ) ).

% subset_imageE
tff(fact_2565_subset__imageE,axiom,
    ! [B4: set_nat,F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat2(F),A))
     => ~ ! [C6: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C6),A)
           => ( B4 != aa_set_nat_set_nat(image_nat_nat2(F),C6) ) ) ) ).

% subset_imageE
tff(fact_2566_image__subsetI,axiom,
    ! [A: set_Code_integer,F: fun_Co8499238673658658103nteger,B4: set_Code_integer] :
      ( ! [X3: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
         => aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(F,X3)),B4) )
     => aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)),B4) ) ).

% image_subsetI
tff(fact_2567_image__subsetI,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer,B4: set_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_set5643387332196575111eger_o(member_Code_integer(aa_nat_Code_integer(F,X3)),B4) )
     => aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)),B4) ) ).

% image_subsetI
tff(fact_2568_image__subsetI,axiom,
    ! [A: set_nat,F: fun_nat_int,B4: set_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_set_int_o(member_int(aa_nat_int(F,X3)),B4) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int2(F,A)),B4) ) ).

% image_subsetI
tff(fact_2569_image__subsetI,axiom,
    ! [A: set_nat,F: fun_nat_o,B4: set_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_set_o_o(member_o(aa_nat_o(F,X3)),B4) )
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_nat_o2(F,A)),B4) ) ).

% image_subsetI
tff(fact_2570_image__subsetI,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit,B4: set_Product_unit] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(F,X3)),B4) )
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_8730104196221521654t_unit(F,A)),B4) ) ).

% image_subsetI
tff(fact_2571_image__subsetI,axiom,
    ! [A: set_int,F: fun_int_int,B4: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_set_int_o(member_int(aa_int_int(F,X3)),B4) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int2(F,A)),B4) ) ).

% image_subsetI
tff(fact_2572_image__subsetI,axiom,
    ! [A: set_int,F: fun_int_o,B4: set_o] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_set_o_o(member_o(aa_int_o(F,X3)),B4) )
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_int_o2(F,A)),B4) ) ).

% image_subsetI
tff(fact_2573_image__subsetI,axiom,
    ! [A: set_int,F: fun_int_Product_unit,B4: set_Product_unit] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(F,X3)),B4) )
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_5846526143423701914t_unit(F,A)),B4) ) ).

% image_subsetI
tff(fact_2574_image__subsetI,axiom,
    ! [A: set_o,F: fun_o_int,B4: set_int] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => aa_set_int_o(member_int(aa_o_int(F,(X3))),B4) )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_o_int2(F,A)),B4) ) ).

% image_subsetI
tff(fact_2575_image__subsetI,axiom,
    ! [A: set_o,F: fun_o_o,B4: set_o] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => aa_set_o_o(member_o(aa_o_o(F,(X3))),B4) )
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o2(F,A)),B4) ) ).

% image_subsetI
tff(fact_2576_image__mono,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B4)
     => aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),B4)) ) ).

% image_mono
tff(fact_2577_image__mono,axiom,
    ! [A: set_nat,B4: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4)) ) ).

% image_mono
tff(fact_2578_image__mono,axiom,
    ! [A: set_nat,B4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),B4)) ) ).

% image_mono
tff(fact_2579_image__mono,axiom,
    ! [A: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)),aa_set_nat_set_nat(image_nat_nat2(F),B4)) ) ).

% image_mono
tff(fact_2580_image__Un,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),sup_su848401254843788991nteger(A,B4)) = sup_su848401254843788991nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),B4)) ).

% image_Un
tff(fact_2581_image__Un,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_nat] : aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = sup_sup_set_set_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4)) ).

% image_Un
tff(fact_2582_image__Un,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_nat] : aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = sup_su848401254843788991nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),B4)) ).

% image_Un
tff(fact_2583_image__Un,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(image_nat_nat2(F),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,aa_set_nat_set_nat(image_nat_nat2(F),A)),aa_set_nat_set_nat(image_nat_nat2(F),B4)) ).

% image_Un
tff(fact_2584_double__diff,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C5)
       => ( aa_set_nat_set_nat(minus_minus_set_nat(B4),aa_set_nat_set_nat(minus_minus_set_nat(C5),A)) = A ) ) ) ).

% double_diff
tff(fact_2585_Diff__subset,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),A) ).

% Diff_subset
tff(fact_2586_Diff__mono,axiom,
    ! [A: set_nat,C5: set_nat,D3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D3),B4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set_nat_set_nat(minus_minus_set_nat(C5),D3)) ) ) ).

% Diff_mono
tff(fact_2587_insert__Diff__if,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      minus_6456595905716665654_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A),B4) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4),minus_6456595905716665654_nat_o(A,B4),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),minus_6456595905716665654_nat_o(A,B4))) ).

% insert_Diff_if
tff(fact_2588_insert__Diff__if,axiom,
    ! [X: int,A: set_int,B4: set_int] :
      minus_minus_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A),B4) = $ite(aa_set_int_o(member_int(X),B4),minus_minus_set_int(A,B4),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),minus_minus_set_int(A,B4))) ).

% insert_Diff_if
tff(fact_2589_insert__Diff__if,axiom,
    ! [X: $o,A: set_o,B4: set_o] :
      minus_minus_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A),B4) = $ite(aa_set_o_o(member_o((X)),B4),minus_minus_set_o(A,B4),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),minus_minus_set_o(A,B4))) ).

% insert_Diff_if
tff(fact_2590_insert__Diff__if,axiom,
    ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      minus_6452836326544984404t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A),B4) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),B4),minus_6452836326544984404t_unit(A,B4),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),minus_6452836326544984404t_unit(A,B4))) ).

% insert_Diff_if
tff(fact_2591_insert__Diff__if,axiom,
    ! [X: nat,A: set_nat,B4: set_nat] :
      aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)),B4) = $ite(aa_set_nat_o(member_nat(X),B4),aa_set_nat_set_nat(minus_minus_set_nat(A),B4),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ).

% insert_Diff_if
tff(fact_2592_Diff__Int__distrib2,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),C5) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C5)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) ).

% Diff_Int_distrib2
tff(fact_2593_Diff__Int__distrib,axiom,
    ! [C5: set_nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C5),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C5),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C5),B4)) ).

% Diff_Int_distrib
tff(fact_2594_Diff__Diff__Int,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ).

% Diff_Diff_Int
tff(fact_2595_Diff__Int2,axiom,
    ! [A: set_nat,C5: set_nat,B4: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C5)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C5)),B4) ).

% Diff_Int2
tff(fact_2596_Int__Diff,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)),C5) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(minus_minus_set_nat(B4),C5)) ).

% Int_Diff
tff(fact_2597_set__diff__diff__left,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),C5) = aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C5)) ).

% set_diff_diff_left
tff(fact_2598_Un__Diff,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)),C5) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),C5)),aa_set_nat_set_nat(minus_minus_set_nat(B4),C5)) ).

% Un_Diff
tff(fact_2599_vimage__Diff,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(vimage_nat_nat(F),A)),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ).

% vimage_Diff
tff(fact_2600_rtrancl__converseI,axiom,
    ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X),transi2905341329935302413cl_nat(R2))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(converse_nat_nat(R2))) ) ).

% rtrancl_converseI
tff(fact_2601_rtrancl__converseI,axiom,
    ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X),transi2902850859426252137cl_int(R2))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(converse_int_int(R2))) ) ).

% rtrancl_converseI
tff(fact_2602_rtrancl__converseD,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(converse_nat_nat(R2)))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X),transi2905341329935302413cl_nat(R2)) ) ).

% rtrancl_converseD
tff(fact_2603_rtrancl__converseD,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(converse_int_int(R2)))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X),transi2902850859426252137cl_int(R2)) ) ).

% rtrancl_converseD
tff(fact_2604_trancl__converseI,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),converse_nat_nat(transi6264000038957366511cl_nat(R2)))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(converse_nat_nat(R2))) ) ).

% trancl_converseI
tff(fact_2605_trancl__converseI,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),converse_int_int(transi6261509568448316235cl_int(R2)))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(converse_int_int(R2))) ) ).

% trancl_converseI
tff(fact_2606_trancl__converseD,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi6264000038957366511cl_nat(converse_nat_nat(R2)))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),converse_nat_nat(transi6264000038957366511cl_nat(R2))) ) ).

% trancl_converseD
tff(fact_2607_trancl__converseD,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi6261509568448316235cl_int(converse_int_int(R2)))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),converse_int_int(transi6261509568448316235cl_int(R2))) ) ).

% trancl_converseD
tff(fact_2608_diff__shunt__var,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( minus_6452836326544984404t_unit(X,Y) = bot_bo3957492148770167129t_unit )
    <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,X),Y) ) ).

% diff_shunt_var
tff(fact_2609_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( minus_minus_set_o(X,Y) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X),Y) ) ).

% diff_shunt_var
tff(fact_2610_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_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_2611_diff__shunt__var,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] :
      ( ( minus_6456595905716665654_nat_o(X,Y) = bot_bo8632255151191677115_nat_o )
    <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(X),Y) ) ).

% diff_shunt_var
tff(fact_2612_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_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_2613_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_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_2614_in__snd__imageE,axiom,
    ! [Y: nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(Y),image_2486076414777270412at_nat(product_snd_nat_nat,S))
     => ~ ! [X3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y),S) ) ).

% in_snd_imageE
tff(fact_2615_in__snd__imageE,axiom,
    ! [Y: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat] :
      ( member6260224972018164377et_nat(Y,image_5643293374885392937et_nat(produc4960056325182568940et_nat,S))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] : ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y),S) ) ).

% in_snd_imageE
tff(fact_2616_in__snd__imageE,axiom,
    ! [Y: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat] :
      ( member4513763875858039863et_nat(Y,image_1425188809386248805et_nat(produc9081944055658670602et_nat,S))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] : ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y),S) ) ).

% in_snd_imageE
tff(fact_2617_in__snd__imageE,axiom,
    ! [Y: set_nat,S: set_Pr3948176798113811640et_nat] :
      ( member_set_nat(Y,image_6088997261112373647et_nat(produc8586169260539613262et_nat,S))
     => ~ ! [X3: heap_e7401611519738050253t_unit] : ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y),S) ) ).

% in_snd_imageE
tff(fact_2618_in__snd__imageE,axiom,
    ! [Y: int,S: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(Y),image_5042161079198086560nt_int(product_snd_int_int,S))
     => ~ ! [X3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y),S) ) ).

% in_snd_imageE
tff(fact_2619_image__Int__subset,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] : aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),inf_in1364745209274528805nteger(A,B4))),inf_in1364745209274528805nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),B4))) ).

% image_Int_subset
tff(fact_2620_image__Int__subset,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_nat] : ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)),inf_inf_set_set_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4))) ).

% image_Int_subset
tff(fact_2621_image__Int__subset,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_nat] : aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),inf_in1364745209274528805nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),B4))) ).

% image_Int_subset
tff(fact_2622_image__Int__subset,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)),aa_set_nat_set_nat(image_nat_nat2(F),B4))) ).

% image_Int_subset
tff(fact_2623_subset__minus__empty,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( minus_6452836326544984404t_unit(A,B4) = bot_bo3957492148770167129t_unit ) ) ).

% subset_minus_empty
tff(fact_2624_subset__minus__empty,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( minus_minus_set_o(A,B4) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_2625_subset__minus__empty,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( minus_minus_set_int(A,B4) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_2626_subset__minus__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
     => ( minus_6456595905716665654_nat_o(A,B4) = bot_bo8632255151191677115_nat_o ) ) ).

% subset_minus_empty
tff(fact_2627_subset__minus__empty,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_set_nat(minus_minus_set_nat(A),B4) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_2628_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)),B4)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),vimage1318793397690208119nteger(F,B4)) ) ).

% image_subset_iff_subset_vimage
tff(fact_2629_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),B4)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),vimage_nat_set_nat(F,B4)) ) ).

% image_subset_iff_subset_vimage
tff(fact_2630_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)),B4)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),vimage6104394758324456170nteger(F,B4)) ) ).

% image_subset_iff_subset_vimage
tff(fact_2631_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)),B4)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(vimage_nat_nat(F),B4)) ) ).

% image_subset_iff_subset_vimage
tff(fact_2632_image__vimage__subset,axiom,
    ! [F: fun_nat_set_nat,A: set_set_nat] : ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),vimage_nat_set_nat(F,A)),A) ).

% image_vimage_subset
tff(fact_2633_image__vimage__subset,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] : aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),vimage1318793397690208119nteger(F,A))),A) ).

% image_vimage_subset
tff(fact_2634_image__vimage__subset,axiom,
    ! [F: fun_nat_Code_integer,A: set_Code_integer] : aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),vimage6104394758324456170nteger(F,A))),A) ).

% image_vimage_subset
tff(fact_2635_image__vimage__subset,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(vimage_nat_nat(F),A))),A) ).

% image_vimage_subset
tff(fact_2636_Diff__insert,axiom,
    ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] : minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = minus_6452836326544984404t_unit(minus_6452836326544984404t_unit(A,B4),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) ).

% Diff_insert
tff(fact_2637_Diff__insert,axiom,
    ! [A: set_o,A2: $o,B4: set_o] : minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = minus_minus_set_o(minus_minus_set_o(A,B4),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) ).

% Diff_insert
tff(fact_2638_Diff__insert,axiom,
    ! [A: set_int,A2: int,B4: set_int] : minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = minus_minus_set_int(minus_minus_set_int(A,B4),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) ).

% Diff_insert
tff(fact_2639_Diff__insert,axiom,
    ! [A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] : minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) = minus_6456595905716665654_nat_o(minus_6456595905716665654_nat_o(A,B4),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) ).

% Diff_insert
tff(fact_2640_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B4: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) ).

% Diff_insert
tff(fact_2641_insert__Diff,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ( aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = A ) ) ).

% insert_Diff
tff(fact_2642_insert__Diff,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = A ) ) ).

% insert_Diff
tff(fact_2643_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = A ) ) ).

% insert_Diff
tff(fact_2644_insert__Diff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ( aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) = A ) ) ).

% insert_Diff
tff(fact_2645_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = A ) ) ).

% insert_Diff
tff(fact_2646_Diff__insert2,axiom,
    ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] : minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4)) = minus_6452836326544984404t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)),B4) ).

% Diff_insert2
tff(fact_2647_Diff__insert2,axiom,
    ! [A: set_o,A2: $o,B4: set_o] : minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4)) = minus_minus_set_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)),B4) ).

% Diff_insert2
tff(fact_2648_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B4: set_int] : minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4)) = minus_minus_set_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)),B4) ).

% Diff_insert2
tff(fact_2649_Diff__insert2,axiom,
    ! [A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] : minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4)) = minus_6456595905716665654_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)),B4) ).

% Diff_insert2
tff(fact_2650_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B4: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))),B4) ).

% Diff_insert2
tff(fact_2651_insert__minus__eq,axiom,
    ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
      ( ( X != Y )
     => ( minus_6452836326544984404t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit)) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit))) ) ) ).

% insert_minus_eq
tff(fact_2652_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ~ ( (X)
        <=> (Y) )
     => ( minus_minus_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o)) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o))) ) ) ).

% insert_minus_eq
tff(fact_2653_insert__minus__eq,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( X != Y )
     => ( minus_minus_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int)) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_2654_insert__minus__eq,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( X != Y )
     => ( minus_6456595905716665654_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o)) = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o))) ) ) ).

% insert_minus_eq
tff(fact_2655_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( X != Y )
     => ( aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_2656_Diff__insert__absorb,axiom,
    ! [X: product_unit,A: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ( minus_6452836326544984404t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = A ) ) ).

% Diff_insert_absorb
tff(fact_2657_Diff__insert__absorb,axiom,
    ! [X: $o,A: set_o] :
      ( ~ aa_set_o_o(member_o((X)),A)
     => ( minus_minus_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = A ) ) ).

% Diff_insert_absorb
tff(fact_2658_Diff__insert__absorb,axiom,
    ! [X: int,A: set_int] :
      ( ~ aa_set_int_o(member_int(X),A)
     => ( minus_minus_set_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = A ) ) ).

% Diff_insert_absorb
tff(fact_2659_Diff__insert__absorb,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ( minus_6456595905716665654_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = A ) ) ).

% Diff_insert_absorb
tff(fact_2660_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = A ) ) ).

% Diff_insert_absorb
tff(fact_2661_set__minus__singleton__eq,axiom,
    ! [X: product_unit,X5: set_Product_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),X5)
     => ( minus_6452836326544984404t_unit(X5,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2662_set__minus__singleton__eq,axiom,
    ! [X: $o,X5: set_o] :
      ( ~ aa_set_o_o(member_o((X)),X5)
     => ( minus_minus_set_o(X5,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2663_set__minus__singleton__eq,axiom,
    ! [X: int,X5: set_int] :
      ( ~ aa_set_int_o(member_int(X),X5)
     => ( minus_minus_set_int(X5,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2664_set__minus__singleton__eq,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,X5: set_fu3555266933746557007_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),X5)
     => ( minus_6456595905716665654_nat_o(X5,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2665_set__minus__singleton__eq,axiom,
    ! [X: nat,X5: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),X5)
     => ( aa_set_nat_set_nat(minus_minus_set_nat(X5),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2666_subset__Diff__insert,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,C5: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),minus_6456595905716665654_nat_o(B4,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),C5)))
    <=> ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),minus_6456595905716665654_nat_o(B4,C5))
        & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A) ) ) ).

% subset_Diff_insert
tff(fact_2667_subset__Diff__insert,axiom,
    ! [A: set_int,B4: set_int,X: int,C5: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),minus_minus_set_int(B4,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),C5)))
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),minus_minus_set_int(B4,C5))
        & ~ aa_set_int_o(member_int(X),A) ) ) ).

% subset_Diff_insert
tff(fact_2668_subset__Diff__insert,axiom,
    ! [A: set_o,B4: set_o,X: $o,C5: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),minus_minus_set_o(B4,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),C5)))
    <=> ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),minus_minus_set_o(B4,C5))
        & ~ aa_set_o_o(member_o((X)),A) ) ) ).

% subset_Diff_insert
tff(fact_2669_subset__Diff__insert,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit,C5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),minus_6452836326544984404t_unit(B4,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),C5)))
    <=> ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),minus_6452836326544984404t_unit(B4,C5))
        & ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A) ) ) ).

% subset_Diff_insert
tff(fact_2670_subset__Diff__insert,axiom,
    ! [A: set_nat,B4: set_nat,X: nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(minus_minus_set_nat(B4),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),C5)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(minus_minus_set_nat(B4),C5))
        & ~ aa_set_nat_o(member_nat(X),A) ) ) ).

% subset_Diff_insert
tff(fact_2671_Diff__triv,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
     => ( minus_6452836326544984404t_unit(A,B4) = A ) ) ).

% Diff_triv
tff(fact_2672_Diff__triv,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
     => ( minus_minus_set_o(A,B4) = A ) ) ).

% Diff_triv
tff(fact_2673_Diff__triv,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
     => ( minus_minus_set_int(A,B4) = A ) ) ).

% Diff_triv
tff(fact_2674_Diff__triv,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o )
     => ( minus_6456595905716665654_nat_o(A,B4) = A ) ) ).

% Diff_triv
tff(fact_2675_Diff__triv,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(minus_minus_set_nat(A),B4) = A ) ) ).

% Diff_triv
tff(fact_2676_Int__Diff__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] : inf_in4660618365625256667t_unit(inf_in4660618365625256667t_unit(A,B4),minus_6452836326544984404t_unit(A,B4)) = bot_bo3957492148770167129t_unit ).

% Int_Diff_disjoint
tff(fact_2677_Int__Diff__disjoint,axiom,
    ! [A: set_o,B4: set_o] : inf_inf_set_o(inf_inf_set_o(A,B4),minus_minus_set_o(A,B4)) = bot_bot_set_o ).

% Int_Diff_disjoint
tff(fact_2678_Int__Diff__disjoint,axiom,
    ! [A: set_int,B4: set_int] : inf_inf_set_int(inf_inf_set_int(A,B4),minus_minus_set_int(A,B4)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_2679_Int__Diff__disjoint,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(inf_in3376754634429691197_nat_o(A,B4),minus_6456595905716665654_nat_o(A,B4)) = bot_bo8632255151191677115_nat_o ).

% Int_Diff_disjoint
tff(fact_2680_Int__Diff__disjoint,axiom,
    ! [A: 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,A),B4)),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_2681_disjoint__alt__simp1,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( minus_6452836326544984404t_unit(A,B4) = A )
    <=> ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit ) ) ).

% disjoint_alt_simp1
tff(fact_2682_disjoint__alt__simp1,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A,B4) = A )
    <=> ( inf_inf_set_o(A,B4) = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
tff(fact_2683_disjoint__alt__simp1,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A,B4) = A )
    <=> ( inf_inf_set_int(A,B4) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_2684_disjoint__alt__simp1,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( minus_6456595905716665654_nat_o(A,B4) = A )
    <=> ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o ) ) ).

% disjoint_alt_simp1
tff(fact_2685_disjoint__alt__simp1,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(minus_minus_set_nat(A),B4) = A )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2686_disjoint__alt__simp2,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( minus_6452836326544984404t_unit(A,B4) != A )
    <=> ( inf_in4660618365625256667t_unit(A,B4) != bot_bo3957492148770167129t_unit ) ) ).

% disjoint_alt_simp2
tff(fact_2687_disjoint__alt__simp2,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A,B4) != A )
    <=> ( inf_inf_set_o(A,B4) != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
tff(fact_2688_disjoint__alt__simp2,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A,B4) != A )
    <=> ( inf_inf_set_int(A,B4) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_2689_disjoint__alt__simp2,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( minus_6456595905716665654_nat_o(A,B4) != A )
    <=> ( inf_in3376754634429691197_nat_o(A,B4) != bot_bo8632255151191677115_nat_o ) ) ).

% disjoint_alt_simp2
tff(fact_2690_disjoint__alt__simp2,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(minus_minus_set_nat(A),B4) != A )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2691_Diff__subset__conv,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),C5)
    <=> 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,B4),C5)) ) ).

% Diff_subset_conv
tff(fact_2692_Diff__partition,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)) = B4 ) ) ).

% Diff_partition
tff(fact_2693_Un__Diff__Int,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) = A ).

% Un_Diff_Int
tff(fact_2694_Int__Diff__Un,axiom,
    ! [A: 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,A),B4)),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) = A ).

% Int_Diff_Un
tff(fact_2695_Diff__Int,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B4),C5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set_nat_set_nat(minus_minus_set_nat(A),C5)) ).

% Diff_Int
tff(fact_2696_Diff__Un,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B4),C5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set_nat_set_nat(minus_minus_set_nat(A),C5)) ).

% Diff_Un
tff(fact_2697_Range__snd,axiom,
    ! [R2: set_Pr3948176798113811640et_nat] : range_9132311542094983838et_nat(R2) = image_6088997261112373647et_nat(produc8586169260539613262et_nat,R2) ).

% Range_snd
tff(fact_2698_Range__snd,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : range_int_int(R2) = image_5042161079198086560nt_int(product_snd_int_int,R2) ).

% Range_snd
tff(fact_2699_snd__eq__Range,axiom,
    ! [R: set_Pr3948176798113811640et_nat] : image_6088997261112373647et_nat(produc8586169260539613262et_nat,R) = range_9132311542094983838et_nat(R) ).

% snd_eq_Range
tff(fact_2700_snd__eq__Range,axiom,
    ! [R: set_Pr958786334691620121nt_int] : image_5042161079198086560nt_int(product_snd_int_int,R) = range_int_int(R) ).

% snd_eq_Range
tff(fact_2701_the__elem__image__unique,axiom,
    ! [A: set_Code_integer,F: fun_Co8499238673658658103nteger,X: code_integer] :
      ( ( A != bot_bo3990330152332043303nteger )
     => ( ! [Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),A)
           => ( aa_Cod1783974652533396728nteger(F,Y2) = aa_Cod1783974652533396728nteger(F,X) ) )
       => ( the_el4656377885804969768nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) = aa_Cod1783974652533396728nteger(F,X) ) ) ) ).

% the_elem_image_unique
tff(fact_2702_the__elem__image__unique,axiom,
    ! [A: set_nat,F: fun_nat_set_nat,X: nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),A)
           => ( aa_nat_set_nat(F,Y2) = aa_nat_set_nat(F,X) ) )
       => ( the_elem_set_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A)) = aa_nat_set_nat(F,X) ) ) ) ).

% the_elem_image_unique
tff(fact_2703_the__elem__image__unique,axiom,
    ! [A: set_nat,F: fun_nat_nat,X: nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),A)
           => ( aa_nat_nat(F,Y2) = aa_nat_nat(F,X) ) )
       => ( aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)) = aa_nat_nat(F,X) ) ) ) ).

% the_elem_image_unique
tff(fact_2704_the__elem__image__unique,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer,X: nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),A)
           => ( aa_nat_Code_integer(F,Y2) = aa_nat_Code_integer(F,X) ) )
       => ( the_el4656377885804969768nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)) = aa_nat_Code_integer(F,X) ) ) ) ).

% the_elem_image_unique
tff(fact_2705_cyclicE,axiom,
    ! [G: set_Pr1261947904930325089at_nat] :
      ( ~ transi3139058337892597825ic_nat(G)
     => ~ ! [X3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3),transi6264000038957366511cl_nat(G)) ) ).

% cyclicE
tff(fact_2706_cyclicE,axiom,
    ! [G: set_Pr958786334691620121nt_int] :
      ( ~ transi3136567867383547549ic_int(G)
     => ~ ! [X3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),X3),transi6261509568448316235cl_int(G)) ) ).

% cyclicE
tff(fact_2707_acyclic__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( transi3139058337892597825ic_nat(R2)
    <=> ! [X2: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),X2),transi6264000038957366511cl_nat(R2)) ) ).

% acyclic_def
tff(fact_2708_acyclic__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( transi3136567867383547549ic_int(R2)
    <=> ! [X2: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),X2),transi6261509568448316235cl_int(R2)) ) ).

% acyclic_def
tff(fact_2709_acyclicI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),X3),transi6264000038957366511cl_nat(R2))
     => transi3139058337892597825ic_nat(R2) ) ).

% acyclicI
tff(fact_2710_acyclicI,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),X3),transi6261509568448316235cl_int(R2))
     => transi3136567867383547549ic_int(R2) ) ).

% acyclicI
tff(fact_2711_snd__image__mp,axiom,
    ! [A: set_Pr1312406793078187350et_nat,B4: set_Pr3948176798113811640et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] :
      ( ord_le8090042543978450200et_nat(image_5643293374885392937et_nat(produc4960056325182568940et_nat,A),B4)
     => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y),A)
       => member6260224972018164377et_nat(Y,B4) ) ) ).

% snd_image_mp
tff(fact_2712_snd__image__mp,axiom,
    ! [A: set_Pr4040394263813967220et_nat,B4: set_Pr1312406793078187350et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] :
      ( ord_le3919382008627836342et_nat(image_1425188809386248805et_nat(produc9081944055658670602et_nat,A),B4)
     => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y),A)
       => member4513763875858039863et_nat(Y,B4) ) ) ).

% snd_image_mp
tff(fact_2713_snd__image__mp,axiom,
    ! [A: set_Pr3948176798113811640et_nat,B4: set_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(image_6088997261112373647et_nat(produc8586169260539613262et_nat,A),B4)
     => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y),A)
       => member_set_nat(Y,B4) ) ) ).

% snd_image_mp
tff(fact_2714_snd__image__mp,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_5042161079198086560nt_int(product_snd_int_int,A)),B4)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),A)
       => aa_set_int_o(member_int(Y),B4) ) ) ).

% snd_image_mp
tff(fact_2715_snd__image__mp,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(product_snd_nat_nat,A)),B4)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),A)
       => aa_set_nat_o(member_nat(Y),B4) ) ) ).

% snd_image_mp
tff(fact_2716_snd__in__Field,axiom,
    ! [R: set_Pr958786334691620121nt_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_5042161079198086560nt_int(product_snd_int_int,R)),field_int(R)) ).

% snd_in_Field
tff(fact_2717_snd__in__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(product_snd_nat_nat,R)),field_nat(R)) ).

% snd_in_Field
tff(fact_2718_Image__subset__snd__image,axiom,
    ! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit] : ord_le6893508408891458716et_nat(image_3942343239045813184et_nat(A,B4),image_6088997261112373647et_nat(produc8586169260539613262et_nat,A)) ).

% Image_subset_snd_image
tff(fact_2719_Image__subset__snd__image,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_int_int(A,B4)),image_5042161079198086560nt_int(product_snd_int_int,A)) ).

% Image_subset_snd_image
tff(fact_2720_infinite__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),bot_bo3990330152332043303nteger))) ) ).

% infinite_remove
tff(fact_2721_infinite__remove,axiom,
    ! [S: set_Product_unit,A2: product_unit] :
      ( ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,minus_6452836326544984404t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) ) ).

% infinite_remove
tff(fact_2722_infinite__remove,axiom,
    ! [S: set_o,A2: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_2723_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_2724_infinite__remove,axiom,
    ! [S: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,minus_6456595905716665654_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) ) ).

% infinite_remove
tff(fact_2725_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(minus_minus_set_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_2726_infinite__coinduct,axiom,
    ! [X5: fun_se7289222521248229150eger_o,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(X5,A)
     => ( ! [A8: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(X5,A8)
           => ? [X4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A8)
                & ( aa_set5643387332196575111eger_o(X5,minus_2355218937544613996nteger(A8,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X4),bot_bo3990330152332043303nteger)))
                  | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A8,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X4),bot_bo3990330152332043303nteger))) ) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% infinite_coinduct
tff(fact_2727_infinite__coinduct,axiom,
    ! [X5: fun_se4342725320470837934unit_o,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(X5,A)
     => ( ! [A8: set_Product_unit] :
            ( aa_set8780210009671629607unit_o(X5,A8)
           => ? [X4: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A8)
                & ( aa_set8780210009671629607unit_o(X5,minus_6452836326544984404t_unit(A8,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X4),bot_bo3957492148770167129t_unit)))
                  | ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,minus_6452836326544984404t_unit(A8,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X4),bot_bo3957492148770167129t_unit))) ) ) )
       => ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A) ) ) ).

% infinite_coinduct
tff(fact_2728_infinite__coinduct,axiom,
    ! [X5: fun_set_o_o,A: set_o] :
      ( aa_set_o_o(X5,A)
     => ( ! [A8: set_o] :
            ( aa_set_o_o(X5,A8)
           => ? [X4: $o] :
                ( aa_set_o_o(member_o((X4)),A8)
                & ( aa_set_o_o(X5,minus_minus_set_o(A8,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X4)),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A8,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X4)),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A) ) ) ).

% infinite_coinduct
tff(fact_2729_infinite__coinduct,axiom,
    ! [X5: fun_set_int_o,A: set_int] :
      ( aa_set_int_o(X5,A)
     => ( ! [A8: set_int] :
            ( aa_set_int_o(X5,A8)
           => ? [X4: int] :
                ( aa_set_int_o(member_int(X4),A8)
                & ( aa_set_int_o(X5,minus_minus_set_int(A8,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X4),bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A8,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X4),bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A) ) ) ).

% infinite_coinduct
tff(fact_2730_infinite__coinduct,axiom,
    ! [X5: fun_se7459991829846211660at_o_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(X5,A)
     => ( ! [A8: set_fu3555266933746557007_nat_o] :
            ( aa_set5091322549519198405at_o_o(X5,A8)
           => ? [X4: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A8)
                & ( aa_set5091322549519198405at_o_o(X5,minus_6456595905716665654_nat_o(A8,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),bot_bo8632255151191677115_nat_o)))
                  | ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,minus_6456595905716665654_nat_o(A8,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),bot_bo8632255151191677115_nat_o))) ) ) )
       => ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A) ) ) ).

% infinite_coinduct
tff(fact_2731_infinite__coinduct,axiom,
    ! [X5: fun_set_nat_o,A: set_nat] :
      ( aa_set_nat_o(X5,A)
     => ( ! [A8: set_nat] :
            ( aa_set_nat_o(X5,A8)
           => ? [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),A8)
                & ( aa_set_nat_o(X5,aa_set_nat_set_nat(minus_minus_set_nat(A8),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X4),bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(minus_minus_set_nat(A8),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X4),bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A) ) ) ).

% infinite_coinduct
tff(fact_2732_finite__empty__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(P,A)
       => ( ! [A3: code_integer,A8: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8)
             => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A8)
               => ( aa_set5643387332196575111eger_o(P,A8)
                 => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A8,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A3),bot_bo3990330152332043303nteger))) ) ) )
         => aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger) ) ) ) ).

% finite_empty_induct
tff(fact_2733_finite__empty__induct,axiom,
    ! [A: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(P,A)
       => ( ! [A3: product_unit,A8: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8)
             => ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A8)
               => ( aa_set8780210009671629607unit_o(P,A8)
                 => aa_set8780210009671629607unit_o(P,minus_6452836326544984404t_unit(A8,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A3),bot_bo3957492148770167129t_unit))) ) ) )
         => aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit) ) ) ) ).

% finite_empty_induct
tff(fact_2734_finite__empty__induct,axiom,
    ! [A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(P,A)
       => ( ! [A3: $o,A8: set_o] :
              ( aa_set_o_o(finite_finite_o,A8)
             => ( aa_set_o_o(member_o((A3)),A8)
               => ( aa_set_o_o(P,A8)
                 => aa_set_o_o(P,minus_minus_set_o(A8,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A3)),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_2735_finite__empty__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,A)
       => ( ! [A3: int,A8: set_int] :
              ( aa_set_int_o(finite_finite_int,A8)
             => ( aa_set_int_o(member_int(A3),A8)
               => ( aa_set_int_o(P,A8)
                 => aa_set_int_o(P,minus_minus_set_int(A8,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A3),bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_2736_finite__empty__induct,axiom,
    ! [A: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(P,A)
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,A8: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A8)
             => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A3),A8)
               => ( aa_set5091322549519198405at_o_o(P,A8)
                 => aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A8,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A3),bot_bo8632255151191677115_nat_o))) ) ) )
         => aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o) ) ) ) ).

% finite_empty_induct
tff(fact_2737_finite__empty__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,A)
       => ( ! [A3: nat,A8: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A8)
             => ( aa_set_nat_o(member_nat(A3),A8)
               => ( aa_set_nat_o(P,A8)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(minus_minus_set_nat(A8),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A3),bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_2738_subset__insert__iff,axiom,
    ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4))
    <=> $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))),B4),aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)) ) ).

% subset_insert_iff
tff(fact_2739_subset__insert__iff,axiom,
    ! [A: set_o,X: $o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4))
    <=> $ite(aa_set_o_o(member_o((X)),A),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))),B4),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)) ) ).

% subset_insert_iff
tff(fact_2740_subset__insert__iff,axiom,
    ! [A: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4))
    <=> $ite(aa_set_int_o(member_int(X),A),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))),B4),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)) ) ).

% subset_insert_iff
tff(fact_2741_subset__insert__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B4))
    <=> $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),B4),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)) ) ).

% subset_insert_iff
tff(fact_2742_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4))
    <=> $ite(aa_set_nat_o(member_nat(X),A),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))),B4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)) ) ).

% subset_insert_iff
tff(fact_2743_Diff__single__insert,axiom,
    ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))),B4)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4)) ) ).

% Diff_single_insert
tff(fact_2744_Diff__single__insert,axiom,
    ! [A: set_o,X: $o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))),B4)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4)) ) ).

% Diff_single_insert
tff(fact_2745_Diff__single__insert,axiom,
    ! [A: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))),B4)
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4)) ) ).

% Diff_single_insert
tff(fact_2746_Diff__single__insert,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),B4)
     => aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B4)) ) ).

% Diff_single_insert
tff(fact_2747_Diff__single__insert,axiom,
    ! [A: 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(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4)) ) ).

% Diff_single_insert
tff(fact_2748_Field__rel__restrict,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,field_nat(rel_restrict_nat(R,A))),aa_set_nat_set_nat(minus_minus_set_nat(field_nat(R)),A)) ).

% Field_rel_restrict
tff(fact_2749_Domain__rel__restrict,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,domain_nat_nat(rel_restrict_nat(R,A))),aa_set_nat_set_nat(minus_minus_set_nat(domain_nat_nat(R)),A)) ).

% Domain_rel_restrict
tff(fact_2750_Range__rel__restrict,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,range_nat_nat(rel_restrict_nat(R,A))),aa_set_nat_set_nat(minus_minus_set_nat(range_nat_nat(R)),A)) ).

% Range_rel_restrict
tff(fact_2751_inf__img__fin__domE,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
             => aa_set_nat_o(finite_finite_nat,vimage6104394758324456170nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2752_inf__img__fin__domE,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage1318793397690208119nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2753_inf__img__fin__domE,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_8730104196221521654t_unit(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),image_8730104196221521654t_unit(F,A))
             => aa_set_nat_o(finite_finite_nat,vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2754_inf__img__fin__domE,axiom,
    ! [F: fun_Co1365676809456677961t_unit,A: set_Code_integer] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_7182656606676592809t_unit(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),image_7182656606676592809t_unit(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage5651688891684837235t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2755_inf__img__fin__domE,axiom,
    ! [F: fun_nat_o,A: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: $o] :
              ( aa_set_o_o(member_o((Y2)),image_nat_o2(F,A))
             => aa_set_nat_o(finite_finite_nat,vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2756_inf__img__fin__domE,axiom,
    ! [F: fun_Code_integer_o,A: set_Code_integer] :
      ( aa_set_o_o(finite_finite_o,image_Code_integer_o(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: $o] :
              ( aa_set_o_o(member_o((Y2)),image_Code_integer_o(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage3005800680117336574eger_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2757_inf__img__fin__domE,axiom,
    ! [F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: nat] :
              ( aa_set_nat_o(member_nat(Y2),aa_set_nat_set_nat(image_nat_nat2(F),A))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2758_inf__img__fin__domE,axiom,
    ! [F: fun_Code_integer_nat,A: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: nat] :
              ( aa_set_nat_o(member_nat(Y2),image_951025933927791156er_nat(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage5839839309545413354er_nat(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2759_inf__img__fin__domE,axiom,
    ! [F: fun_nat_int,A: set_nat] :
      ( aa_set_int_o(finite_finite_int,image_nat_int2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ~ ! [Y2: int] :
              ( aa_set_int_o(member_int(Y2),image_nat_int2(F,A))
             => aa_set_nat_o(finite_finite_nat,vimage_nat_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2760_inf__img__fin__domE,axiom,
    ! [F: fun_Code_integer_int,A: set_Code_integer] :
      ( aa_set_int_o(finite_finite_int,image_948535463418740880er_int(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ~ ! [Y2: int] :
              ( aa_set_int_o(member_int(Y2),image_948535463418740880er_int(F,A))
             => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage5837348839036363078er_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int))) ) ) ) ).

% inf_img_fin_domE
tff(fact_2761_inf__img__fin__dom,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
            & ~ aa_set_nat_o(finite_finite_nat,vimage6104394758324456170nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),bot_bo3990330152332043303nteger))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2762_inf__img__fin__dom,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage1318793397690208119nteger(F,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),bot_bo3990330152332043303nteger))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2763_inf__img__fin__dom,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_8730104196221521654t_unit(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),image_8730104196221521654t_unit(F,A))
            & ~ aa_set_nat_o(finite_finite_nat,vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2764_inf__img__fin__dom,axiom,
    ! [F: fun_Co1365676809456677961t_unit,A: set_Code_integer] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,image_7182656606676592809t_unit(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),image_7182656606676592809t_unit(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage5651688891684837235t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2765_inf__img__fin__dom,axiom,
    ! [F: fun_nat_o,A: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),image_nat_o2(F,A))
            & ~ aa_set_nat_o(finite_finite_nat,vimage_nat_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2766_inf__img__fin__dom,axiom,
    ! [F: fun_Code_integer_o,A: set_Code_integer] :
      ( aa_set_o_o(finite_finite_o,image_Code_integer_o(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),image_Code_integer_o(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage3005800680117336574eger_o(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2767_inf__img__fin__dom,axiom,
    ! [F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(image_nat_nat2(F),A))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2768_inf__img__fin__dom,axiom,
    ! [F: fun_Code_integer_nat,A: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),image_951025933927791156er_nat(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage5839839309545413354er_nat(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2769_inf__img__fin__dom,axiom,
    ! [F: fun_nat_int,A: set_nat] :
      ( aa_set_int_o(finite_finite_int,image_nat_int2(F,A))
     => ( ~ aa_set_nat_o(finite_finite_nat,A)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),image_nat_int2(F,A))
            & ~ aa_set_nat_o(finite_finite_nat,vimage_nat_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2770_inf__img__fin__dom,axiom,
    ! [F: fun_Code_integer_int,A: set_Code_integer] :
      ( aa_set_int_o(finite_finite_int,image_948535463418740880er_int(F,A))
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),image_948535463418740880er_int(F,A))
            & ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,vimage5837348839036363078er_int(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int))) ) ) ) ).

% inf_img_fin_dom
tff(fact_2771_acyclic__insert__cyclic,axiom,
    ! [G: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( transi3139058337892597825ic_nat(G)
     => ( ~ transi3139058337892597825ic_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),G))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X),transi2905341329935302413cl_nat(G)) ) ) ).

% acyclic_insert_cyclic
tff(fact_2772_acyclic__insert__cyclic,axiom,
    ! [G: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( transi3136567867383547549ic_int(G)
     => ( ~ transi3136567867383547549ic_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),G))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X),transi2902850859426252137cl_int(G)) ) ) ).

% acyclic_insert_cyclic
tff(fact_2773_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_set_nat_set_nat,N: set_set_nat] :
      ( ! [X3: set_nat,Y2: set_nat] : aa_set_nat_set_nat(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(Ha,X3)),aa_set_nat_set_nat(Ha,Y2))
     => ( finite1152437895449049373et_nat(N)
       => ( ( N != bot_bot_set_set_nat )
         => ( aa_set_nat_set_nat(Ha,lattic3014633134055518761et_nat(N)) = lattic3014633134055518761et_nat(image_7916887816326733075et_nat(Ha,N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2774_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_assn_assn,N: set_assn] :
      ( ! [X3: assn,Y2: assn] : aa_assn_assn(Ha,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X3),Y2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(Ha,X3)),aa_assn_assn(Ha,Y2))
     => ( finite_finite_assn(N)
       => ( ( N != bot_bot_set_assn )
         => ( aa_assn_assn(Ha,lattic47131356835913163n_assn(N)) = lattic47131356835913163n_assn(image_assn_assn(Ha,N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2775_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_Pr2191293291710260995t_unit,N: set_Product_unit] :
      ( ! [X3: product_unit,Y2: product_unit] : aa_Pro8294289767544744956t_unit(Ha,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X3),Y2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(Ha,X3)),aa_Pro8294289767544744956t_unit(Ha,Y2))
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,N)
       => ( ( N != bot_bo3957492148770167129t_unit )
         => ( aa_Pro8294289767544744956t_unit(Ha,lattic1263872656861969706t_unit(N)) = lattic1263872656861969706t_unit(image_405062704495631173t_unit(Ha,N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2776_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_o_o,N: set_o] :
      ( ! [X3: $o,Y2: $o] :
          ( aa_o_o(Ha,aa_o_o(aa_o_fun_o_o(inf_inf_o,(X3)),(Y2)))
        <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_o_o(Ha,(X3))),aa_o_o(Ha,(Y2))) )
     => ( aa_set_o_o(finite_finite_o,N)
       => ( ( N != bot_bot_set_o )
         => ( aa_o_o(Ha,aa_set_o_o(lattic4107685809792843317_fin_o,N))
          <=> aa_set_o_o(lattic4107685809792843317_fin_o,image_o_o2(Ha,N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2777_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_nat_nat,N: set_nat] :
      ( ! [X3: nat,Y2: nat] : aa_nat_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X3),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(Ha,X3)),aa_nat_nat(Ha,Y2))
     => ( aa_set_nat_o(finite_finite_nat,N)
       => ( ( N != bot_bot_set_nat )
         => ( aa_nat_nat(Ha,aa_set_nat_nat(lattic5238388535129920115in_nat,N)) = aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(image_nat_nat2(Ha),N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2778_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_int_int,N: set_int] :
      ( ! [X3: int,Y2: int] : aa_int_int(Ha,aa_int_int(aa_int_fun_int_int(inf_inf_int,X3),Y2)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(Ha,X3)),aa_int_int(Ha,Y2))
     => ( aa_set_int_o(finite_finite_int,N)
       => ( ( N != bot_bot_set_int )
         => ( aa_int_int(Ha,lattic5235898064620869839in_int(N)) = lattic5235898064620869839in_int(image_int_int2(Ha,N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2779_Inf__fin_Ohom__commute,axiom,
    ! [Ha: fun_fu8419763957888848259_nat_o,N: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_fun2638059832795814268_nat_o(Ha,aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X3),Y2)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(aa_fun2638059832795814268_nat_o(Ha,X3)),aa_fun2638059832795814268_nat_o(Ha,Y2))
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,N)
       => ( ( N != bot_bo8632255151191677115_nat_o )
         => ( aa_fun2638059832795814268_nat_o(Ha,lattic6089491107740071820_nat_o(N)) = lattic6089491107740071820_nat_o(image_6551304387237727173_nat_o(Ha,N)) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_2780_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_set_nat_set_nat,N: set_set_nat] :
      ( ! [X3: set_nat,Y2: set_nat] : aa_set_nat_set_nat(Ha,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(Ha,X3)),aa_set_nat_set_nat(Ha,Y2))
     => ( finite1152437895449049373et_nat(N)
       => ( ( N != bot_bot_set_set_nat )
         => ( aa_set_nat_set_nat(Ha,lattic3835124923745554447et_nat(N)) = lattic3835124923745554447et_nat(image_7916887816326733075et_nat(Ha,N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2781_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_assn_assn,N: set_assn] :
      ( ! [X3: assn,Y2: assn] : aa_assn_assn(Ha,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X3),Y2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(Ha,X3)),aa_assn_assn(Ha,Y2))
     => ( finite_finite_assn(N)
       => ( ( N != bot_bot_set_assn )
         => ( aa_assn_assn(Ha,lattic2150320897289308081n_assn(N)) = lattic2150320897289308081n_assn(image_assn_assn(Ha,N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2782_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_Pr2191293291710260995t_unit,N: set_Product_unit] :
      ( ! [X3: product_unit,Y2: product_unit] : aa_Pro8294289767544744956t_unit(Ha,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X3),Y2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(Ha,X3)),aa_Pro8294289767544744956t_unit(Ha,Y2))
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,N)
       => ( ( N != bot_bo3957492148770167129t_unit )
         => ( aa_Pro8294289767544744956t_unit(Ha,lattic5294303975357428420t_unit(N)) = lattic5294303975357428420t_unit(image_405062704495631173t_unit(Ha,N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2783_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_o_o,N: set_o] :
      ( ! [X3: $o,Y2: $o] :
          ( aa_o_o(Ha,aa_o_o(aa_o_fun_o_o(sup_sup_o,(X3)),(Y2)))
        <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_o_o(Ha,(X3))),aa_o_o(Ha,(Y2))) )
     => ( aa_set_o_o(finite_finite_o,N)
       => ( ( N != bot_bot_set_o )
         => ( aa_o_o(Ha,aa_set_o_o(lattic1508158080041050831_fin_o,N))
          <=> aa_set_o_o(lattic1508158080041050831_fin_o,image_o_o2(Ha,N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2784_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_nat_nat,N: set_nat] :
      ( ! [X3: nat,Y2: nat] : aa_nat_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X3),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(Ha,X3)),aa_nat_nat(Ha,Y2))
     => ( aa_set_nat_o(finite_finite_nat,N)
       => ( ( N != bot_bot_set_nat )
         => ( aa_nat_nat(Ha,aa_set_nat_nat(lattic1093996805478795353in_nat,N)) = aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(image_nat_nat2(Ha),N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2785_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_int_int,N: set_int] :
      ( ! [X3: int,Y2: int] : aa_int_int(Ha,aa_int_int(aa_int_fun_int_int(sup_sup_int,X3),Y2)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(Ha,X3)),aa_int_int(Ha,Y2))
     => ( aa_set_int_o(finite_finite_int,N)
       => ( ( N != bot_bot_set_int )
         => ( aa_int_int(Ha,lattic1091506334969745077in_int(N)) = lattic1091506334969745077in_int(image_int_int2(Ha,N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2786_Sup__fin_Ohom__commute,axiom,
    ! [Ha: fun_fu8419763957888848259_nat_o,N: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] : aa_fun2638059832795814268_nat_o(Ha,aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X3),Y2)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(aa_fun2638059832795814268_nat_o(Ha,X3)),aa_fun2638059832795814268_nat_o(Ha,Y2))
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,N)
       => ( ( N != bot_bo8632255151191677115_nat_o )
         => ( aa_fun2638059832795814268_nat_o(Ha,lattic7388253182610234406_nat_o(N)) = lattic7388253182610234406_nat_o(image_6551304387237727173_nat_o(Ha,N)) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_2787_finite__remove__induct,axiom,
    ! [B4: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [A8: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8)
             => ( ( A8 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A8),B4)
                 => ( ! [X4: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A8)
                       => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A8,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X4),bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P,A8) ) ) ) )
         => aa_set5643387332196575111eger_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_2788_finite__remove__induct,axiom,
    ! [B4: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [A8: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8)
             => ( ( A8 != bot_bo3957492148770167129t_unit )
               => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A8),B4)
                 => ( ! [X4: product_unit] :
                        ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A8)
                       => aa_set8780210009671629607unit_o(P,minus_6452836326544984404t_unit(A8,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X4),bot_bo3957492148770167129t_unit))) )
                   => aa_set8780210009671629607unit_o(P,A8) ) ) ) )
         => aa_set8780210009671629607unit_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_2789_finite__remove__induct,axiom,
    ! [B4: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [A8: set_o] :
              ( aa_set_o_o(finite_finite_o,A8)
             => ( ( A8 != bot_bot_set_o )
               => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A8),B4)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A8)
                       => aa_set_o_o(P,minus_minus_set_o(A8,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X4)),bot_bot_set_o))) )
                   => aa_set_o_o(P,A8) ) ) ) )
         => aa_set_o_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_2790_finite__remove__induct,axiom,
    ! [B4: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A8: set_int] :
              ( aa_set_int_o(finite_finite_int,A8)
             => ( ( A8 != bot_bot_set_int )
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A8),B4)
                 => ( ! [X4: int] :
                        ( aa_set_int_o(member_int(X4),A8)
                       => aa_set_int_o(P,minus_minus_set_int(A8,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X4),bot_bot_set_int))) )
                   => aa_set_int_o(P,A8) ) ) ) )
         => aa_set_int_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_2791_finite__remove__induct,axiom,
    ! [B4: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [A8: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A8)
             => ( ( A8 != bot_bo8632255151191677115_nat_o )
               => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A8),B4)
                 => ( ! [X4: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A8)
                       => aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A8,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),bot_bo8632255151191677115_nat_o))) )
                   => aa_set5091322549519198405at_o_o(P,A8) ) ) ) )
         => aa_set5091322549519198405at_o_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_2792_finite__remove__induct,axiom,
    ! [B4: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A8: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A8)
             => ( ( A8 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A8),B4)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat(X4),A8)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(minus_minus_set_nat(A8),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X4),bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A8) ) ) ) )
         => aa_set_nat_o(P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_2793_remove__induct,axiom,
    ! [P: fun_se7289222521248229150eger_o,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
     => ( ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_set5643387332196575111eger_o(P,B4) )
       => ( ! [A8: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8)
             => ( ( A8 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A8),B4)
                 => ( ! [X4: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A8)
                       => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A8,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X4),bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P,A8) ) ) ) )
         => aa_set5643387332196575111eger_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_2794_remove__induct,axiom,
    ! [P: fun_se4342725320470837934unit_o,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
     => ( ( ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => aa_set8780210009671629607unit_o(P,B4) )
       => ( ! [A8: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8)
             => ( ( A8 != bot_bo3957492148770167129t_unit )
               => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A8),B4)
                 => ( ! [X4: product_unit] :
                        ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A8)
                       => aa_set8780210009671629607unit_o(P,minus_6452836326544984404t_unit(A8,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X4),bot_bo3957492148770167129t_unit))) )
                   => aa_set8780210009671629607unit_o(P,A8) ) ) ) )
         => aa_set8780210009671629607unit_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_2795_remove__induct,axiom,
    ! [P: fun_set_o_o,B4: set_o] :
      ( aa_set_o_o(P,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B4)
         => aa_set_o_o(P,B4) )
       => ( ! [A8: set_o] :
              ( aa_set_o_o(finite_finite_o,A8)
             => ( ( A8 != bot_bot_set_o )
               => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A8),B4)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A8)
                       => aa_set_o_o(P,minus_minus_set_o(A8,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X4)),bot_bot_set_o))) )
                   => aa_set_o_o(P,A8) ) ) ) )
         => aa_set_o_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_2796_remove__induct,axiom,
    ! [P: fun_set_int_o,B4: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B4)
         => aa_set_int_o(P,B4) )
       => ( ! [A8: set_int] :
              ( aa_set_int_o(finite_finite_int,A8)
             => ( ( A8 != bot_bot_set_int )
               => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A8),B4)
                 => ( ! [X4: int] :
                        ( aa_set_int_o(member_int(X4),A8)
                       => aa_set_int_o(P,minus_minus_set_int(A8,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X4),bot_bot_set_int))) )
                   => aa_set_int_o(P,A8) ) ) ) )
         => aa_set_int_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_2797_remove__induct,axiom,
    ! [P: fun_se7459991829846211660at_o_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
     => ( ( ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
         => aa_set5091322549519198405at_o_o(P,B4) )
       => ( ! [A8: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A8)
             => ( ( A8 != bot_bo8632255151191677115_nat_o )
               => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A8),B4)
                 => ( ! [X4: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A8)
                       => aa_set5091322549519198405at_o_o(P,minus_6456595905716665654_nat_o(A8,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),bot_bo8632255151191677115_nat_o))) )
                   => aa_set5091322549519198405at_o_o(P,A8) ) ) ) )
         => aa_set5091322549519198405at_o_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_2798_remove__induct,axiom,
    ! [P: fun_set_nat_o,B4: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B4)
         => aa_set_nat_o(P,B4) )
       => ( ! [A8: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A8)
             => ( ( A8 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A8),B4)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat(X4),A8)
                       => aa_set_nat_o(P,aa_set_nat_set_nat(minus_minus_set_nat(A8),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X4),bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A8) ) ) ) )
         => aa_set_nat_o(P,B4) ) ) ) ).

% remove_induct
tff(fact_2799_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( order_53248997263766692_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
          <=> ~ member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,A2),minus_2488345188983292974_nat_o(R2,id_fun4653463382387092555_nat_o)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_2800_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( order_6238756238976269133r_on_o(field_o(R2),R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
          <=> ~ member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(A2)),minus_7052513330764642158od_o_o(R2,id_o2)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_2801_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( order_888282469287196226t_unit(field_Product_unit(R2),R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)
          <=> ~ member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),A2),minus_6985556733573110190t_unit(R2,id_Product_unit)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_2802_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( order_4473980167227706203on_nat(field_nat(R2),R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
          <=> ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),minus_1356011639430497352at_nat(R2,id_nat)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_2803_Linear__order__in__diff__Id,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( order_4471489696718655927on_int(field_int(R2),R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
          <=> ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),minus_1052850069191792384nt_int(R2,id_int)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_2804_max__ext_Omax__extI,axiom,
    ! [X5: set_Code_integer,Y5: set_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X5)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Y5)
       => ( ( Y5 != bot_bo3990330152332043303nteger )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X5)
               => ? [Xa3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),Y5)
                    & member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X3),Xa3),R) ) )
           => member4307123515891402160nteger(produc7443773368509356479nteger(X5,Y5),max_ext_Code_integer(R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_2805_max__ext_Omax__extI,axiom,
    ! [X5: set_Product_unit,Y5: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,Y5)
       => ( ( Y5 != bot_bo3957492148770167129t_unit )
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
               => ? [Xa3: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(Xa3),Y5)
                    & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X3),Xa3),R) ) )
           => member4189953012975926160t_unit(produc7269801343787132631t_unit(X5,Y5),max_ext_Product_unit(R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_2806_max__ext_Omax__extI,axiom,
    ! [X5: set_o,Y5: set_o,R: set_Product_prod_o_o] :
      ( aa_set_o_o(finite_finite_o,X5)
     => ( aa_set_o_o(finite_finite_o,Y5)
       => ( ( Y5 != bot_bot_set_o )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),X5)
               => ? [Xa3: $o] :
                    ( aa_set_o_o(member_o((Xa3)),Y5)
                    & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X3)),(Xa3)),R) ) )
           => member9116954335612470352_set_o(produc5838405689764958487_set_o(X5,Y5),max_ext_o(R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_2807_max__ext_Omax__extI,axiom,
    ! [X5: set_nat,Y5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( aa_set_nat_o(finite_finite_nat,Y5)
       => ( ( Y5 != bot_bot_set_nat )
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),X5)
               => ? [Xa3: nat] :
                    ( aa_set_nat_o(member_nat(Xa3),Y5)
                    & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3),R) ) )
           => member8277197624267554838et_nat(produc4532415448927165861et_nat(X5,Y5),max_ext_nat(R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_2808_max__ext_Omax__extI,axiom,
    ! [X5: set_int,Y5: set_int,R: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( aa_set_int_o(finite_finite_int,Y5)
       => ( ( Y5 != bot_bot_set_int )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),X5)
               => ? [Xa3: int] :
                    ( aa_set_int_o(member_int(Xa3),Y5)
                    & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Xa3),R) ) )
           => member2572552093476627150et_int(produc6363374080413544029et_int(X5,Y5),max_ext_int(R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_2809_max__ext_Omax__extI,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,Y5: set_fu3555266933746557007_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,X5)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,Y5)
       => ( ( Y5 != bot_bo8632255151191677115_nat_o )
         => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),X5)
               => ? [Xa3: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa3),Y5)
                    & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X3,Xa3),R) ) )
           => member8095380121567751440_nat_o(produc5434073240646464855_nat_o(X5,Y5),max_ex8279603704637224490_nat_o(R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_2810_max__ext_Osimps,axiom,
    ! [A1: set_Code_integer,A22: set_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member4307123515891402160nteger(produc7443773368509356479nteger(A1,A22),max_ext_Code_integer(R))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A1)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A22)
        & ( A22 != bot_bo3990330152332043303nteger )
        & ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A1)
           => ? [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A22)
                & member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X2),Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_2811_max__ext_Osimps,axiom,
    ! [A1: set_Product_unit,A22: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
      ( member4189953012975926160t_unit(produc7269801343787132631t_unit(A1,A22),max_ext_Product_unit(R))
    <=> ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A1)
        & aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A22)
        & ( A22 != bot_bo3957492148770167129t_unit )
        & ! [X2: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A1)
           => ? [Xa2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),A22)
                & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X2),Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_2812_max__ext_Osimps,axiom,
    ! [A1: set_o,A22: set_o,R: set_Product_prod_o_o] :
      ( member9116954335612470352_set_o(produc5838405689764958487_set_o(A1,A22),max_ext_o(R))
    <=> ( aa_set_o_o(finite_finite_o,A1)
        & aa_set_o_o(finite_finite_o,A22)
        & ( A22 != bot_bot_set_o )
        & ! [X2: $o] :
            ( aa_set_o_o(member_o((X2)),A1)
           => ? [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A22)
                & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X2)),(Xa2)),R) ) ) ) ) ).

% max_ext.simps
tff(fact_2813_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A1,A22),max_ext_nat(R))
    <=> ( aa_set_nat_o(finite_finite_nat,A1)
        & aa_set_nat_o(finite_finite_nat,A22)
        & ( A22 != bot_bot_set_nat )
        & ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A1)
           => ? [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A22)
                & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_2814_max__ext_Osimps,axiom,
    ! [A1: set_int,A22: set_int,R: set_Pr958786334691620121nt_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A1,A22),max_ext_int(R))
    <=> ( aa_set_int_o(finite_finite_int,A1)
        & aa_set_int_o(finite_finite_int,A22)
        & ( A22 != bot_bot_set_int )
        & ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A1)
           => ? [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A22)
                & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_2815_max__ext_Osimps,axiom,
    ! [A1: set_fu3555266933746557007_nat_o,A22: set_fu3555266933746557007_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( member8095380121567751440_nat_o(produc5434073240646464855_nat_o(A1,A22),max_ex8279603704637224490_nat_o(R))
    <=> ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A1)
        & aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A22)
        & ( A22 != bot_bo8632255151191677115_nat_o )
        & ! [X2: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A1)
           => ? [Xa2: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa2),A22)
                & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X2,Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_2816_max__ext_Ocases,axiom,
    ! [A1: set_Code_integer,A22: set_Code_integer,R: set_Pr4811707699266497531nteger] :
      ( member4307123515891402160nteger(produc7443773368509356479nteger(A1,A22),max_ext_Code_integer(R))
     => ~ ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A1)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A22)
           => ( ( A22 != bot_bo3990330152332043303nteger )
             => ~ ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A1)
                   => ? [Xa4: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(Xa4),A22)
                        & member157494554546826820nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X4),Xa4),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_2817_max__ext_Ocases,axiom,
    ! [A1: set_Product_unit,A22: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
      ( member4189953012975926160t_unit(produc7269801343787132631t_unit(A1,A22),max_ext_Product_unit(R))
     => ~ ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A1)
         => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A22)
           => ( ( A22 != bot_bo3957492148770167129t_unit )
             => ~ ! [X4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A1)
                   => ? [Xa4: product_unit] :
                        ( aa_set8780210009671629607unit_o(member_Product_unit(Xa4),A22)
                        & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X4),Xa4),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_2818_max__ext_Ocases,axiom,
    ! [A1: set_o,A22: set_o,R: set_Product_prod_o_o] :
      ( member9116954335612470352_set_o(produc5838405689764958487_set_o(A1,A22),max_ext_o(R))
     => ~ ( aa_set_o_o(finite_finite_o,A1)
         => ( aa_set_o_o(finite_finite_o,A22)
           => ( ( A22 != bot_bot_set_o )
             => ~ ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A1)
                   => ? [Xa4: $o] :
                        ( aa_set_o_o(member_o((Xa4)),A22)
                        & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X4)),(Xa4)),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_2819_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A1,A22),max_ext_nat(R))
     => ~ ( aa_set_nat_o(finite_finite_nat,A1)
         => ( aa_set_nat_o(finite_finite_nat,A22)
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A1)
                   => ? [Xa4: nat] :
                        ( aa_set_nat_o(member_nat(Xa4),A22)
                        & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa4),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_2820_max__ext_Ocases,axiom,
    ! [A1: set_int,A22: set_int,R: set_Pr958786334691620121nt_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A1,A22),max_ext_int(R))
     => ~ ( aa_set_int_o(finite_finite_int,A1)
         => ( aa_set_int_o(finite_finite_int,A22)
           => ( ( A22 != bot_bot_set_int )
             => ~ ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A1)
                   => ? [Xa4: int] :
                        ( aa_set_int_o(member_int(Xa4),A22)
                        & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X4),Xa4),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_2821_max__ext_Ocases,axiom,
    ! [A1: set_fu3555266933746557007_nat_o,A22: set_fu3555266933746557007_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( member8095380121567751440_nat_o(produc5434073240646464855_nat_o(A1,A22),max_ex8279603704637224490_nat_o(R))
     => ~ ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A1)
         => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A22)
           => ( ( A22 != bot_bo8632255151191677115_nat_o )
             => ~ ! [X4: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A1)
                   => ? [Xa4: fun_Pr6460944699146269935_nat_o] :
                        ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa4),A22)
                        & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X4,Xa4),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_2822_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
     => ( ! [X3: rat,K: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
       => ! [X4: rat,K2: rat] :
            ( ( aa_rat_o(P,X4)
              | aa_rat_o(Q,X4) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_2823_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X3: int,K: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
     => ( ! [X3: int,K: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
       => ! [X4: int,K2: int] :
            ( ( aa_int_o(P,X4)
              | aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3)))
              | aa_int_o(Q,aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_2824_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
     => ( ! [X3: rat,K: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
       => ! [X4: rat,K2: rat] :
            ( ( aa_rat_o(P,X4)
              & aa_rat_o(Q,X4) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_2825_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X3: int,K: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
     => ( ! [X3: int,K: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
       => ! [X4: int,K2: int] :
            ( ( aa_int_o(P,X4)
              & aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3)))
              & aa_int_o(Q,aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_2826_left__diff__distrib,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A2),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% left_diff_distrib
tff(fact_2827_left__diff__distrib,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(A2),B)),C) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_2828_right__diff__distrib,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% right_diff_distrib
tff(fact_2829_right__diff__distrib,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% right_diff_distrib
tff(fact_2830_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),C)),A2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)) ).

% left_diff_distrib'
tff(fact_2831_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(B),C)),A2) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)) ).

% left_diff_distrib'
tff(fact_2832_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(B),C)),A2) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)) ).

% left_diff_distrib'
tff(fact_2833_right__diff__distrib_H,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% right_diff_distrib'
tff(fact_2834_right__diff__distrib_H,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(minus_minus_nat(B),C)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) ).

% right_diff_distrib'
tff(fact_2835_right__diff__distrib_H,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% right_diff_distrib'
tff(fact_2836_in__lex__prod,axiom,
    ! [A2: nat,B: int,A4: nat,B3: int,R2: set_Pr1261947904930325089at_nat,S2: set_Pr958786334691620121nt_int] :
      ( member4619661753630440592at_int(produc886916002739626967at_int(product_Pair_nat_int(A2,B),product_Pair_nat_int(A4,B3)),lex_prod_nat_int(R2,S2))
    <=> ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A4),R2)
        | ( ( A2 = A4 )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2837_in__lex__prod,axiom,
    ! [A2: int,B: nat,A4: int,B3: nat,R2: set_Pr958786334691620121nt_int,S2: set_Pr1261947904930325089at_nat] :
      ( member2930414080797523088nt_nat(produc8921240378546987991nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A2),B),aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A4),B3)),lex_prod_int_nat(R2,S2))
    <=> ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A4),R2)
        | ( ( A2 = A4 )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2838_in__lex__prod,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A4: heap_e7401611519738050253t_unit,B3: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr5488025237498180813et_nat] :
      ( member6099555550032318734et_nat(produc8199053930788261021et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A4),B3)),lex_pr538634069021852900et_nat(R2,S2))
    <=> ( member5046970461208427408t_unit(produc8256076736985659095t_unit(A2,A4),R2)
        | ( ( A2 = A4 )
          & member8277197624267554838et_nat(produc4532415448927165861et_nat(B,B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2839_in__lex__prod,axiom,
    ! [A2: nat,B: nat,A4: nat,B3: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3)),lex_prod_nat_nat(R2,S2))
    <=> ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A4),R2)
        | ( ( A2 = A4 )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2840_in__lex__prod,axiom,
    ! [A2: int,B: int,A4: int,B3: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),aa_int7462565939943104719nt_int(product_Pair_int_int(A4),B3)),lex_prod_int_int(R2,S2))
    <=> ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A4),R2)
        | ( ( A2 = A4 )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2841_in__lex__prod,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A4: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr719794911490849221et_nat] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A4),B3)),lex_pr6479962325641483138et_nat(R2,S2))
    <=> ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,A4),R2)
        | ( ( A2 = A4 )
          & member6099555550032318734et_nat(produc8199053930788261021et_nat(B,B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2842_in__lex__prod,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A4: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat,R2: set_Pr2890355626087285447_nat_o,S2: set_Pr9066344408807579265et_nat] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A4),B3)),lex_pr1264321499788159136et_nat(R2,S2))
    <=> ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,A4),R2)
        | ( ( A2 = A4 )
          & member2216856746449346506et_nat(produc8050396023069295193et_nat(B,B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_2843_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr5094982260447487303t_unit] :
      ( order_888282469287196226t_unit(field_Product_unit(R2),R2)
     => ( wf_Product_unit(minus_6985556733573110190t_unit(R2,id_Product_unit))
      <=> ! [A9: set_Product_unit] :
            ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A9),field_Product_unit(R2))
           => ( ( A9 != bot_bo3957492148770167129t_unit )
             => ? [X2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A9)
                  & ! [Xa2: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),A9)
                     => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X2),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_2844_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( order_6238756238976269133r_on_o(field_o(R2),R2)
     => ( wf_o(minus_7052513330764642158od_o_o(R2,id_o2))
      <=> ! [A9: set_o] :
            ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A9),field_o(R2))
           => ( ( A9 != bot_bot_set_o )
             => ? [X2: $o] :
                  ( aa_set_o_o(member_o((X2)),A9)
                  & ! [Xa2: $o] :
                      ( aa_set_o_o(member_o((Xa2)),A9)
                     => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X2)),(Xa2)),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_2845_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( order_4471489696718655927on_int(field_int(R2),R2)
     => ( wf_int(minus_1052850069191792384nt_int(R2,id_int))
      <=> ! [A9: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A9),field_int(R2))
           => ( ( A9 != bot_bot_set_int )
             => ? [X2: int] :
                  ( aa_set_int_o(member_int(X2),A9)
                  & ! [Xa2: int] :
                      ( aa_set_int_o(member_int(Xa2),A9)
                     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_2846_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o] :
      ( order_53248997263766692_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( wf_fun7865846636378006616_nat_o(minus_2488345188983292974_nat_o(R2,id_fun4653463382387092555_nat_o))
      <=> ! [A9: set_fu3555266933746557007_nat_o] :
            ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A9),field_48612661033931452_nat_o(R2))
           => ( ( A9 != bot_bo8632255151191677115_nat_o )
             => ? [X2: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A9)
                  & ! [Xa2: fun_Pr6460944699146269935_nat_o] :
                      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa2),A9)
                     => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X2,Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_2847_Linear__order__wf__diff__Id,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( order_4473980167227706203on_nat(field_nat(R2),R2)
     => ( wf_nat(minus_1356011639430497352at_nat(R2,id_nat))
      <=> ! [A9: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A9),field_nat(R2))
           => ( ( A9 != bot_bot_set_nat )
             => ? [X2: nat] :
                  ( aa_set_nat_o(member_nat(X2),A9)
                  & ! [Xa2: nat] :
                      ( aa_set_nat_o(member_nat(Xa2),A9)
                     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_2848_remove__def,axiom,
    ! [X: product_unit,A: set_Product_unit] : remove_Product_unit(X,A) = minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) ).

% remove_def
tff(fact_2849_remove__def,axiom,
    ! [X: $o,A: set_o] : remove_o((X),A) = minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) ).

% remove_def
tff(fact_2850_remove__def,axiom,
    ! [X: int,A: set_int] : remove_int(X,A) = minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) ).

% remove_def
tff(fact_2851_remove__def,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] : remove8193825956935957076_nat_o(X,A) = minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) ).

% remove_def
tff(fact_2852_remove__def,axiom,
    ! [X: nat,A: set_nat] : aa_set_nat_set_nat(remove_nat(X),A) = aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) ).

% remove_def
tff(fact_2853_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: fun_Pr1520749766685617582unit_o] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))),field_Product_unit(R2))
       => ( ( ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),minus_6985556733573110190t_unit(R2,id_Product_unit))
              | member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),A2),minus_6985556733573110190t_unit(R2,id_Product_unit)) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Phi,A2),B) )
         => ( ( ( A2 = B )
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Phi,A2),B) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_2854_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: fun_o_fun_o_o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))),field_o(R2))
       => ( ( ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),minus_7052513330764642158od_o_o(R2,id_o2))
              | member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(A2)),minus_7052513330764642158od_o_o(R2,id_o2)) )
           => aa_o_o(aa_o_fun_o_o(Phi,(A2)),(B)) )
         => ( ( ( (A2)
              <=> (B) )
             => aa_o_o(aa_o_fun_o_o(Phi,(A2)),(B)) )
           => aa_o_o(aa_o_fun_o_o(Phi,(A2)),(B)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_2855_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: fun_int_fun_int_o] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))),field_int(R2))
       => ( ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),minus_1052850069191792384nt_int(R2,id_int))
              | member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),minus_1052850069191792384nt_int(R2,id_int)) )
           => aa_int_o(aa_int_fun_int_o(Phi,A2),B) )
         => ( ( ( A2 = B )
             => aa_int_o(aa_int_fun_int_o(Phi,A2),B) )
           => aa_int_o(aa_int_fun_int_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_2856_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,Phi: fun_fu4164567611050918574at_o_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))),field_48612661033931452_nat_o(R2))
       => ( ( ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),minus_2488345188983292974_nat_o(R2,id_fun4653463382387092555_nat_o))
              | member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,A2),minus_2488345188983292974_nat_o(R2,id_fun4653463382387092555_nat_o)) )
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Phi,A2),B) )
         => ( ( ( A2 = B )
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Phi,A2),B) )
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_2857_wo__rel_Ocases__Total3,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: fun_nat_fun_nat_o] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))),field_nat(R2))
       => ( ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),minus_1356011639430497352at_nat(R2,id_nat))
              | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),minus_1356011639430497352at_nat(R2,id_nat)) )
           => aa_nat_o(aa_nat_fun_nat_o(Phi,A2),B) )
         => ( ( ( A2 = B )
             => aa_nat_o(aa_nat_fun_nat_o(Phi,A2),B) )
           => aa_nat_o(aa_nat_fun_nat_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_2858_wf__max,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] :
      ( wf_fun7865846636378006616_nat_o(conver8092695895528356987_nat_o(R))
     => ( ( R != bot_bo6763312114214544691_nat_o )
       => ~ ! [M2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(M2),minus_6456595905716665654_nat_o(range_307841406276416053_nat_o(R),domain5580505327626972364_nat_o(R))) ) ) ).

% wf_max
tff(fact_2859_wf__max,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( wf_int(converse_int_int(R))
     => ( ( R != bot_bo1796632182523588997nt_int )
       => ~ ! [M2: int] : ~ aa_set_int_o(member_int(M2),minus_minus_set_int(range_int_int(R),domain_int_int(R))) ) ) ).

% wf_max
tff(fact_2860_wf__max,axiom,
    ! [R: set_Product_prod_o_o] :
      ( wf_o(converse_o_o(R))
     => ( ( R != bot_bo7073875226086086771od_o_o )
       => ~ ! [M2: $o] : ~ aa_set_o_o(member_o((M2)),minus_minus_set_o(range_o_o(R),domain_o_o(R))) ) ) ).

% wf_max
tff(fact_2861_wf__max,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( wf_Product_unit(conver2495244158383847931t_unit(R))
     => ( ( R != bot_bo1654022762305692595t_unit )
       => ~ ! [M2: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(M2),minus_6452836326544984404t_unit(range_6617724647562260917t_unit(R),domain4292024440186670924t_unit(R))) ) ) ).

% wf_max
tff(fact_2862_wf__max,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( wf_nat(converse_nat_nat(R))
     => ( ( R != bot_bo2099793752762293965at_nat )
       => ~ ! [M2: nat] : ~ aa_set_nat_o(member_nat(M2),aa_set_nat_set_nat(minus_minus_set_nat(range_nat_nat(R)),domain_nat_nat(R))) ) ) ).

% wf_max
tff(fact_2863_wf__Un,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,S2: set_Pr5094982260447487303t_unit] :
      ( wf_Product_unit(R2)
     => ( wf_Product_unit(S2)
       => ( ( inf_in4660618365625256667t_unit(domain4292024440186670924t_unit(R2),range_6617724647562260917t_unit(S2)) = bot_bo3957492148770167129t_unit )
         => wf_Product_unit(sup_su6481247232241170203t_unit(R2,S2)) ) ) ) ).

% wf_Un
tff(fact_2864_wf__Un,axiom,
    ! [R2: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
      ( wf_o(R2)
     => ( wf_o(S2)
       => ( ( inf_inf_set_o(domain_o_o(R2),range_o_o(S2)) = bot_bot_set_o )
         => wf_o(sup_su5769328420594410459od_o_o(R2,S2)) ) ) ) ).

% wf_Un
tff(fact_2865_wf__Un,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( wf_nat(R2)
     => ( wf_nat(S2)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,domain_nat_nat(R2)),range_nat_nat(S2)) = bot_bot_set_nat )
         => wf_nat(sup_su6327502436637775413at_nat(R2,S2)) ) ) ) ).

% wf_Un
tff(fact_2866_wf__Un,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( wf_int(R2)
     => ( wf_int(S2)
       => ( ( inf_inf_set_int(domain_int_int(R2),range_int_int(S2)) = bot_bot_set_int )
         => wf_int(sup_su6024340866399070445nt_int(R2,S2)) ) ) ) ).

% wf_Un
tff(fact_2867_wf__Un,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,S2: set_Pr2890355626087285447_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R2)
     => ( wf_fun7865846636378006616_nat_o(S2)
       => ( ( inf_in3376754634429691197_nat_o(domain5580505327626972364_nat_o(R2),range_307841406276416053_nat_o(S2)) = bot_bo8632255151191677115_nat_o )
         => wf_fun7865846636378006616_nat_o(sup_su6171071944722280603_nat_o(R2,S2)) ) ) ) ).

% wf_Un
tff(fact_2868_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr5094982260447487303t_unit] :
      ( order_888282469287196226t_unit(field_Product_unit(R2),R2)
     => ( order_3164716736300062379t_unit(field_Product_unit(R2),R2)
      <=> ! [A9: set_Product_unit] :
            ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A9),field_Product_unit(R2))
           => ( ( A9 != bot_bo3957492148770167129t_unit )
             => ? [X2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A9)
                  & ! [Xa2: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),A9)
                     => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(X2),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_2869_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( order_6238756238976269133r_on_o(field_o(R2),R2)
     => ( order_2615775362358406454r_on_o(field_o(R2),R2)
      <=> ! [A9: set_o] :
            ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A9),field_o(R2))
           => ( ( A9 != bot_bot_set_o )
             => ? [X2: $o] :
                  ( aa_set_o_o(member_o((X2)),A9)
                  & ! [Xa2: $o] :
                      ( aa_set_o_o(member_o((Xa2)),A9)
                     => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X2)),(Xa2)),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_2870_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( order_4471489696718655927on_int(field_int(R2),R2)
     => ( order_2886507596567047182on_int(field_int(R2),R2)
      <=> ! [A9: set_int] :
            ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A9),field_int(R2))
           => ( ( A9 != bot_bot_set_int )
             => ? [X2: int] :
                  ( aa_set_int_o(member_int(X2),A9)
                  & ! [Xa2: int] :
                      ( aa_set_int_o(member_int(Xa2),A9)
                     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_2871_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o] :
      ( order_53248997263766692_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( order_4477081128474576525_nat_o(field_48612661033931452_nat_o(R2),R2)
      <=> ! [A9: set_fu3555266933746557007_nat_o] :
            ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A9),field_48612661033931452_nat_o(R2))
           => ( ( A9 != bot_bo8632255151191677115_nat_o )
             => ? [X2: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A9)
                  & ! [Xa2: fun_Pr6460944699146269935_nat_o] :
                      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa2),A9)
                     => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(X2,Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_2872_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( order_4473980167227706203on_nat(field_nat(R2),R2)
     => ( order_2888998067076097458on_nat(field_nat(R2),R2)
      <=> ! [A9: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A9),field_nat(R2))
           => ( ( A9 != bot_bot_set_nat )
             => ? [X2: nat] :
                  ( aa_set_nat_o(member_nat(X2),A9)
                  & ! [Xa2: nat] :
                      ( aa_set_nat_o(member_nat(Xa2),A9)
                     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_2873_underS__incr,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( trans_int(R2)
     => ( antisym_int(R2)
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
         => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(order_underS_int(R2),A2)),aa_int_set_int(order_underS_int(R2),B)) ) ) ) ).

% underS_incr
tff(fact_2874_underS__incr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( trans_nat(R2)
     => ( antisym_nat(R2)
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(order_underS_nat(R2),A2)),aa_nat_set_nat(order_underS_nat(R2),B)) ) ) ) ).

% underS_incr
tff(fact_2875_member__remove,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),remove8193825956935957076_nat_o(Y,A))
    <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_2876_member__remove,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(remove_nat(Y),A))
    <=> ( aa_set_nat_o(member_nat(X),A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_2877_member__remove,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( aa_set_int_o(member_int(X),remove_int(Y,A))
    <=> ( aa_set_int_o(member_int(X),A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_2878_member__remove,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( aa_set_o_o(member_o((X)),remove_o((Y),A))
    <=> ( aa_set_o_o(member_o((X)),A)
        & ~ ( (X)
          <=> (Y) ) ) ) ).

% member_remove
tff(fact_2879_member__remove,axiom,
    ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),remove_Product_unit(Y,A))
    <=> ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_2880_wf__insert,axiom,
    ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
      ( wf_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X),R2))
    <=> ( wf_nat(R2)
        & ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R2)) ) ) ).

% wf_insert
tff(fact_2881_wf__insert,axiom,
    ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
      ( wf_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X),R2))
    <=> ( wf_int(R2)
        & ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R2)) ) ) ).

% wf_insert
tff(fact_2882_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P: fun_nat_o,A2: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( ( Y4 != X3 )
                  & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),X3),R2) )
               => aa_nat_o(P,Y4) )
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,A2) ) ) ).

% wo_rel.well_order_induct
tff(fact_2883_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr958786334691620121nt_int,P: fun_int_o,A2: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( ! [X3: int] :
            ( ! [Y4: int] :
                ( ( ( Y4 != X3 )
                  & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y4),X3),R2) )
               => aa_int_o(P,Y4) )
           => aa_int_o(P,X3) )
       => aa_int_o(P,A2) ) ) ).

% wo_rel.well_order_induct
tff(fact_2884_wf__induct__rule,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P: fun_nat_o,A2: nat] :
      ( wf_nat(R2)
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),X3),R2)
               => aa_nat_o(P,Y4) )
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,A2) ) ) ).

% wf_induct_rule
tff(fact_2885_wf__induct__rule,axiom,
    ! [R2: set_Pr958786334691620121nt_int,P: fun_int_o,A2: int] :
      ( wf_int(R2)
     => ( ! [X3: int] :
            ( ! [Y4: int] :
                ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y4),X3),R2)
               => aa_int_o(P,Y4) )
           => aa_int_o(P,X3) )
       => aa_int_o(P,A2) ) ) ).

% wf_induct_rule
tff(fact_2886_wf__eq__minimal,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R2)
    <=> ! [Q4: set_fu3555266933746557007_nat_o] :
          ( ? [X2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Q4)
         => ? [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),Q4)
              & ! [Y3: fun_Pr6460944699146269935_nat_o] :
                  ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Y3,X2),R2)
                 => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y3),Q4) ) ) ) ) ).

% wf_eq_minimal
tff(fact_2887_wf__eq__minimal,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( wf_o(R2)
    <=> ! [Q4: set_o] :
          ( ? [X2: $o] : aa_set_o_o(member_o((X2)),Q4)
         => ? [X2: $o] :
              ( aa_set_o_o(member_o((X2)),Q4)
              & ! [Y3: $o] :
                  ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y3)),(X2)),R2)
                 => ~ aa_set_o_o(member_o((Y3)),Q4) ) ) ) ) ).

% wf_eq_minimal
tff(fact_2888_wf__eq__minimal,axiom,
    ! [R2: set_Pr5094982260447487303t_unit] :
      ( wf_Product_unit(R2)
    <=> ! [Q4: set_Product_unit] :
          ( ? [X2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(X2),Q4)
         => ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),Q4)
              & ! [Y3: product_unit] :
                  ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Y3),X2),R2)
                 => ~ aa_set8780210009671629607unit_o(member_Product_unit(Y3),Q4) ) ) ) ) ).

% wf_eq_minimal
tff(fact_2889_wf__eq__minimal,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( wf_nat(R2)
    <=> ! [Q4: set_nat] :
          ( ? [X2: nat] : aa_set_nat_o(member_nat(X2),Q4)
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),Q4)
              & ! [Y3: nat] :
                  ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),X2),R2)
                 => ~ aa_set_nat_o(member_nat(Y3),Q4) ) ) ) ) ).

% wf_eq_minimal
tff(fact_2890_wf__eq__minimal,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( wf_int(R2)
    <=> ! [Q4: set_int] :
          ( ? [X2: int] : aa_set_int_o(member_int(X2),Q4)
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),Q4)
              & ! [Y3: int] :
                  ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y3),X2),R2)
                 => ~ aa_set_int_o(member_int(Y3),Q4) ) ) ) ) ).

% wf_eq_minimal
tff(fact_2891_wf__not__refl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( wf_nat(R2)
     => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),R2) ) ).

% wf_not_refl
tff(fact_2892_wf__not__refl,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int] :
      ( wf_int(R2)
     => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),R2) ) ).

% wf_not_refl
tff(fact_2893_wf__not__sym,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
      ( wf_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),X),R2)
       => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),A2),R2) ) ) ).

% wf_not_sym
tff(fact_2894_wf__not__sym,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
      ( wf_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),X),R2)
       => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),A2),R2) ) ) ).

% wf_not_sym
tff(fact_2895_wf__irrefl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( wf_nat(R2)
     => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),A2),R2) ) ).

% wf_irrefl
tff(fact_2896_wf__irrefl,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int] :
      ( wf_int(R2)
     => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),A2),R2) ) ).

% wf_irrefl
tff(fact_2897_wf__induct,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P: fun_nat_o,A2: nat] :
      ( wf_nat(R2)
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),X3),R2)
               => aa_nat_o(P,Y4) )
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,A2) ) ) ).

% wf_induct
tff(fact_2898_wf__induct,axiom,
    ! [R2: set_Pr958786334691620121nt_int,P: fun_int_o,A2: int] :
      ( wf_int(R2)
     => ( ! [X3: int] :
            ( ! [Y4: int] :
                ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y4),X3),R2)
               => aa_int_o(P,Y4) )
           => aa_int_o(P,X3) )
       => aa_int_o(P,A2) ) ) ).

% wf_induct
tff(fact_2899_wf__asym,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
      ( wf_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),X),R2)
       => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),A2),R2) ) ) ).

% wf_asym
tff(fact_2900_wf__asym,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
      ( wf_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),X),R2)
       => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),A2),R2) ) ) ).

% wf_asym
tff(fact_2901_wfUNIVI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [P3: fun_nat_o,X3: nat] :
          ( ! [Xa3: nat] :
              ( ! [Y2: nat] :
                  ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Xa3),R2)
                 => aa_nat_o(P3,Y2) )
             => aa_nat_o(P3,Xa3) )
         => aa_nat_o(P3,X3) )
     => wf_nat(R2) ) ).

% wfUNIVI
tff(fact_2902_wfUNIVI,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( ! [P3: fun_int_o,X3: int] :
          ( ! [Xa3: int] :
              ( ! [Y2: int] :
                  ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Xa3),R2)
                 => aa_int_o(P3,Y2) )
             => aa_int_o(P3,Xa3) )
         => aa_int_o(P3,X3) )
     => wf_int(R2) ) ).

% wfUNIVI
tff(fact_2903_wfI__min,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Q2: set_fu3555266933746557007_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),Q2)
         => ? [Xa3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa3),Q2)
              & ! [Y2: fun_Pr6460944699146269935_nat_o] :
                  ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Y2,Xa3),R)
                 => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),Q2) ) ) )
     => wf_fun7865846636378006616_nat_o(R) ) ).

% wfI_min
tff(fact_2904_wfI__min,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ! [X3: $o,Q2: set_o] :
          ( aa_set_o_o(member_o((X3)),Q2)
         => ? [Xa3: $o] :
              ( aa_set_o_o(member_o((Xa3)),Q2)
              & ! [Y2: $o] :
                  ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y2)),(Xa3)),R)
                 => ~ aa_set_o_o(member_o((Y2)),Q2) ) ) )
     => wf_o(R) ) ).

% wfI_min
tff(fact_2905_wfI__min,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( ! [X3: product_unit,Q2: set_Product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),Q2)
         => ? [Xa3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Xa3),Q2)
              & ! [Y2: product_unit] :
                  ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Y2),Xa3),R)
                 => ~ aa_set8780210009671629607unit_o(member_Product_unit(Y2),Q2) ) ) )
     => wf_Product_unit(R) ) ).

% wfI_min
tff(fact_2906_wfI__min,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Q2: set_nat] :
          ( aa_set_nat_o(member_nat(X3),Q2)
         => ? [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),Q2)
              & ! [Y2: nat] :
                  ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Xa3),R)
                 => ~ aa_set_nat_o(member_nat(Y2),Q2) ) ) )
     => wf_nat(R) ) ).

% wfI_min
tff(fact_2907_wfI__min,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Q2: set_int] :
          ( aa_set_int_o(member_int(X3),Q2)
         => ? [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),Q2)
              & ! [Y2: int] :
                  ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),Xa3),R)
                 => ~ aa_set_int_o(member_int(Y2),Q2) ) ) )
     => wf_int(R) ) ).

% wfI_min
tff(fact_2908_wfE__min,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,X: fun_Pr6460944699146269935_nat_o,Q: set_fu3555266933746557007_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),Q)
       => ~ ! [Z3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Z3),Q)
             => ~ ! [Y4: fun_Pr6460944699146269935_nat_o] :
                    ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Y4,Z3),R)
                   => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),Q) ) ) ) ) ).

% wfE_min
tff(fact_2909_wfE__min,axiom,
    ! [R: set_Product_prod_o_o,X: $o,Q: set_o] :
      ( wf_o(R)
     => ( aa_set_o_o(member_o((X)),Q)
       => ~ ! [Z3: $o] :
              ( aa_set_o_o(member_o((Z3)),Q)
             => ~ ! [Y4: $o] :
                    ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y4)),(Z3)),R)
                   => ~ aa_set_o_o(member_o((Y4)),Q) ) ) ) ) ).

% wfE_min
tff(fact_2910_wfE__min,axiom,
    ! [R: set_Pr5094982260447487303t_unit,X: product_unit,Q: set_Product_unit] :
      ( wf_Product_unit(R)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),Q)
       => ~ ! [Z3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Z3),Q)
             => ~ ! [Y4: product_unit] :
                    ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Y4),Z3),R)
                   => ~ aa_set8780210009671629607unit_o(member_Product_unit(Y4),Q) ) ) ) ) ).

% wfE_min
tff(fact_2911_wfE__min,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Q: set_nat] :
      ( wf_nat(R)
     => ( aa_set_nat_o(member_nat(X),Q)
       => ~ ! [Z3: nat] :
              ( aa_set_nat_o(member_nat(Z3),Q)
             => ~ ! [Y4: nat] :
                    ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),Z3),R)
                   => ~ aa_set_nat_o(member_nat(Y4),Q) ) ) ) ) ).

% wfE_min
tff(fact_2912_wfE__min,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int,Q: set_int] :
      ( wf_int(R)
     => ( aa_set_int_o(member_int(X),Q)
       => ~ ! [Z3: int] :
              ( aa_set_int_o(member_int(Z3),Q)
             => ~ ! [Y4: int] :
                    ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y4),Z3),R)
                   => ~ aa_set_int_o(member_int(Y4),Q) ) ) ) ) ).

% wfE_min
tff(fact_2913_wf__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( wf_nat(R2)
    <=> ! [P4: fun_nat_o] :
          ( ! [X2: nat] :
              ( ! [Y3: nat] :
                  ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),X2),R2)
                 => aa_nat_o(P4,Y3) )
             => aa_nat_o(P4,X2) )
         => ! [X_12: nat] : aa_nat_o(P4,X_12) ) ) ).

% wf_def
tff(fact_2914_wf__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( wf_int(R2)
    <=> ! [P4: fun_int_o] :
          ( ! [X2: int] :
              ( ! [Y3: int] :
                  ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y3),X2),R2)
                 => aa_int_o(P4,Y3) )
             => aa_int_o(P4,X2) )
         => ! [X_12: int] : aa_int_o(P4,X_12) ) ) ).

% wf_def
tff(fact_2915_well__order__on__domain,axiom,
    ! [A: set_fu3555266933746557007_nat_o,R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( order_4477081128474576525_nat_o(A,R2)
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
          & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A) ) ) ) ).

% well_order_on_domain
tff(fact_2916_well__order__on__domain,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( order_2615775362358406454r_on_o(A,R2)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
       => ( aa_set_o_o(member_o((A2)),A)
          & aa_set_o_o(member_o((B)),A) ) ) ) ).

% well_order_on_domain
tff(fact_2917_well__order__on__domain,axiom,
    ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( order_3164716736300062379t_unit(A,R2)
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
          & aa_set8780210009671629607unit_o(member_Product_unit(B),A) ) ) ) ).

% well_order_on_domain
tff(fact_2918_well__order__on__domain,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( order_2888998067076097458on_nat(A,R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => ( aa_set_nat_o(member_nat(A2),A)
          & aa_set_nat_o(member_nat(B),A) ) ) ) ).

% well_order_on_domain
tff(fact_2919_well__order__on__domain,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( order_2886507596567047182on_int(A,R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => ( aa_set_int_o(member_int(A2),A)
          & aa_set_int_o(member_int(B),A) ) ) ) ).

% well_order_on_domain
tff(fact_2920_underS__E,axiom,
    ! [I2: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o,J: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I2),order_7714984474677380533_nat_o(R,J))
     => ( ( I2 != J )
        & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(I2,J),R) ) ) ).

% underS_E
tff(fact_2921_underS__E,axiom,
    ! [I2: $o,R: set_Product_prod_o_o,J: $o] :
      ( aa_set_o_o(member_o((I2)),aa_o_set_o(order_underS_o(R),(J)))
     => ( ~ ( (I2)
          <=> (J) )
        & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((I2)),(J)),R) ) ) ).

% underS_E
tff(fact_2922_underS__E,axiom,
    ! [I2: product_unit,R: set_Pr5094982260447487303t_unit,J: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(I2),aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R),J))
     => ( ( I2 != J )
        & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(I2),J),R) ) ) ).

% underS_E
tff(fact_2923_underS__E,axiom,
    ! [I2: nat,R: set_Pr1261947904930325089at_nat,J: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(order_underS_nat(R),J))
     => ( ( I2 != J )
        & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I2),J),R) ) ) ).

% underS_E
tff(fact_2924_underS__E,axiom,
    ! [I2: int,R: set_Pr958786334691620121nt_int,J: int] :
      ( aa_set_int_o(member_int(I2),aa_int_set_int(order_underS_int(R),J))
     => ( ( I2 != J )
        & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J),R) ) ) ).

% underS_E
tff(fact_2925_underS__I,axiom,
    ! [I2: fun_Pr6460944699146269935_nat_o,J: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o] :
      ( ( I2 != J )
     => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(I2,J),R)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I2),order_7714984474677380533_nat_o(R,J)) ) ) ).

% underS_I
tff(fact_2926_underS__I,axiom,
    ! [I2: $o,J: $o,R: set_Product_prod_o_o] :
      ( ~ ( (I2)
        <=> (J) )
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((I2)),(J)),R)
       => aa_set_o_o(member_o((I2)),aa_o_set_o(order_underS_o(R),(J))) ) ) ).

% underS_I
tff(fact_2927_underS__I,axiom,
    ! [I2: product_unit,J: product_unit,R: set_Pr5094982260447487303t_unit] :
      ( ( I2 != J )
     => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(I2),J),R)
       => aa_set8780210009671629607unit_o(member_Product_unit(I2),aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R),J)) ) ) ).

% underS_I
tff(fact_2928_underS__I,axiom,
    ! [I2: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( I2 != J )
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I2),J),R)
       => aa_set_nat_o(member_nat(I2),aa_nat_set_nat(order_underS_nat(R),J)) ) ) ).

% underS_I
tff(fact_2929_underS__I,axiom,
    ! [I2: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( ( I2 != J )
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J),R)
       => aa_set_int_o(member_int(I2),aa_int_set_int(order_underS_int(R),J)) ) ) ).

% underS_I
tff(fact_2930_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: fun_Pr6460944699146269935_nat_o,R: set_Pr2890355626087285447_nat_o,J: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I2),order_7714984474677380533_nat_o(R,J))
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I2),field_48612661033931452_nat_o(R)) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_2931_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: nat,R: set_Pr1261947904930325089at_nat,J: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(order_underS_nat(R),J))
     => aa_set_nat_o(member_nat(I2),field_nat(R)) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_2932_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: int,R: set_Pr958786334691620121nt_int,J: int] :
      ( aa_set_int_o(member_int(I2),aa_int_set_int(order_underS_int(R),J))
     => aa_set_int_o(member_int(I2),field_int(R)) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_2933_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: $o,R: set_Product_prod_o_o,J: $o] :
      ( aa_set_o_o(member_o((I2)),aa_o_set_o(order_underS_o(R),(J)))
     => aa_set_o_o(member_o((I2)),field_o(R)) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_2934_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: product_unit,R: set_Pr5094982260447487303t_unit,J: product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(I2),aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R),J))
     => aa_set8780210009671629607unit_o(member_Product_unit(I2),field_Product_unit(R)) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_2935_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),field_nat(R2))
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),field_nat(R2))
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3),R2)
                | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa3),X4),R2) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_2936_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ! [X4: int] :
          ( aa_set_int_o(member_int(X4),field_int(R2))
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),field_int(R2))
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X4),Xa3),R2)
                | member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa3),X4),R2) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_2937_wfE__min_H,axiom,
    ! [R: set_Pr5094982260447487303t_unit,Q: set_Product_unit] :
      ( wf_Product_unit(R)
     => ( ( Q != bot_bo3957492148770167129t_unit )
       => ~ ! [Z3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Z3),Q)
             => ~ ! [Y4: product_unit] :
                    ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Y4),Z3),R)
                   => ~ aa_set8780210009671629607unit_o(member_Product_unit(Y4),Q) ) ) ) ) ).

% wfE_min'
tff(fact_2938_wfE__min_H,axiom,
    ! [R: set_Product_prod_o_o,Q: set_o] :
      ( wf_o(R)
     => ( ( Q != bot_bot_set_o )
       => ~ ! [Z3: $o] :
              ( aa_set_o_o(member_o((Z3)),Q)
             => ~ ! [Y4: $o] :
                    ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y4)),(Z3)),R)
                   => ~ aa_set_o_o(member_o((Y4)),Q) ) ) ) ) ).

% wfE_min'
tff(fact_2939_wfE__min_H,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Q: set_nat] :
      ( wf_nat(R)
     => ( ( Q != bot_bot_set_nat )
       => ~ ! [Z3: nat] :
              ( aa_set_nat_o(member_nat(Z3),Q)
             => ~ ! [Y4: nat] :
                    ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),Z3),R)
                   => ~ aa_set_nat_o(member_nat(Y4),Q) ) ) ) ) ).

% wfE_min'
tff(fact_2940_wfE__min_H,axiom,
    ! [R: set_Pr958786334691620121nt_int,Q: set_int] :
      ( wf_int(R)
     => ( ( Q != bot_bot_set_int )
       => ~ ! [Z3: int] :
              ( aa_set_int_o(member_int(Z3),Q)
             => ~ ! [Y4: int] :
                    ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y4),Z3),R)
                   => ~ aa_set_int_o(member_int(Y4),Q) ) ) ) ) ).

% wfE_min'
tff(fact_2941_wfE__min_H,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,Q: set_fu3555266933746557007_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R)
     => ( ( Q != bot_bo8632255151191677115_nat_o )
       => ~ ! [Z3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Z3),Q)
             => ~ ! [Y4: fun_Pr6460944699146269935_nat_o] :
                    ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Y4,Z3),R)
                   => ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),Q) ) ) ) ) ).

% wfE_min'
tff(fact_2942_well__order__on__empty,axiom,
    order_3164716736300062379t_unit(bot_bo3957492148770167129t_unit,bot_bo1654022762305692595t_unit) ).

% well_order_on_empty
tff(fact_2943_well__order__on__empty,axiom,
    order_2615775362358406454r_on_o(bot_bot_set_o,bot_bo7073875226086086771od_o_o) ).

% well_order_on_empty
tff(fact_2944_well__order__on__empty,axiom,
    order_2888998067076097458on_nat(bot_bot_set_nat,bot_bo2099793752762293965at_nat) ).

% well_order_on_empty
tff(fact_2945_well__order__on__empty,axiom,
    order_2886507596567047182on_int(bot_bot_set_int,bot_bo1796632182523588997nt_int) ).

% well_order_on_empty
tff(fact_2946_well__order__on__empty,axiom,
    order_4477081128474576525_nat_o(bot_bo8632255151191677115_nat_o,bot_bo6763312114214544691_nat_o) ).

% well_order_on_empty
tff(fact_2947_well__order__induct__imp,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,P: fun_fu5135969500543783852at_o_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( ! [Y4: fun_Pr6460944699146269935_nat_o] :
                ( ( ( Y4 != X3 )
                  & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Y4,X3),R2) )
               => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y4),field_48612661033931452_nat_o(R2))
                 => aa_fun1675921005893230629at_o_o(P,Y4) ) )
           => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),field_48612661033931452_nat_o(R2))
             => aa_fun1675921005893230629at_o_o(P,X3) ) )
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
         => aa_fun1675921005893230629at_o_o(P,A2) ) ) ) ).

% well_order_induct_imp
tff(fact_2948_well__order__induct__imp,axiom,
    ! [R2: set_Product_prod_o_o,P: fun_o_o,A2: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( ! [X3: $o] :
            ( ! [Y4: $o] :
                ( ( ~ ( (Y4)
                    <=> (X3) )
                  & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y4)),(X3)),R2) )
               => ( aa_set_o_o(member_o((Y4)),field_o(R2))
                 => aa_o_o(P,(Y4)) ) )
           => ( aa_set_o_o(member_o((X3)),field_o(R2))
             => aa_o_o(P,(X3)) ) )
       => ( aa_set_o_o(member_o((A2)),field_o(R2))
         => aa_o_o(P,(A2)) ) ) ) ).

% well_order_induct_imp
tff(fact_2949_well__order__induct__imp,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,P: fun_Product_unit_o,A2: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( ! [X3: product_unit] :
            ( ! [Y4: product_unit] :
                ( ( ( Y4 != X3 )
                  & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Y4),X3),R2) )
               => ( aa_set8780210009671629607unit_o(member_Product_unit(Y4),field_Product_unit(R2))
                 => aa_Product_unit_o(P,Y4) ) )
           => ( aa_set8780210009671629607unit_o(member_Product_unit(X3),field_Product_unit(R2))
             => aa_Product_unit_o(P,X3) ) )
       => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
         => aa_Product_unit_o(P,A2) ) ) ) ).

% well_order_induct_imp
tff(fact_2950_well__order__induct__imp,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P: fun_nat_o,A2: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( ( Y4 != X3 )
                  & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),X3),R2) )
               => ( aa_set_nat_o(member_nat(Y4),field_nat(R2))
                 => aa_nat_o(P,Y4) ) )
           => ( aa_set_nat_o(member_nat(X3),field_nat(R2))
             => aa_nat_o(P,X3) ) )
       => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
         => aa_nat_o(P,A2) ) ) ) ).

% well_order_induct_imp
tff(fact_2951_well__order__induct__imp,axiom,
    ! [R2: set_Pr958786334691620121nt_int,P: fun_int_o,A2: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( ! [X3: int] :
            ( ! [Y4: int] :
                ( ( ( Y4 != X3 )
                  & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y4),X3),R2) )
               => ( aa_set_int_o(member_int(Y4),field_int(R2))
                 => aa_int_o(P,Y4) ) )
           => ( aa_set_int_o(member_int(X3),field_int(R2))
             => aa_int_o(P,X3) ) )
       => ( aa_set_int_o(member_int(A2),field_int(R2))
         => aa_int_o(P,A2) ) ) ) ).

% well_order_induct_imp
tff(fact_2952_underS__empty,axiom,
    ! [A2: product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
     => ( aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R2),A2) = bot_bo3957492148770167129t_unit ) ) ).

% underS_empty
tff(fact_2953_underS__empty,axiom,
    ! [A2: $o,R2: set_Product_prod_o_o] :
      ( ~ aa_set_o_o(member_o((A2)),field_o(R2))
     => ( aa_o_set_o(order_underS_o(R2),(A2)) = bot_bot_set_o ) ) ).

% underS_empty
tff(fact_2954_underS__empty,axiom,
    ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set_nat_o(member_nat(A2),field_nat(R2))
     => ( aa_nat_set_nat(order_underS_nat(R2),A2) = bot_bot_set_nat ) ) ).

% underS_empty
tff(fact_2955_underS__empty,axiom,
    ! [A2: int,R2: set_Pr958786334691620121nt_int] :
      ( ~ aa_set_int_o(member_int(A2),field_int(R2))
     => ( aa_int_set_int(order_underS_int(R2),A2) = bot_bot_set_int ) ) ).

% underS_empty
tff(fact_2956_underS__empty,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,R2: set_Pr2890355626087285447_nat_o] :
      ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
     => ( order_7714984474677380533_nat_o(R2,A2) = bot_bo8632255151191677115_nat_o ) ) ).

% underS_empty
tff(fact_2957_wfI__pf,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( ! [A8: set_Product_unit] :
          ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A8),image_8431784415008483091t_unit(R,A8))
         => ( A8 = bot_bo3957492148770167129t_unit ) )
     => wf_Product_unit(R) ) ).

% wfI_pf
tff(fact_2958_wfI__pf,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ! [A8: set_o] :
          ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A8),image_o_o(R,A8))
         => ( A8 = bot_bot_set_o ) )
     => wf_o(R) ) ).

% wfI_pf
tff(fact_2959_wfI__pf,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ! [A8: set_int] :
          ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A8),image_int_int(R,A8))
         => ( A8 = bot_bot_set_int ) )
     => wf_int(R) ) ).

% wfI_pf
tff(fact_2960_wfI__pf,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] :
      ( ! [A8: set_fu3555266933746557007_nat_o] :
          ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A8),image_2188787866847195539_nat_o(R,A8))
         => ( A8 = bot_bo8632255151191677115_nat_o ) )
     => wf_fun7865846636378006616_nat_o(R) ) ).

% wfI_pf
tff(fact_2961_wfI__pf,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [A8: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A8),aa_set_nat_set_nat(image_nat_nat(R),A8))
         => ( A8 = bot_bot_set_nat ) )
     => wf_nat(R) ) ).

% wfI_pf
tff(fact_2962_wfE__pf,axiom,
    ! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
      ( wf_Product_unit(R)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),image_8431784415008483091t_unit(R,A))
       => ( A = bot_bo3957492148770167129t_unit ) ) ) ).

% wfE_pf
tff(fact_2963_wfE__pf,axiom,
    ! [R: set_Product_prod_o_o,A: set_o] :
      ( wf_o(R)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),image_o_o(R,A))
       => ( A = bot_bot_set_o ) ) ) ).

% wfE_pf
tff(fact_2964_wfE__pf,axiom,
    ! [R: set_Pr958786334691620121nt_int,A: set_int] :
      ( wf_int(R)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),image_int_int(R,A))
       => ( A = bot_bot_set_int ) ) ) ).

% wfE_pf
tff(fact_2965_wfE__pf,axiom,
    ! [R: set_Pr2890355626087285447_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),image_2188787866847195539_nat_o(R,A))
       => ( A = bot_bo8632255151191677115_nat_o ) ) ) ).

% wfE_pf
tff(fact_2966_wfE__pf,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat] :
      ( wf_nat(R)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(image_nat_nat(R),A))
       => ( A = bot_bot_set_nat ) ) ) ).

% wfE_pf
tff(fact_2967_wf__eq__minimal2,axiom,
    ! [R2: set_Pr5094982260447487303t_unit] :
      ( wf_Product_unit(R2)
    <=> ! [A9: set_Product_unit] :
          ( ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A9),field_Product_unit(R2))
            & ( A9 != bot_bo3957492148770167129t_unit ) )
         => ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A9)
              & ! [Xa2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),A9)
                 => ~ member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(Xa2),X2),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_2968_wf__eq__minimal2,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( wf_o(R2)
    <=> ! [A9: set_o] :
          ( ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A9),field_o(R2))
            & ( A9 != bot_bot_set_o ) )
         => ? [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A9)
              & ! [Xa2: $o] :
                  ( aa_set_o_o(member_o((Xa2)),A9)
                 => ~ member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa2)),(X2)),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_2969_wf__eq__minimal2,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( wf_int(R2)
    <=> ! [A9: set_int] :
          ( ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A9),field_int(R2))
            & ( A9 != bot_bot_set_int ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A9)
              & ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),A9)
                 => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa2),X2),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_2970_wf__eq__minimal2,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R2)
    <=> ! [A9: set_fu3555266933746557007_nat_o] :
          ( ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A9),field_48612661033931452_nat_o(R2))
            & ( A9 != bot_bo8632255151191677115_nat_o ) )
         => ? [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A9)
              & ! [Xa2: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa2),A9)
                 => ~ member3012145478770660624_nat_o(produc4714251849521319255_nat_o(Xa2,X2),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_2971_wf__eq__minimal2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( wf_nat(R2)
    <=> ! [A9: set_nat] :
          ( ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A9),field_nat(R2))
            & ( A9 != bot_bot_set_nat ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A9)
              & ! [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),A9)
                 => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),X2),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_2972_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: fun_Pr1520749766685617582unit_o] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))),field_Product_unit(R2))
       => ( ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Phi,A2),B) )
         => ( ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),A2),R2)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Phi,A2),B) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_2973_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: fun_o_fun_o_o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))),field_o(R2))
       => ( ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2)
           => aa_o_o(aa_o_fun_o_o(Phi,(A2)),(B)) )
         => ( ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(A2)),R2)
             => aa_o_o(aa_o_fun_o_o(Phi,(A2)),(B)) )
           => aa_o_o(aa_o_fun_o_o(Phi,(A2)),(B)) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_2974_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: fun_int_fun_int_o] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))),field_int(R2))
       => ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
           => aa_int_o(aa_int_fun_int_o(Phi,A2),B) )
         => ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),R2)
             => aa_int_o(aa_int_fun_int_o(Phi,A2),B) )
           => aa_int_o(aa_int_fun_int_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_2975_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o,Phi: fun_fu4164567611050918574at_o_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))),field_48612661033931452_nat_o(R2))
       => ( ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2)
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Phi,A2),B) )
         => ( ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,A2),R2)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Phi,A2),B) )
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_2976_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: fun_nat_fun_nat_o] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))),field_nat(R2))
       => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
           => aa_nat_o(aa_nat_fun_nat_o(Phi,A2),B) )
         => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),R2)
             => aa_nat_o(aa_nat_fun_nat_o(Phi,A2),B) )
           => aa_nat_o(aa_nat_fun_nat_o(Phi,A2),B) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_2977_wf__no__path,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( ( inf_in4660618365625256667t_unit(domain4292024440186670924t_unit(R),range_6617724647562260917t_unit(R)) = bot_bo3957492148770167129t_unit )
     => wf_Product_unit(R) ) ).

% wf_no_path
tff(fact_2978_wf__no__path,axiom,
    ! [R: set_Product_prod_o_o] :
      ( ( inf_inf_set_o(domain_o_o(R),range_o_o(R)) = bot_bot_set_o )
     => wf_o(R) ) ).

% wf_no_path
tff(fact_2979_wf__no__path,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,domain_nat_nat(R)),range_nat_nat(R)) = bot_bot_set_nat )
     => wf_nat(R) ) ).

% wf_no_path
tff(fact_2980_wf__no__path,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ( inf_inf_set_int(domain_int_int(R),range_int_int(R)) = bot_bot_set_int )
     => wf_int(R) ) ).

% wf_no_path
tff(fact_2981_wf__no__path,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(domain5580505327626972364_nat_o(R),range_307841406276416053_nat_o(R)) = bot_bo8632255151191677115_nat_o )
     => wf_fun7865846636378006616_nat_o(R) ) ).

% wf_no_path
tff(fact_2982_wf__min,axiom,
    ! [R: set_Pr2890355626087285447_nat_o] :
      ( wf_fun7865846636378006616_nat_o(R)
     => ( ( R != bot_bo6763312114214544691_nat_o )
       => ~ ! [M2: fun_Pr6460944699146269935_nat_o] : ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(M2),minus_6456595905716665654_nat_o(domain5580505327626972364_nat_o(R),range_307841406276416053_nat_o(R))) ) ) ).

% wf_min
tff(fact_2983_wf__min,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( wf_int(R)
     => ( ( R != bot_bo1796632182523588997nt_int )
       => ~ ! [M2: int] : ~ aa_set_int_o(member_int(M2),minus_minus_set_int(domain_int_int(R),range_int_int(R))) ) ) ).

% wf_min
tff(fact_2984_wf__min,axiom,
    ! [R: set_Product_prod_o_o] :
      ( wf_o(R)
     => ( ( R != bot_bo7073875226086086771od_o_o )
       => ~ ! [M2: $o] : ~ aa_set_o_o(member_o((M2)),minus_minus_set_o(domain_o_o(R),range_o_o(R))) ) ) ).

% wf_min
tff(fact_2985_wf__min,axiom,
    ! [R: set_Pr5094982260447487303t_unit] :
      ( wf_Product_unit(R)
     => ( ( R != bot_bo1654022762305692595t_unit )
       => ~ ! [M2: product_unit] : ~ aa_set8780210009671629607unit_o(member_Product_unit(M2),minus_6452836326544984404t_unit(domain4292024440186670924t_unit(R),range_6617724647562260917t_unit(R))) ) ) ).

% wf_min
tff(fact_2986_wf__min,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( wf_nat(R)
     => ( ( R != bot_bo2099793752762293965at_nat )
       => ~ ! [M2: nat] : ~ aa_set_nat_o(member_nat(M2),aa_set_nat_set_nat(minus_minus_set_nat(domain_nat_nat(R)),range_nat_nat(R))) ) ) ).

% wf_min
tff(fact_2987_underS__incl__iff,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( order_53248997263766692_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(order_7714984474677380533_nat_o(R2,A2)),order_7714984474677380533_nat_o(R2,B))
          <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_2988_underS__incl__iff,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( order_6238756238976269133r_on_o(field_o(R2),R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,aa_o_set_o(order_underS_o(R2),(A2))),aa_o_set_o(order_underS_o(R2),(B)))
          <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_2989_underS__incl__iff,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( order_888282469287196226t_unit(field_Product_unit(R2),R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R2),A2)),aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R2),B))
          <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_2990_underS__incl__iff,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( order_4471489696718655927on_int(field_int(R2),R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(order_underS_int(R2),A2)),aa_int_set_int(order_underS_int(R2),B))
          <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_2991_underS__incl__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( order_4473980167227706203on_nat(field_nat(R2),R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(order_underS_nat(R2),A2)),aa_nat_set_nat(order_underS_nat(R2),B))
          <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_2992_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),field_Product_unit(R2))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ? [X_1: product_unit] : aa_Product_unit_o(bNF_We909621271159000003t_unit(R2,B4),X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_2993_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R2: set_Product_prod_o_o,B4: set_o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),field_o(R2))
       => ( ( B4 != bot_bot_set_o )
         => ? [X_1: $o] : aa_o_o(bNF_We4647864369647134286inim_o(R2,B4),(X_1)) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_2994_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
       => ( ( B4 != bot_bot_set_int )
         => ? [X_1: int] : aa_int_o(bNF_We6511469253196982518im_int(R2,B4),X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_2995_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),field_48612661033931452_nat_o(R2))
       => ( ( B4 != bot_bo8632255151191677115_nat_o )
         => ? [X_1: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(bNF_We6692912380981919653_nat_o(R2,B4),X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_2996_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
       => ( ( B4 != bot_bot_set_nat )
         => ? [X_1: nat] : aa_nat_o(bNF_We6513959723706032794im_nat(R2,B4),X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_2997_Refl__under__underS,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
      ( refl_on_Product_unit(field_Product_unit(R2),R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_Pro234974305641502940t_unit(order_6949322460238004850t_unit(R2),A2) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R2),A2)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)) ) ) ) ).

% Refl_under_underS
tff(fact_2998_Refl__under__underS,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o] :
      ( refl_on_o(field_o(R2),R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_o_set_o(order_under_o(R2),(A2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_o_set_o(order_underS_o(R2),(A2))),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)) ) ) ) ).

% Refl_under_underS
tff(fact_2999_Refl__under__underS,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( refl_on_nat(field_nat(R2),R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_nat_set_nat(order_under_nat(R2),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(order_underS_nat(R2),A2)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat)) ) ) ) ).

% Refl_under_underS
tff(fact_3000_Refl__under__underS,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int] :
      ( refl_on_int(field_int(R2),R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_int_set_int(order_under_int(R2),A2) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_int_set_int(order_underS_int(R2),A2)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)) ) ) ) ).

% Refl_under_underS
tff(fact_3001_Refl__under__underS,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( refl_o5094713388964525678_nat_o(field_48612661033931452_nat_o(R2),R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( order_7476434973341305044_nat_o(R2,A2) = aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,order_7714984474677380533_nat_o(R2,A2)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o)) ) ) ) ).

% Refl_under_underS
tff(fact_3002_wo__rel_Ominim__inField,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),field_Product_unit(R2))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => aa_set8780210009671629607unit_o(member_Product_unit(bNF_We7185707985517244377t_unit(R2,B4)),field_Product_unit(R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_3003_wo__rel_Ominim__inField,axiom,
    ! [R2: set_Product_prod_o_o,B4: set_o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),field_o(R2))
       => ( ( B4 != bot_bot_set_o )
         => aa_set_o_o(member_o(aa_set_o_o(bNF_We537455116209922916inim_o(R2),B4)),field_o(R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_3004_wo__rel_Ominim__inField,axiom,
    ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
       => ( ( B4 != bot_bot_set_int )
         => aa_set_int_o(member_int(bNF_We3272633801743244640im_int(R2,B4)),field_int(R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_3005_wo__rel_Ominim__inField,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),field_48612661033931452_nat_o(R2))
       => ( ( B4 != bot_bo8632255151191677115_nat_o )
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(bNF_We3125803481452543163_nat_o(R2,B4)),field_48612661033931452_nat_o(R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_3006_wo__rel_Ominim__inField,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
       => ( ( B4 != bot_bot_set_nat )
         => aa_set_nat_o(member_nat(aa_set_nat_nat(bNF_We3275124272252294916im_nat(R2),B4)),field_nat(R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_3007_wo__rel_Ominim__in,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),field_Product_unit(R2))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => aa_set8780210009671629607unit_o(member_Product_unit(bNF_We7185707985517244377t_unit(R2,B4)),B4) ) ) ) ).

% wo_rel.minim_in
tff(fact_3008_wo__rel_Ominim__in,axiom,
    ! [R2: set_Product_prod_o_o,B4: set_o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),field_o(R2))
       => ( ( B4 != bot_bot_set_o )
         => aa_set_o_o(member_o(aa_set_o_o(bNF_We537455116209922916inim_o(R2),B4)),B4) ) ) ) ).

% wo_rel.minim_in
tff(fact_3009_wo__rel_Ominim__in,axiom,
    ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
       => ( ( B4 != bot_bot_set_int )
         => aa_set_int_o(member_int(bNF_We3272633801743244640im_int(R2,B4)),B4) ) ) ) ).

% wo_rel.minim_in
tff(fact_3010_wo__rel_Ominim__in,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),field_48612661033931452_nat_o(R2))
       => ( ( B4 != bot_bo8632255151191677115_nat_o )
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(bNF_We3125803481452543163_nat_o(R2,B4)),B4) ) ) ) ).

% wo_rel.minim_in
tff(fact_3011_wo__rel_Ominim__in,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
       => ( ( B4 != bot_bot_set_nat )
         => aa_set_nat_o(member_nat(aa_set_nat_nat(bNF_We3275124272252294916im_nat(R2),B4)),B4) ) ) ) ).

% wo_rel.minim_in
tff(fact_3012_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,B4: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
         => ( ! [B2: fun_Pr6460944699146269935_nat_o] :
                ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B2),B4)
               => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B2),R2) )
           => ( A2 = bNF_We3125803481452543163_nat_o(R2,B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_3013_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Product_prod_o_o,B4: set_o,A2: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),field_o(R2))
       => ( aa_set_o_o(member_o((A2)),B4)
         => ( ! [B2: $o] :
                ( aa_set_o_o(member_o((B2)),B4)
               => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B2)),R2) )
           => ( (A2)
            <=> aa_set_o_o(bNF_We537455116209922916inim_o(R2),B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_3014_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,A2: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
         => ( ! [B2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(B2),B4)
               => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B2),R2) )
           => ( A2 = bNF_We7185707985517244377t_unit(R2,B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_3015_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr958786334691620121nt_int,B4: set_int,A2: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
       => ( aa_set_int_o(member_int(A2),B4)
         => ( ! [B2: int] :
                ( aa_set_int_o(member_int(B2),B4)
               => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),R2) )
           => ( A2 = bNF_We3272633801743244640im_int(R2,B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_3016_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,A2: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
       => ( aa_set_nat_o(member_nat(A2),B4)
         => ( ! [B2: nat] :
                ( aa_set_nat_o(member_nat(B2),B4)
               => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),R2) )
           => ( A2 = aa_set_nat_nat(bNF_We3275124272252294916im_nat(R2),B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_3017_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,B4: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),B4)
         => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(bNF_We3125803481452543163_nat_o(R2,B4),B),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_3018_wo__rel_Ominim__least,axiom,
    ! [R2: set_Product_prod_o_o,B4: set_o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),field_o(R2))
       => ( aa_set_o_o(member_o((B)),B4)
         => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o(aa_set_o_o(bNF_We537455116209922916inim_o(R2),B4)),(B)),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_3019_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),B4)
         => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(bNF_We7185707985517244377t_unit(R2,B4)),B),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_3020_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr958786334691620121nt_int,B4: set_int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
       => ( aa_set_int_o(member_int(B),B4)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(bNF_We3272633801743244640im_int(R2,B4)),B),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_3021_wo__rel_Ominim__least,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),B4)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_set_nat_nat(bNF_We3275124272252294916im_nat(R2),B4)),B),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_3022_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B)),R2)
            & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B)),R2)
            & aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_3023_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))),R2)
            & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))),R2)
            & aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_3024_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B)),R2)
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B)),R2)
            & aa_set_nat_o(member_nat(aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_3025_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B)),R2)
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B)),R2)
            & aa_set_int_o(member_int(aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_3026_wo__rel_Omax2__greater__among,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B)),R2)
            & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B)),R2)
            & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_3027_wo__rel_Omax2__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B) = $ite(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2),B,A2) ) ) ).

% wo_rel.max2_def
tff(fact_3028_wo__rel_Omax2__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B) = $ite(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2),B,A2) ) ) ).

% wo_rel.max2_def
tff(fact_3029_under__incr,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( trans_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2)
       => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,aa_int_set_int(order_under_int(R2),A2)),aa_int_set_int(order_under_int(R2),B)) ) ) ).

% under_incr
tff(fact_3030_under__incr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( trans_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(order_under_nat(R2),A2)),aa_nat_set_nat(order_under_nat(R2),B)) ) ) ).

% under_incr
tff(fact_3031_wo__rel_Ominim__isMinim,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),field_Product_unit(R2))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => aa_Product_unit_o(bNF_We909621271159000003t_unit(R2,B4),bNF_We7185707985517244377t_unit(R2,B4)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_3032_wo__rel_Ominim__isMinim,axiom,
    ! [R2: set_Product_prod_o_o,B4: set_o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),field_o(R2))
       => ( ( B4 != bot_bot_set_o )
         => aa_o_o(bNF_We4647864369647134286inim_o(R2,B4),aa_set_o_o(bNF_We537455116209922916inim_o(R2),B4)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_3033_wo__rel_Ominim__isMinim,axiom,
    ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),field_int(R2))
       => ( ( B4 != bot_bot_set_int )
         => aa_int_o(bNF_We6511469253196982518im_int(R2,B4),bNF_We3272633801743244640im_int(R2,B4)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_3034_wo__rel_Ominim__isMinim,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(B4),field_48612661033931452_nat_o(R2))
       => ( ( B4 != bot_bo8632255151191677115_nat_o )
         => aa_fun1675921005893230629at_o_o(bNF_We6692912380981919653_nat_o(R2,B4),bNF_We3125803481452543163_nat_o(R2,B4)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_3035_wo__rel_Ominim__isMinim,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),field_nat(R2))
       => ( ( B4 != bot_bot_set_nat )
         => aa_nat_o(bNF_We6513959723706032794im_nat(R2,B4),aa_set_nat_nat(bNF_We3275124272252294916im_nat(R2),B4)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_3036_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_fun1675921005893230629at_o_o(bNF_We6692912380981919653_nat_o(R2,A),B)
      <=> ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A)
          & ! [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),A)
             => member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,X2),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_3037_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Product_prod_o_o,A: set_o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_o_o(bNF_We4647864369647134286inim_o(R2,A),(B))
      <=> ( aa_set_o_o(member_o((B)),A)
          & ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(X2)),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_3038_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A: set_Product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_Product_unit_o(bNF_We909621271159000003t_unit(R2,A),B)
      <=> ( aa_set8780210009671629607unit_o(member_Product_unit(B),A)
          & ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),X2),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_3039_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_nat_o(bNF_We6513959723706032794im_nat(R2,A),B)
      <=> ( aa_set_nat_o(member_nat(B),A)
          & ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),X2),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_3040_wo__rel_OisMinim__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A: set_int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_int_o(bNF_We6511469253196982518im_int(R2,A),B)
      <=> ( aa_set_int_o(member_int(B),A)
          & ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),X2),R2) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_3041_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B)),R2)
            & member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B)),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_3042_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))),R2)
            & member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_3043_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B)),R2)
            & member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B)),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_3044_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B)),R2)
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B)),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_3045_wo__rel_Omax2__greater,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B)),R2)
            & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B)),R2) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_3046_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( ( aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B) = B )
          <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(A2,B),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_3047_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( ( aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))
            <=> (B) )
          <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A2)),(B)),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_3048_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( ( aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B) = B )
          <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(A2),B),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_3049_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( ( aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B) = B )
          <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_3050_wo__rel_Omax2__equals2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( ( aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B) = B )
          <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R2) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_3051_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => ( ( aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B) = A2 )
          <=> member3012145478770660624_nat_o(produc4714251849521319255_nat_o(B,A2),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_3052_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => ( ( aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))
            <=> (A2) )
          <=> member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(A2)),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_3053_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => ( ( aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B) = A2 )
          <=> member7821138191923116944t_unit(aa_Pro457057934763163478t_unit(produc5692694829384537815t_unit(B),A2),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_3054_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => ( ( aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B) = A2 )
          <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),A2),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_3055_wo__rel_Omax2__equals1,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => ( ( aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B) = A2 )
          <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B),A2),R2) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_3056_wo__rel_Omax2__among,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
      ( bNF_We2091578341578594809t_unit(R2)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),field_Product_unit(R2))
       => ( aa_set8780210009671629607unit_o(member_Product_unit(B),field_Product_unit(R2))
         => aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(bNF_We7204996500647184731t_unit(R2,A2),B)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit))) ) ) ) ).

% wo_rel.max2_among
tff(fact_3057_wo__rel_Omax2__among,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( bNF_We6437023980930612612_rel_o(R2)
     => ( aa_set_o_o(member_o((A2)),field_o(R2))
       => ( aa_set_o_o(member_o((B)),field_o(R2))
         => aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(bNF_We3104086342435940326max2_o(R2),(A2)),(B))),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o))) ) ) ) ).

% wo_rel.max2_among
tff(fact_3058_wo__rel_Omax2__among,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( bNF_We3818239936649020644el_nat(R2)
     => ( aa_set_nat_o(member_nat(A2),field_nat(R2))
       => ( aa_set_nat_o(member_nat(B),field_nat(R2))
         => aa_set_nat_o(member_nat(aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A2),B)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat))) ) ) ) ).

% wo_rel.max2_among
tff(fact_3059_wo__rel_Omax2__among,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
      ( bNF_We3815749466139970368el_int(R2)
     => ( aa_set_int_o(member_int(A2),field_int(R2))
       => ( aa_set_int_o(member_int(B),field_int(R2))
         => aa_set_int_o(member_int(aa_int_int(bNF_We2378448595015613790x2_int(R2,A2),B)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int))) ) ) ) ).

% wo_rel.max2_among
tff(fact_3060_wo__rel_Omax2__among,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( bNF_We3617305991974360795_nat_o(R2)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),field_48612661033931452_nat_o(R2))
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),field_48612661033931452_nat_o(R2))
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(bNF_We5143902115314319677_nat_o(R2,A2),B)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o))) ) ) ) ).

% wo_rel.max2_among
tff(fact_3061_chains__extend,axiom,
    ! [C: set_set_nat,S: set_set_nat,Z: set_nat] :
      ( member_set_set_nat(C,chains_nat2(S))
     => ( member_set_nat(Z,S)
       => ( ! [X3: set_nat] :
              ( member_set_nat(X3,C)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Z) )
         => member_set_set_nat(sup_sup_set_set_nat(insert_set_nat(Z,bot_bot_set_set_nat),C),chains_nat2(S)) ) ) ) ).

% chains_extend
tff(fact_3062_Zorns__po__lemma,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( order_5251275573222108571on_nat(field_nat(R2),R2)
     => ( ! [C6: set_nat] :
            ( member_set_nat(C6,chains_nat(R2))
           => ? [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),field_nat(R2))
                & ! [Xa4: nat] :
                    ( aa_set_nat_o(member_nat(Xa4),C6)
                   => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa4),X4),R2) ) ) )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),field_nat(R2))
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),field_nat(R2))
               => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3),R2)
                 => ( Xa3 = X3 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_3063_Zorns__po__lemma,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( order_5248785102713058295on_int(field_int(R2),R2)
     => ( ! [C6: set_int] :
            ( member_set_int(C6,chains_int(R2))
           => ? [X4: int] :
                ( aa_set_int_o(member_int(X4),field_int(R2))
                & ! [Xa4: int] :
                    ( aa_set_int_o(member_int(Xa4),C6)
                   => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Xa4),X4),R2) ) ) )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),field_int(R2))
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),field_int(R2))
               => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Xa3),R2)
                 => ( Xa3 = X3 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_3064_bsqr__max2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
      ( order_2888998067076097458on_nat(field_nat(R2),R2)
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B1),B22)),bNF_We1732890022510290291qr_nat(R2))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,A1),A22)),aa_nat_nat(bNF_We2380939065524664066x2_nat(R2,B1),B22)),R2) ) ) ).

% bsqr_max2
tff(fact_3065_bsqr__max2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A1: int,A22: int,B1: int,B22: int] :
      ( order_2886507596567047182on_int(field_int(R2),R2)
     => ( member8566619992076573584nt_int(produc3646306378393792727nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A1),A22),aa_int7462565939943104719nt_int(product_Pair_int_int(B1),B22)),bNF_We1730399552001240015qr_int(R2))
       => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(bNF_We2378448595015613790x2_int(R2,A1),A22)),aa_int_int(bNF_We2378448595015613790x2_int(R2,B1),B22)),R2) ) ) ).

% bsqr_max2
tff(fact_3066_wf__Union,axiom,
    ! [R: set_se8994758365886191399t_unit] :
      ( ! [X3: set_Pr5094982260447487303t_unit] :
          ( member1248447801810324848t_unit(X3,R)
         => wf_Product_unit(X3) )
     => ( ! [X3: set_Pr5094982260447487303t_unit] :
            ( member1248447801810324848t_unit(X3,R)
           => ! [Xa4: set_Pr5094982260447487303t_unit] :
                ( member1248447801810324848t_unit(Xa4,R)
               => ( ( X3 != Xa4 )
                 => ( inf_in4660618365625256667t_unit(domain4292024440186670924t_unit(X3),range_6617724647562260917t_unit(Xa4)) = bot_bo3957492148770167129t_unit ) ) ) )
       => wf_Product_unit(comple2798140142350441396t_unit(R)) ) ) ).

% wf_Union
tff(fact_3067_wf__Union,axiom,
    ! [R: set_se491199571785101031od_o_o] :
      ( ! [X3: set_Product_prod_o_o] :
          ( member3157015407008874800od_o_o(X3,R)
         => wf_o(X3) )
     => ( ! [X3: set_Product_prod_o_o] :
            ( member3157015407008874800od_o_o(X3,R)
           => ! [Xa4: set_Product_prod_o_o] :
                ( member3157015407008874800od_o_o(Xa4,R)
               => ( ( X3 != Xa4 )
                 => ( inf_inf_set_o(domain_o_o(X3),range_o_o(Xa4)) = bot_bot_set_o ) ) ) )
       => wf_o(comple2567785453016644468od_o_o(R)) ) ) ).

% wf_Union
tff(fact_3068_wf__Union,axiom,
    ! [R: set_se7855581050983116737at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X3,R)
         => wf_nat(X3) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(X3,R)
           => ! [Xa4: set_Pr1261947904930325089at_nat] :
                ( member2643936169264416010at_nat(Xa4,R)
               => ( ( X3 != Xa4 )
                 => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,domain_nat_nat(X3)),range_nat_nat(Xa4)) = bot_bot_set_nat ) ) ) )
       => wf_nat(comple5685304695842803022at_nat(R)) ) ) ).

% wf_Union
tff(fact_3069_wf__Union,axiom,
    ! [R: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( member2340774599025711042nt_int(X3,R)
         => wf_int(X3) )
     => ( ! [X3: set_Pr958786334691620121nt_int] :
            ( member2340774599025711042nt_int(X3,R)
           => ! [Xa4: set_Pr958786334691620121nt_int] :
                ( member2340774599025711042nt_int(Xa4,R)
               => ( ( X3 != Xa4 )
                 => ( inf_inf_set_int(domain_int_int(X3),range_int_int(Xa4)) = bot_bot_set_int ) ) ) )
       => wf_int(comple5382143125604098054nt_int(R)) ) ) ).

% wf_Union
tff(fact_3070_wf__Union,axiom,
    ! [R: set_se7341084505648470951_nat_o] :
      ( ! [X3: set_Pr2890355626087285447_nat_o] :
          ( member1641360573044772336_nat_o(X3,R)
         => wf_fun7865846636378006616_nat_o(X3) )
     => ( ! [X3: set_Pr2890355626087285447_nat_o] :
            ( member1641360573044772336_nat_o(X3,R)
           => ! [Xa4: set_Pr2890355626087285447_nat_o] :
                ( member1641360573044772336_nat_o(Xa4,R)
               => ( ( X3 != Xa4 )
                 => ( inf_in3376754634429691197_nat_o(domain5580505327626972364_nat_o(X3),range_307841406276416053_nat_o(Xa4)) = bot_bo8632255151191677115_nat_o ) ) ) )
       => wf_fun7865846636378006616_nat_o(comple3438388261678958644_nat_o(R)) ) ) ).

% wf_Union
tff(fact_3071_Pow__singleton__iff,axiom,
    ! [X5: set_Product_unit,Y5: set_Product_unit] :
      ( ( pow_Product_unit(X5) = insert8986028188087627965t_unit(Y5,bot_bo6284674557190688313t_unit) )
    <=> ( ( X5 = bot_bo3957492148770167129t_unit )
        & ( Y5 = bot_bo3957492148770167129t_unit ) ) ) ).

% Pow_singleton_iff
tff(fact_3072_Pow__singleton__iff,axiom,
    ! [X5: set_o,Y5: set_o] :
      ( ( pow_o(X5) = insert_set_o(Y5,bot_bot_set_set_o) )
    <=> ( ( X5 = bot_bot_set_o )
        & ( Y5 = bot_bot_set_o ) ) ) ).

% Pow_singleton_iff
tff(fact_3073_Pow__singleton__iff,axiom,
    ! [X5: set_nat,Y5: set_nat] :
      ( ( pow_nat(X5) = insert_set_nat(Y5,bot_bot_set_set_nat) )
    <=> ( ( X5 = bot_bot_set_nat )
        & ( Y5 = bot_bot_set_nat ) ) ) ).

% Pow_singleton_iff
tff(fact_3074_Pow__singleton__iff,axiom,
    ! [X5: set_int,Y5: set_int] :
      ( ( pow_int(X5) = insert_set_int(Y5,bot_bot_set_set_int) )
    <=> ( ( X5 = bot_bot_set_int )
        & ( Y5 = bot_bot_set_int ) ) ) ).

% Pow_singleton_iff
tff(fact_3075_Pow__singleton__iff,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,Y5: set_fu3555266933746557007_nat_o] :
      ( ( pow_fu4138109184033261068_nat_o(X5) = insert755438361227728415_nat_o(Y5,bot_bo4069457504818663451_nat_o) )
    <=> ( ( X5 = bot_bo8632255151191677115_nat_o )
        & ( Y5 = bot_bo8632255151191677115_nat_o ) ) ) ).

% Pow_singleton_iff
tff(fact_3076_Pow__empty,axiom,
    pow_Product_unit(bot_bo3957492148770167129t_unit) = insert8986028188087627965t_unit(bot_bo3957492148770167129t_unit,bot_bo6284674557190688313t_unit) ).

% Pow_empty
tff(fact_3077_Pow__empty,axiom,
    pow_o(bot_bot_set_o) = insert_set_o(bot_bot_set_o,bot_bot_set_set_o) ).

% Pow_empty
tff(fact_3078_Pow__empty,axiom,
    pow_nat(bot_bot_set_nat) = insert_set_nat(bot_bot_set_nat,bot_bot_set_set_nat) ).

% Pow_empty
tff(fact_3079_Pow__empty,axiom,
    pow_int(bot_bot_set_int) = insert_set_int(bot_bot_set_int,bot_bot_set_set_int) ).

% Pow_empty
tff(fact_3080_Pow__empty,axiom,
    pow_fu4138109184033261068_nat_o(bot_bo8632255151191677115_nat_o) = insert755438361227728415_nat_o(bot_bo8632255151191677115_nat_o,bot_bo4069457504818663451_nat_o) ).

% Pow_empty
tff(fact_3081_rtrancl__mapI,axiom,
    ! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(E))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,A2)),aa_nat_nat(F,B)),transi2905341329935302413cl_nat(image_5168914502847457605at_nat(pairself_nat_nat(F),E))) ) ).

% rtrancl_mapI
tff(fact_3082_rtrancl__mapI,axiom,
    ! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: fun_nat_int] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),transi2905341329935302413cl_nat(E))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,A2)),aa_nat_int(F,B)),transi2902850859426252137cl_int(image_1990417195239315453nt_int(pairself_nat_int(F),E))) ) ).

% rtrancl_mapI
tff(fact_3083_rtrancl__mapI,axiom,
    ! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(E))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,A2)),aa_int_nat(F,B)),transi2905341329935302413cl_nat(image_5831868185956570253at_nat(pairself_int_nat(F),E))) ) ).

% rtrancl_mapI
tff(fact_3084_rtrancl__mapI,axiom,
    ! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: fun_int_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),transi2902850859426252137cl_int(E))
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,A2)),aa_int_int(F,B)),transi2902850859426252137cl_int(image_2653370878348428101nt_int(pairself_int_int(F),E))) ) ).

% rtrancl_mapI
tff(fact_3085_disjoint__image__subset,axiom,
    ! [A11: set_set_nat,F: fun_set_nat_set_nat] :
      ( pairwise_set_nat(disjnt_nat,A11)
     => ( ! [X7: set_nat] :
            ( member_set_nat(X7,A11)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(F,X7)),X7) )
       => pairwise_set_nat(disjnt_nat,image_7916887816326733075et_nat(F,A11)) ) ) ).

% disjoint_image_subset
tff(fact_3086_pairself_Opelims,axiom,
    ! [X: fun_nat_nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro262675715343573244at_nat(pairself_nat_nat(X),Xa) = Y )
     => ( accp_P952520887328684368at_nat(pairself_rel_nat_nat,produc7556547088534825035at_nat(X,Xa))
       => ~ ! [A3: nat,B2: nat] :
              ( ( Xa = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) )
             => ( ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(X,A3)),aa_nat_nat(X,B2)) )
               => ~ accp_P952520887328684368at_nat(pairself_rel_nat_nat,produc7556547088534825035at_nat(X,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2))) ) ) ) ) ).

% pairself.pelims
tff(fact_3087_pairself_Opelims,axiom,
    ! [X: fun_nat_int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
      ( ( aa_Pro6307550444590206900nt_int(pairself_nat_int(X),Xa) = Y )
     => ( accp_P9078809943539461620at_nat(pairself_rel_nat_int,produc6459464107890826479at_nat(X,Xa))
       => ~ ! [A3: nat,B2: nat] :
              ( ( Xa = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) )
             => ( ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(X,A3)),aa_nat_int(X,B2)) )
               => ~ accp_P9078809943539461620at_nat(pairself_rel_nat_int,produc6459464107890826479at_nat(X,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2))) ) ) ) ) ).

% pairself.pelims
tff(fact_3088_pairself_Opelims,axiom,
    ! [X: fun_int_nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
      ( ( aa_Pro925629398452685892at_nat(pairself_int_nat(X),Xa) = Y )
     => ( accp_P2409395980843090604nt_int(pairself_rel_int_nat,produc6138086444679794087nt_int(X,Xa))
       => ~ ! [A3: int,B2: int] :
              ( ( Xa = aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2) )
             => ( ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(X,A3)),aa_int_nat(X,B2)) )
               => ~ accp_P2409395980843090604nt_int(pairself_rel_int_nat,produc6138086444679794087nt_int(X,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2))) ) ) ) ) ).

% pairself.pelims
tff(fact_3089_pairself_Opelims,axiom,
    ! [X: fun_int_int,Xa: product_prod_int_int,Y: product_prod_int_int] :
      ( ( aa_Pro6970504127699319548nt_int(pairself_int_int(X),Xa) = Y )
     => ( accp_P1312313000199092048nt_int(pairself_rel_int_int,produc5041003464035795531nt_int(X,Xa))
       => ~ ! [A3: int,B2: int] :
              ( ( Xa = aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2) )
             => ( ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(X,A3)),aa_int_int(X,B2)) )
               => ~ accp_P1312313000199092048nt_int(pairself_rel_int_int,produc5041003464035795531nt_int(X,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2))) ) ) ) ) ).

% pairself.pelims
tff(fact_3090_Pow__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( member_set_nat(A,pow_nat(B4))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ).

% Pow_iff
tff(fact_3091_PowI,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => member_set_nat(A,pow_nat(B4)) ) ).

% PowI
tff(fact_3092_Pow__Int__eq,axiom,
    ! [A: set_nat,B4: set_nat] : pow_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) = inf_inf_set_set_nat(pow_nat(A),pow_nat(B4)) ).

% Pow_Int_eq
tff(fact_3093_pairwiseI,axiom,
    ! [S: set_fu3555266933746557007_nat_o,R: fun_fu4164567611050918574at_o_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),S)
         => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),S)
           => ( ( X3 != Y2 )
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X3),Y2) ) ) )
     => aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(R),S) ) ).

% pairwiseI
tff(fact_3094_pairwiseI,axiom,
    ! [S: set_nat,R: fun_nat_fun_nat_o] :
      ( ! [X3: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X3),S)
         => ( aa_set_nat_o(member_nat(Y2),S)
           => ( ( X3 != Y2 )
             => aa_nat_o(aa_nat_fun_nat_o(R,X3),Y2) ) ) )
     => aa_set_nat_o(pairwise_nat(R),S) ) ).

% pairwiseI
tff(fact_3095_pairwiseI,axiom,
    ! [S: set_int,R: fun_int_fun_int_o] :
      ( ! [X3: int,Y2: int] :
          ( aa_set_int_o(member_int(X3),S)
         => ( aa_set_int_o(member_int(Y2),S)
           => ( ( X3 != Y2 )
             => aa_int_o(aa_int_fun_int_o(R,X3),Y2) ) ) )
     => aa_set_int_o(pairwise_int(R),S) ) ).

% pairwiseI
tff(fact_3096_pairwiseI,axiom,
    ! [S: set_o,R: fun_o_fun_o_o] :
      ( ! [X3: $o,Y2: $o] :
          ( aa_set_o_o(member_o((X3)),S)
         => ( aa_set_o_o(member_o((Y2)),S)
           => ( ~ ( (X3)
                <=> (Y2) )
             => aa_o_o(aa_o_fun_o_o(R,(X3)),(Y2)) ) ) )
     => aa_set_o_o(pairwise_o(R),S) ) ).

% pairwiseI
tff(fact_3097_pairwiseI,axiom,
    ! [S: set_Product_unit,R: fun_Pr1520749766685617582unit_o] :
      ( ! [X3: product_unit,Y2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),S)
         => ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),S)
           => ( ( X3 != Y2 )
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X3),Y2) ) ) )
     => aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(R),S) ) ).

% pairwiseI
tff(fact_3098_pairwiseD,axiom,
    ! [R: fun_fu4164567611050918574at_o_o,S: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(R),S)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),S)
         => ( ( X != Y )
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X),Y) ) ) ) ) ).

% pairwiseD
tff(fact_3099_pairwiseD,axiom,
    ! [R: fun_nat_fun_nat_o,S: set_nat,X: nat,Y: nat] :
      ( aa_set_nat_o(pairwise_nat(R),S)
     => ( aa_set_nat_o(member_nat(X),S)
       => ( aa_set_nat_o(member_nat(Y),S)
         => ( ( X != Y )
           => aa_nat_o(aa_nat_fun_nat_o(R,X),Y) ) ) ) ) ).

% pairwiseD
tff(fact_3100_pairwiseD,axiom,
    ! [R: fun_int_fun_int_o,S: set_int,X: int,Y: int] :
      ( aa_set_int_o(pairwise_int(R),S)
     => ( aa_set_int_o(member_int(X),S)
       => ( aa_set_int_o(member_int(Y),S)
         => ( ( X != Y )
           => aa_int_o(aa_int_fun_int_o(R,X),Y) ) ) ) ) ).

% pairwiseD
tff(fact_3101_pairwiseD,axiom,
    ! [R: fun_o_fun_o_o,S: set_o,X: $o,Y: $o] :
      ( aa_set_o_o(pairwise_o(R),S)
     => ( aa_set_o_o(member_o((X)),S)
       => ( aa_set_o_o(member_o((Y)),S)
         => ( ~ ( (X)
              <=> (Y) )
           => aa_o_o(aa_o_fun_o_o(R,(X)),(Y)) ) ) ) ) ).

% pairwiseD
tff(fact_3102_pairwiseD,axiom,
    ! [R: fun_Pr1520749766685617582unit_o,S: set_Product_unit,X: product_unit,Y: product_unit] :
      ( aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(R),S)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),S)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),S)
         => ( ( X != Y )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X),Y) ) ) ) ) ).

% pairwiseD
tff(fact_3103_refl__on__UNION,axiom,
    ! [S: set_nat,A: fun_nat_set_nat,R2: fun_na7044673412300895210at_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),S)
         => refl_on_nat(aa_nat_set_nat(A,X3),aa_nat7062706319057040595at_nat(R2,X3)) )
     => refl_on_nat(comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A),S)),comple5685304695842803022at_nat(image_7178329752028323786at_nat(R2,S))) ) ).

% refl_on_UNION
tff(fact_3104_Pow__bottom,axiom,
    ! [B4: set_Product_unit] : member5877623283571906838t_unit(bot_bo3957492148770167129t_unit,pow_Product_unit(B4)) ).

% Pow_bottom
tff(fact_3105_Pow__bottom,axiom,
    ! [B4: set_o] : member_set_o(bot_bot_set_o,pow_o(B4)) ).

% Pow_bottom
tff(fact_3106_Pow__bottom,axiom,
    ! [B4: set_nat] : member_set_nat(bot_bot_set_nat,pow_nat(B4)) ).

% Pow_bottom
tff(fact_3107_Pow__bottom,axiom,
    ! [B4: set_int] : member_set_int(bot_bot_set_int,pow_int(B4)) ).

% Pow_bottom
tff(fact_3108_Pow__bottom,axiom,
    ! [B4: set_fu3555266933746557007_nat_o] : member361801372707117048_nat_o(bot_bo8632255151191677115_nat_o,pow_fu4138109184033261068_nat_o(B4)) ).

% Pow_bottom
tff(fact_3109_Pow__mono,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ord_le6893508408891458716et_nat(pow_nat(A),pow_nat(B4)) ) ).

% Pow_mono
tff(fact_3110_PowD,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( member_set_nat(A,pow_nat(B4))
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ).

% PowD
tff(fact_3111_Un__Pow__subset,axiom,
    ! [A: set_nat,B4: set_nat] : ord_le6893508408891458716et_nat(sup_sup_set_set_nat(pow_nat(A),pow_nat(B4)),pow_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))) ).

% Un_Pow_subset
tff(fact_3112_pairwise__imageI,axiom,
    ! [A: set_Code_integer,F: fun_Co8499238673658658103nteger,P: fun_Co2081214437782444048eger_o] :
      ( ! [X3: code_integer,Y2: code_integer] :
          ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
         => ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),A)
           => ( ( X3 != Y2 )
             => ( ( aa_Cod1783974652533396728nteger(F,X3) != aa_Cod1783974652533396728nteger(F,Y2) )
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Cod1783974652533396728nteger(F,X3)),aa_Cod1783974652533396728nteger(F,Y2)) ) ) ) )
     => aa_set5643387332196575111eger_o(pairwi4741737654671186738nteger(P),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) ) ).

% pairwise_imageI
tff(fact_3113_pairwise__imageI,axiom,
    ! [A: set_nat,F: fun_nat_set_nat,P: fun_se2170385732985038782_nat_o] :
      ( ! [X3: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_set_nat_o(member_nat(Y2),A)
           => ( ( X3 != Y2 )
             => ( ( aa_nat_set_nat(F,X3) != aa_nat_set_nat(F,Y2) )
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(P,aa_nat_set_nat(F,X3)),aa_nat_set_nat(F,Y2)) ) ) ) )
     => pairwise_set_nat(P,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A)) ) ).

% pairwise_imageI
tff(fact_3114_pairwise__imageI,axiom,
    ! [A: set_nat,F: fun_nat_nat,P: fun_nat_fun_nat_o] :
      ( ! [X3: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_set_nat_o(member_nat(Y2),A)
           => ( ( X3 != Y2 )
             => ( ( aa_nat_nat(F,X3) != aa_nat_nat(F,Y2) )
               => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) ) ) ) )
     => aa_set_nat_o(pairwise_nat(P),aa_set_nat_set_nat(image_nat_nat2(F),A)) ) ).

% pairwise_imageI
tff(fact_3115_pairwise__imageI,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer,P: fun_Co2081214437782444048eger_o] :
      ( ! [X3: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_set_nat_o(member_nat(Y2),A)
           => ( ( X3 != Y2 )
             => ( ( aa_nat_Code_integer(F,X3) != aa_nat_Code_integer(F,Y2) )
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) ) ) ) )
     => aa_set5643387332196575111eger_o(pairwi4741737654671186738nteger(P),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)) ) ).

% pairwise_imageI
tff(fact_3116_pairwise__empty,axiom,
    ! [P: fun_Pr1520749766685617582unit_o] : aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(P),bot_bo3957492148770167129t_unit) ).

% pairwise_empty
tff(fact_3117_pairwise__empty,axiom,
    ! [P: fun_o_fun_o_o] : aa_set_o_o(pairwise_o(P),bot_bot_set_o) ).

% pairwise_empty
tff(fact_3118_pairwise__empty,axiom,
    ! [P: fun_nat_fun_nat_o] : aa_set_nat_o(pairwise_nat(P),bot_bot_set_nat) ).

% pairwise_empty
tff(fact_3119_pairwise__empty,axiom,
    ! [P: fun_int_fun_int_o] : aa_set_int_o(pairwise_int(P),bot_bot_set_int) ).

% pairwise_empty
tff(fact_3120_pairwise__empty,axiom,
    ! [P: fun_fu4164567611050918574at_o_o] : aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(P),bot_bo8632255151191677115_nat_o) ).

% pairwise_empty
tff(fact_3121_pairwise__subset,axiom,
    ! [P: fun_nat_fun_nat_o,S: set_nat,T2: set_nat] :
      ( aa_set_nat_o(pairwise_nat(P),S)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,T2),S)
       => aa_set_nat_o(pairwise_nat(P),T2) ) ) ).

% pairwise_subset
tff(fact_3122_pairwise__mono,axiom,
    ! [P: fun_nat_fun_nat_o,A: set_nat,Q: fun_nat_fun_nat_o,B4: set_nat] :
      ( aa_set_nat_o(pairwise_nat(P),A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,X3),Y2)
           => aa_nat_o(aa_nat_fun_nat_o(Q,X3),Y2) )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
         => aa_set_nat_o(pairwise_nat(Q),B4) ) ) ) ).

% pairwise_mono
tff(fact_3123_pairwise__insert,axiom,
    ! [R2: fun_fu4164567611050918574at_o_o,X: fun_Pr6460944699146269935_nat_o,S2: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(R2),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),S2))
    <=> ( ! [Y3: fun_Pr6460944699146269935_nat_o] :
            ( ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y3),S2)
              & ( Y3 != X ) )
           => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R2,X),Y3)
              & aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R2,Y3),X) ) )
        & aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(R2),S2) ) ) ).

% pairwise_insert
tff(fact_3124_pairwise__insert,axiom,
    ! [R2: fun_nat_fun_nat_o,X: nat,S2: set_nat] :
      ( aa_set_nat_o(pairwise_nat(R2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),S2))
    <=> ( ! [Y3: nat] :
            ( ( aa_set_nat_o(member_nat(Y3),S2)
              & ( Y3 != X ) )
           => ( aa_nat_o(aa_nat_fun_nat_o(R2,X),Y3)
              & aa_nat_o(aa_nat_fun_nat_o(R2,Y3),X) ) )
        & aa_set_nat_o(pairwise_nat(R2),S2) ) ) ).

% pairwise_insert
tff(fact_3125_pairwise__insert,axiom,
    ! [R2: fun_int_fun_int_o,X: int,S2: set_int] :
      ( aa_set_int_o(pairwise_int(R2),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),S2))
    <=> ( ! [Y3: int] :
            ( ( aa_set_int_o(member_int(Y3),S2)
              & ( Y3 != X ) )
           => ( aa_int_o(aa_int_fun_int_o(R2,X),Y3)
              & aa_int_o(aa_int_fun_int_o(R2,Y3),X) ) )
        & aa_set_int_o(pairwise_int(R2),S2) ) ) ).

% pairwise_insert
tff(fact_3126_pairwise__insert,axiom,
    ! [R2: fun_o_fun_o_o,X: $o,S2: set_o] :
      ( aa_set_o_o(pairwise_o(R2),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),S2))
    <=> ( ! [Y3: $o] :
            ( ( aa_set_o_o(member_o((Y3)),S2)
              & ~ ( (Y3)
                <=> (X) ) )
           => ( aa_o_o(aa_o_fun_o_o(R2,(X)),(Y3))
              & aa_o_o(aa_o_fun_o_o(R2,(Y3)),(X)) ) )
        & aa_set_o_o(pairwise_o(R2),S2) ) ) ).

% pairwise_insert
tff(fact_3127_pairwise__insert,axiom,
    ! [R2: fun_Pr1520749766685617582unit_o,X: product_unit,S2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(R2),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),S2))
    <=> ( ! [Y3: product_unit] :
            ( ( aa_set8780210009671629607unit_o(member_Product_unit(Y3),S2)
              & ( Y3 != X ) )
           => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R2,X),Y3)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R2,Y3),X) ) )
        & aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(R2),S2) ) ) ).

% pairwise_insert
tff(fact_3128_finite__UNION__then__finite,axiom,
    ! [B4: fun_fu2098174926252251144et_nat,A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(image_7230913979079189040et_nat(B4,A)))
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
       => aa_set_nat_o(finite_finite_nat,aa_fun7872747175131507769et_nat(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3129_finite__UNION__then__finite,axiom,
    ! [B4: fun_nat_set_nat,A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(B4),A)))
     => ( aa_set_nat_o(member_nat(A2),A)
       => aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3130_finite__UNION__then__finite,axiom,
    ! [B4: fun_int_set_nat,A: set_int,A2: int] :
      ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(image_int_set_nat(B4,A)))
     => ( aa_set_int_o(member_int(A2),A)
       => aa_set_nat_o(finite_finite_nat,aa_int_set_nat(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3131_finite__UNION__then__finite,axiom,
    ! [B4: fun_o_set_nat,A: set_o,A2: $o] :
      ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(image_o_set_nat(B4,A)))
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set_nat_o(finite_finite_nat,aa_o_set_nat(B4,(A2))) ) ) ).

% finite_UNION_then_finite
tff(fact_3132_finite__UNION__then__finite,axiom,
    ! [B4: fun_Pr1628945989756059814et_nat,A: set_Product_unit,A2: product_unit] :
      ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(image_8173849991827986510et_nat(B4,A)))
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
       => aa_set_nat_o(finite_finite_nat,aa_Pro6064029160526250199et_nat(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3133_finite__UNION__then__finite,axiom,
    ! [B4: fun_fu9113425075065601941nteger,A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(image_427149922635871293nteger(B4,A)))
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_fun2356954173479049798nteger(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3134_finite__UNION__then__finite,axiom,
    ! [B4: fun_na9118744110877893344nteger,A: set_nat,A2: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(image_2385391725447763818nteger(B4,A)))
     => ( aa_set_nat_o(member_nat(A2),A)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_nat8922193080109606305nteger(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3135_finite__UNION__then__finite,axiom,
    ! [B4: fun_in6808926178784942396nteger,A: set_int,A2: int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(image_75573793354812870nteger(B4,A)))
     => ( aa_set_int_o(member_int(A2),A)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_int6612375148016655357nteger(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3136_finite__UNION__then__finite,axiom,
    ! [B4: fun_o_5240169576699714110nteger,A: set_o,A2: $o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(image_6491638074533506790nteger(B4,A)))
     => ( aa_set_o_o(member_o((A2)),A)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_o_s5402425747378369775nteger(B4,(A2))) ) ) ).

% finite_UNION_then_finite
tff(fact_3137_finite__UNION__then__finite,axiom,
    ! [B4: fun_Pr1789781366282006579nteger,A: set_Product_unit,A2: product_unit] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(image_4571654851662262875nteger(B4,A)))
     => ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_Pro8006899289762625252nteger(B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3138_image__Pow__surj,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_set_nat] :
      ( ( aa_set7246812951530217472et_nat(image_nat_set_nat(F),A) = B4 )
     => ( image_6725021117256019401et_nat(image_nat_set_nat(F),pow_nat(A)) = pow_set_nat(B4) ) ) ).

% image_Pow_surj
tff(fact_3139_image__Pow__surj,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat2(F),A) = B4 )
     => ( image_7916887816326733075et_nat(image_nat_nat2(F),pow_nat(A)) = pow_nat(B4) ) ) ).

% image_Pow_surj
tff(fact_3140_image__Pow__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A) = B4 )
     => ( image_6270750266955655469nteger(image_4470545334726330049nteger(F),pow_Code_integer(A)) = pow_Code_integer(B4) ) ) ).

% image_Pow_surj
tff(fact_3141_image__Pow__surj,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_Code_integer] :
      ( ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A) = B4 )
     => ( image_6916300684244607776nteger(image_1215581382706833972nteger(F),pow_nat(A)) = pow_Code_integer(B4) ) ) ).

% image_Pow_surj
tff(fact_3142_Pow__insert,axiom,
    ! [A2: nat,A: set_nat] : pow_nat(aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)) = sup_sup_set_set_nat(pow_nat(A),image_7916887816326733075et_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),pow_nat(A))) ).

% Pow_insert
tff(fact_3143_Pow__insert,axiom,
    ! [A2: int,A: set_int] : pow_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A)) = sup_sup_set_set_int(pow_int(A),image_524474410958335435et_int(aa_int7955932875426912299et_int(insert_int,A2),pow_int(A))) ).

% Pow_insert
tff(fact_3144_Pow__insert,axiom,
    ! [A2: $o,A: set_o] : pow_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A)) = sup_sup_set_set_o(pow_o(A),image_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),pow_o(A))) ).

% Pow_insert
tff(fact_3145_Pow__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] : pow_Product_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) = sup_su5975557051335651745t_unit(pow_Product_unit(A),image_4019945301284682565t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),pow_Product_unit(A))) ).

% Pow_insert
tff(fact_3146_pairself_Osimps,axiom,
    ! [F: fun_nat_nat,A2: nat,B: nat] : aa_Pro262675715343573244at_nat(pairself_nat_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,A2)),aa_nat_nat(F,B)) ).

% pairself.simps
tff(fact_3147_pairself_Osimps,axiom,
    ! [F: fun_nat_int,A2: nat,B: nat] : aa_Pro6307550444590206900nt_int(pairself_nat_int(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,A2)),aa_nat_int(F,B)) ).

% pairself.simps
tff(fact_3148_pairself_Osimps,axiom,
    ! [F: fun_int_nat,A2: int,B: int] : aa_Pro925629398452685892at_nat(pairself_int_nat(F),aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,A2)),aa_int_nat(F,B)) ).

% pairself.simps
tff(fact_3149_pairself_Osimps,axiom,
    ! [F: fun_int_int,A2: int,B: int] : aa_Pro6970504127699319548nt_int(pairself_int_int(F),aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,A2)),aa_int_int(F,B)) ).

% pairself.simps
tff(fact_3150_pairself_Oelims,axiom,
    ! [X: fun_nat_nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro262675715343573244at_nat(pairself_nat_nat(X),Xa) = Y )
     => ~ ! [A3: nat,B2: nat] :
            ( ( Xa = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) )
           => ( Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(X,A3)),aa_nat_nat(X,B2)) ) ) ) ).

% pairself.elims
tff(fact_3151_pairself_Oelims,axiom,
    ! [X: fun_nat_int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
      ( ( aa_Pro6307550444590206900nt_int(pairself_nat_int(X),Xa) = Y )
     => ~ ! [A3: nat,B2: nat] :
            ( ( Xa = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) )
           => ( Y != aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(X,A3)),aa_nat_int(X,B2)) ) ) ) ).

% pairself.elims
tff(fact_3152_pairself_Oelims,axiom,
    ! [X: fun_int_nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
      ( ( aa_Pro925629398452685892at_nat(pairself_int_nat(X),Xa) = Y )
     => ~ ! [A3: int,B2: int] :
            ( ( Xa = aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2) )
           => ( Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(X,A3)),aa_int_nat(X,B2)) ) ) ) ).

% pairself.elims
tff(fact_3153_pairself_Oelims,axiom,
    ! [X: fun_int_int,Xa: product_prod_int_int,Y: product_prod_int_int] :
      ( ( aa_Pro6970504127699319548nt_int(pairself_int_int(X),Xa) = Y )
     => ~ ! [A3: int,B2: int] :
            ( ( Xa = aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2) )
           => ( Y != aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(X,A3)),aa_int_int(X,B2)) ) ) ) ).

% pairself.elims
tff(fact_3154_pairwise__singleton,axiom,
    ! [P: fun_Pr1520749766685617582unit_o,A: product_unit] : aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(P),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A),bot_bo3957492148770167129t_unit)) ).

% pairwise_singleton
tff(fact_3155_pairwise__singleton,axiom,
    ! [P: fun_o_fun_o_o,A: $o] : aa_set_o_o(pairwise_o(P),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A)),bot_bot_set_o)) ).

% pairwise_singleton
tff(fact_3156_pairwise__singleton,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat] : aa_set_nat_o(pairwise_nat(P),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A),bot_bot_set_nat)) ).

% pairwise_singleton
tff(fact_3157_pairwise__singleton,axiom,
    ! [P: fun_int_fun_int_o,A: int] : aa_set_int_o(pairwise_int(P),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A),bot_bot_set_int)) ).

% pairwise_singleton
tff(fact_3158_pairwise__singleton,axiom,
    ! [P: fun_fu4164567611050918574at_o_o,A: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(P),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A),bot_bo8632255151191677115_nat_o)) ).

% pairwise_singleton
tff(fact_3159_finite__Sup__in,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( member_set_nat(X3,A)
             => ( member_set_nat(Y2,A)
               => member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X3),Y2),A) ) )
         => member_set_nat(comple7399068483239264473et_nat(A),A) ) ) ) ).

% finite_Sup_in
tff(fact_3160_finite__Sup__in,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y2: $o] :
              ( aa_set_o_o(member_o((X3)),A)
             => ( aa_set_o_o(member_o((Y2)),A)
               => aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(sup_sup_o,(X3)),(Y2))),A) ) )
         => aa_set_o_o(member_o(aa_set_o_o(complete_Sup_Sup_o,A)),A) ) ) ) ).

% finite_Sup_in
tff(fact_3161_finite__Sup__in,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
             => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
               => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(X3),Y2)),A) ) )
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(comple7878897588548607196_nat_o(A)),A) ) ) ) ).

% finite_Sup_in
tff(fact_3162_finite__Sup__in,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit,Y2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
             => ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),A)
               => aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X3),Y2)),A) ) )
         => aa_set8780210009671629607unit_o(member_Product_unit(comple4687483117567863418t_unit(A)),A) ) ) ) ).

% finite_Sup_in
tff(fact_3163_Sup__fin__Sup,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(lattic1508158080041050831_fin_o,A)
        <=> aa_set_o_o(complete_Sup_Sup_o,A) ) ) ) ).

% Sup_fin_Sup
tff(fact_3164_Sup__fin__Sup,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( lattic7388253182610234406_nat_o(A) = comple7878897588548607196_nat_o(A) ) ) ) ).

% Sup_fin_Sup
tff(fact_3165_Sup__fin__Sup,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic5294303975357428420t_unit(A) = comple4687483117567863418t_unit(A) ) ) ) ).

% Sup_fin_Sup
tff(fact_3166_insert__partition,axiom,
    ! [X: set_Product_unit,F2: set_set_Product_unit] :
      ( ~ member5877623283571906838t_unit(X,F2)
     => ( ! [X3: set_Product_unit] :
            ( member5877623283571906838t_unit(X3,insert8986028188087627965t_unit(X,F2))
           => ! [Xa4: set_Product_unit] :
                ( member5877623283571906838t_unit(Xa4,insert8986028188087627965t_unit(X,F2))
               => ( ( X3 != Xa4 )
                 => ( inf_in4660618365625256667t_unit(X3,Xa4) = bot_bo3957492148770167129t_unit ) ) ) )
       => ( inf_in4660618365625256667t_unit(X,comple3349607820504768346t_unit(F2)) = bot_bo3957492148770167129t_unit ) ) ) ).

% insert_partition
tff(fact_3167_insert__partition,axiom,
    ! [X: set_o,F2: set_set_o] :
      ( ~ member_set_o(X,F2)
     => ( ! [X3: set_o] :
            ( member_set_o(X3,insert_set_o(X,F2))
           => ! [Xa4: set_o] :
                ( member_set_o(Xa4,insert_set_o(X,F2))
               => ( ( X3 != Xa4 )
                 => ( inf_inf_set_o(X3,Xa4) = bot_bot_set_o ) ) ) )
       => ( inf_inf_set_o(X,comple90263536869209701_set_o(F2)) = bot_bot_set_o ) ) ) ).

% insert_partition
tff(fact_3168_insert__partition,axiom,
    ! [X: set_nat,F2: set_set_nat] :
      ( ~ member_set_nat(X,F2)
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,insert_set_nat(X,F2))
           => ! [Xa4: set_nat] :
                ( member_set_nat(Xa4,insert_set_nat(X,F2))
               => ( ( X3 != Xa4 )
                 => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Xa4) = bot_bot_set_nat ) ) ) )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),comple7399068483239264473et_nat(F2)) = bot_bot_set_nat ) ) ) ).

% insert_partition
tff(fact_3169_insert__partition,axiom,
    ! [X: set_int,F2: set_set_int] :
      ( ~ member_set_int(X,F2)
     => ( ! [X3: set_int] :
            ( member_set_int(X3,insert_set_int(X,F2))
           => ! [Xa4: set_int] :
                ( member_set_int(Xa4,insert_set_int(X,F2))
               => ( ( X3 != Xa4 )
                 => ( inf_inf_set_int(X3,Xa4) = bot_bot_set_int ) ) ) )
       => ( inf_inf_set_int(X,comple3221217463730067765et_int(F2)) = bot_bot_set_int ) ) ) ).

% insert_partition
tff(fact_3170_insert__partition,axiom,
    ! [X: set_fu3555266933746557007_nat_o,F2: set_se6690607124073039791_nat_o] :
      ( ~ member361801372707117048_nat_o(X,F2)
     => ( ! [X3: set_fu3555266933746557007_nat_o] :
            ( member361801372707117048_nat_o(X3,insert755438361227728415_nat_o(X,F2))
           => ! [Xa4: set_fu3555266933746557007_nat_o] :
                ( member361801372707117048_nat_o(Xa4,insert755438361227728415_nat_o(X,F2))
               => ( ( X3 != Xa4 )
                 => ( inf_in3376754634429691197_nat_o(X3,Xa4) = bot_bo8632255151191677115_nat_o ) ) ) )
       => ( inf_in3376754634429691197_nat_o(X,comple4665849098253521980_nat_o(F2)) = bot_bo8632255151191677115_nat_o ) ) ) ).

% insert_partition
tff(fact_3171_image__Pow__mono,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A),B4)
     => ord_le9131159989063066194et_nat(image_6725021117256019401et_nat(image_nat_set_nat(F),pow_nat(A)),pow_set_nat(B4)) ) ).

% image_Pow_mono
tff(fact_3172_image__Pow__mono,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)),B4)
     => ord_le1914454125413604393nteger(image_6270750266955655469nteger(image_4470545334726330049nteger(F),pow_Code_integer(A)),pow_Code_integer(B4)) ) ).

% image_Pow_mono
tff(fact_3173_image__Pow__mono,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A)),B4)
     => ord_le1914454125413604393nteger(image_6916300684244607776nteger(image_1215581382706833972nteger(F),pow_nat(A)),pow_Code_integer(B4)) ) ).

% image_Pow_mono
tff(fact_3174_image__Pow__mono,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)),B4)
     => ord_le6893508408891458716et_nat(image_7916887816326733075et_nat(image_nat_nat2(F),pow_nat(A)),pow_nat(B4)) ) ).

% image_Pow_mono
tff(fact_3175_pairwise__alt,axiom,
    ! [R: fun_Pr1520749766685617582unit_o,S: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(pairwi7919127943033184120t_unit(R),S)
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),S)
         => ! [Xa2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),minus_6452836326544984404t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X2),bot_bo3957492148770167129t_unit)))
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(R,X2),Xa2) ) ) ) ).

% pairwise_alt
tff(fact_3176_pairwise__alt,axiom,
    ! [R: fun_o_fun_o_o,S: set_o] :
      ( aa_set_o_o(pairwise_o(R),S)
    <=> ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),S)
         => ! [Xa2: $o] :
              ( aa_set_o_o(member_o((Xa2)),minus_minus_set_o(S,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X2)),bot_bot_set_o)))
             => aa_o_o(aa_o_fun_o_o(R,(X2)),(Xa2)) ) ) ) ).

% pairwise_alt
tff(fact_3177_pairwise__alt,axiom,
    ! [R: fun_int_fun_int_o,S: set_int] :
      ( aa_set_int_o(pairwise_int(R),S)
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),S)
         => ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),minus_minus_set_int(S,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X2),bot_bot_set_int)))
             => aa_int_o(aa_int_fun_int_o(R,X2),Xa2) ) ) ) ).

% pairwise_alt
tff(fact_3178_pairwise__alt,axiom,
    ! [R: fun_fu4164567611050918574at_o_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(pairwi7285160467931473626_nat_o(R),S)
    <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S)
         => ! [Xa2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa2),minus_6456595905716665654_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),bot_bo8632255151191677115_nat_o)))
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(R,X2),Xa2) ) ) ) ).

% pairwise_alt
tff(fact_3179_pairwise__alt,axiom,
    ! [R: fun_nat_fun_nat_o,S: set_nat] :
      ( aa_set_nat_o(pairwise_nat(R),S)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S)
         => ! [Xa2: nat] :
              ( aa_set_nat_o(member_nat(Xa2),aa_set_nat_set_nat(minus_minus_set_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X2),bot_bot_set_nat)))
             => aa_nat_o(aa_nat_fun_nat_o(R,X2),Xa2) ) ) ) ).

% pairwise_alt
tff(fact_3180_Sup__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(A2)),aa_set_o_o(complete_Sup_Sup_o,A)) ) ).

% Sup_insert
tff(fact_3181_Sup__insert,axiom,
    ! [A2: set_nat,A: set_set_nat] : comple7399068483239264473et_nat(insert_set_nat(A2,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(A)) ).

% Sup_insert
tff(fact_3182_Sup__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] : comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),comple4687483117567863418t_unit(A)) ).

% Sup_insert
tff(fact_3183_ccpo__Sup__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% ccpo_Sup_singleton
tff(fact_3184_ccpo__Sup__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : comple7878897588548607196_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X ).

% ccpo_Sup_singleton
tff(fact_3185_ccpo__Sup__singleton,axiom,
    ! [X: product_unit] : comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% ccpo_Sup_singleton
tff(fact_3186_cSup__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% cSup_singleton
tff(fact_3187_cSup__singleton,axiom,
    ! [X: int] : complete_Sup_Sup_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% cSup_singleton
tff(fact_3188_cSup__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : comple7878897588548607196_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X ).

% cSup_singleton
tff(fact_3189_cSup__singleton,axiom,
    ! [X: nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% cSup_singleton
tff(fact_3190_cSup__singleton,axiom,
    ! [X: product_unit] : comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% cSup_singleton
tff(fact_3191_Sup__empty,axiom,
    comple3349607820504768346t_unit(bot_bo6284674557190688313t_unit) = bot_bo3957492148770167129t_unit ).

% Sup_empty
tff(fact_3192_Sup__empty,axiom,
    comple90263536869209701_set_o(bot_bot_set_set_o) = bot_bot_set_o ).

% Sup_empty
tff(fact_3193_Sup__empty,axiom,
    comple7399068483239264473et_nat(bot_bot_set_set_nat) = bot_bot_set_nat ).

% Sup_empty
tff(fact_3194_Sup__empty,axiom,
    comple3221217463730067765et_int(bot_bot_set_set_int) = bot_bot_set_int ).

% Sup_empty
tff(fact_3195_Sup__empty,axiom,
    comple4665849098253521980_nat_o(bot_bo4069457504818663451_nat_o) = bot_bo8632255151191677115_nat_o ).

% Sup_empty
tff(fact_3196_Sup__empty,axiom,
    ( aa_set_o_o(complete_Sup_Sup_o,bot_bot_set_o)
  <=> bot_bot_o ) ).

% Sup_empty
tff(fact_3197_Sup__empty,axiom,
    comple7878897588548607196_nat_o(bot_bo8632255151191677115_nat_o) = bot_bo8756837403692309851_nat_o ).

% Sup_empty
tff(fact_3198_Sup__empty,axiom,
    comple4687483117567863418t_unit(bot_bo3957492148770167129t_unit) = bot_bot_Product_unit ).

% Sup_empty
tff(fact_3199_Sup__bot__conv_I1_J,axiom,
    ! [A: set_set_Product_unit] :
      ( ( comple3349607820504768346t_unit(A) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: set_Product_unit] :
          ( member5877623283571906838t_unit(X2,A)
         => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3200_Sup__bot__conv_I1_J,axiom,
    ! [A: set_set_o] :
      ( ( comple90263536869209701_set_o(A) = bot_bot_set_o )
    <=> ! [X2: set_o] :
          ( member_set_o(X2,A)
         => ( X2 = bot_bot_set_o ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3201_Sup__bot__conv_I1_J,axiom,
    ! [A: set_set_nat] :
      ( ( comple7399068483239264473et_nat(A) = bot_bot_set_nat )
    <=> ! [X2: set_nat] :
          ( member_set_nat(X2,A)
         => ( X2 = bot_bot_set_nat ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3202_Sup__bot__conv_I1_J,axiom,
    ! [A: set_set_int] :
      ( ( comple3221217463730067765et_int(A) = bot_bot_set_int )
    <=> ! [X2: set_int] :
          ( member_set_int(X2,A)
         => ( X2 = bot_bot_set_int ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3203_Sup__bot__conv_I1_J,axiom,
    ! [A: set_se6690607124073039791_nat_o] :
      ( ( comple4665849098253521980_nat_o(A) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(X2,A)
         => ( X2 = bot_bo8632255151191677115_nat_o ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3204_Sup__bot__conv_I1_J,axiom,
    ! [A: set_Product_unit] :
      ( ( comple4687483117567863418t_unit(A) = bot_bot_Product_unit )
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
         => ( X2 = bot_bot_Product_unit ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3205_Sup__bot__conv_I2_J,axiom,
    ! [A: set_set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = comple3349607820504768346t_unit(A) )
    <=> ! [X2: set_Product_unit] :
          ( member5877623283571906838t_unit(X2,A)
         => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3206_Sup__bot__conv_I2_J,axiom,
    ! [A: set_set_o] :
      ( ( bot_bot_set_o = comple90263536869209701_set_o(A) )
    <=> ! [X2: set_o] :
          ( member_set_o(X2,A)
         => ( X2 = bot_bot_set_o ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3207_Sup__bot__conv_I2_J,axiom,
    ! [A: set_set_nat] :
      ( ( bot_bot_set_nat = comple7399068483239264473et_nat(A) )
    <=> ! [X2: set_nat] :
          ( member_set_nat(X2,A)
         => ( X2 = bot_bot_set_nat ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3208_Sup__bot__conv_I2_J,axiom,
    ! [A: set_set_int] :
      ( ( bot_bot_set_int = comple3221217463730067765et_int(A) )
    <=> ! [X2: set_int] :
          ( member_set_int(X2,A)
         => ( X2 = bot_bot_set_int ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3209_Sup__bot__conv_I2_J,axiom,
    ! [A: set_se6690607124073039791_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = comple4665849098253521980_nat_o(A) )
    <=> ! [X2: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(X2,A)
         => ( X2 = bot_bo8632255151191677115_nat_o ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3210_Sup__bot__conv_I2_J,axiom,
    ! [A: set_Product_unit] :
      ( ( bot_bot_Product_unit = comple4687483117567863418t_unit(A) )
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
         => ( X2 = bot_bot_Product_unit ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3211_cSup__eq__Sup__fin,axiom,
    ! [X5: set_o] :
      ( aa_set_o_o(finite_finite_o,X5)
     => ( ( X5 != bot_bot_set_o )
       => ( aa_set_o_o(complete_Sup_Sup_o,X5)
        <=> aa_set_o_o(lattic1508158080041050831_fin_o,X5) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_3212_cSup__eq__Sup__fin,axiom,
    ! [X5: set_int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( ( X5 != bot_bot_set_int )
       => ( complete_Sup_Sup_int(X5) = lattic1091506334969745077in_int(X5) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_3213_cSup__eq__Sup__fin,axiom,
    ! [X5: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,X5)
     => ( ( X5 != bot_bo8632255151191677115_nat_o )
       => ( comple7878897588548607196_nat_o(X5) = lattic7388253182610234406_nat_o(X5) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_3214_cSup__eq__Sup__fin,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( ( X5 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X5) = aa_set_nat_nat(lattic1093996805478795353in_nat,X5) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_3215_cSup__eq__Sup__fin,axiom,
    ! [X5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ( comple4687483117567863418t_unit(X5) = lattic5294303975357428420t_unit(X5) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_3216_Union__image__insert,axiom,
    ! [F: fun_nat_set_nat,A2: nat,B4: set_nat] : comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_nat_set_nat(F,A2)),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4))) ).

% Union_image_insert
tff(fact_3217_Union__image__insert,axiom,
    ! [F: fun_int_set_nat,A2: int,B4: set_int] : comple7399068483239264473et_nat(image_int_set_nat(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_int_set_nat(F,A2)),comple7399068483239264473et_nat(image_int_set_nat(F,B4))) ).

% Union_image_insert
tff(fact_3218_Union__image__insert,axiom,
    ! [F: fun_o_set_nat,A2: $o,B4: set_o] : comple7399068483239264473et_nat(image_o_set_nat(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_o_set_nat(F,(A2))),comple7399068483239264473et_nat(image_o_set_nat(F,B4))) ).

% Union_image_insert
tff(fact_3219_Union__image__insert,axiom,
    ! [F: fun_Pr1628945989756059814et_nat,A2: product_unit,B4: set_Product_unit] : comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_Pro6064029160526250199et_nat(F,A2)),comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,B4))) ).

% Union_image_insert
tff(fact_3220_Union__image__empty,axiom,
    ! [A: set_nat,F: fun_Pr1628945989756059814et_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,bot_bo3957492148770167129t_unit))) = A ).

% Union_image_empty
tff(fact_3221_Union__image__empty,axiom,
    ! [A: set_nat,F: fun_o_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),comple7399068483239264473et_nat(image_o_set_nat(F,bot_bot_set_o))) = A ).

% Union_image_empty
tff(fact_3222_Union__image__empty,axiom,
    ! [A: set_nat,F: fun_nat_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),bot_bot_set_nat))) = A ).

% Union_image_empty
tff(fact_3223_Union__image__empty,axiom,
    ! [A: set_nat,F: fun_int_set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),comple7399068483239264473et_nat(image_int_set_nat(F,bot_bot_set_int))) = A ).

% Union_image_empty
tff(fact_3224_Union__image__empty,axiom,
    ! [A: set_nat,F: fun_fu2098174926252251144et_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),comple7399068483239264473et_nat(image_7230913979079189040et_nat(F,bot_bo8632255151191677115_nat_o))) = A ).

% Union_image_empty
tff(fact_3225_Sup__SUP__eq,axiom,
    ! [S: set_fu8039672399046406370at_o_o,X4: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(comple3647288994071827583at_o_o(S),X4)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),comple4665849098253521980_nat_o(image_401909237675433180_nat_o(collec6407654713642994138_nat_o,S))) ) ).

% Sup_SUP_eq
tff(fact_3226_Sup__SUP__eq,axiom,
    ! [S: set_fun_nat_o,X4: nat] :
      ( aa_nat_o(comple5348352744826190222_nat_o(S),X4)
    <=> aa_set_nat_o(member_nat(X4),comple7399068483239264473et_nat(image_4516309135669182690et_nat(collect_nat,S))) ) ).

% Sup_SUP_eq
tff(fact_3227_Sup__SUP__eq,axiom,
    ! [S: set_fun_int_o,X4: int] :
      ( aa_int_o(comple3527310399393502258_int_o(S),X4)
    <=> aa_set_int_o(member_int(X4),comple3221217463730067765et_int(image_3849661908695225314et_int(collect_int,S))) ) ).

% Sup_SUP_eq
tff(fact_3228_Sup__SUP__eq,axiom,
    ! [S: set_fun_o_o,X4: $o] :
      ( aa_o_o(comple7262668920879777174un_o_o(S),(X4))
    <=> aa_set_o_o(member_o((X4)),comple90263536869209701_set_o(image_fun_o_o_set_o(collect_o,S))) ) ).

% Sup_SUP_eq
tff(fact_3229_Sup__SUP__eq,axiom,
    ! [S: set_fu7570441001188634436unit_o,X4: product_unit] :
      ( aa_Product_unit_o(comple72412983790229473unit_o(S),X4)
    <=> aa_set8780210009671629607unit_o(member_Product_unit(X4),comple3349607820504768346t_unit(image_2610332678118349720t_unit(collect_Product_unit,S))) ) ).

% Sup_SUP_eq
tff(fact_3230_empty__Union__conv,axiom,
    ! [A: set_set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = comple3349607820504768346t_unit(A) )
    <=> ! [X2: set_Product_unit] :
          ( member5877623283571906838t_unit(X2,A)
         => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ).

% empty_Union_conv
tff(fact_3231_empty__Union__conv,axiom,
    ! [A: set_set_o] :
      ( ( bot_bot_set_o = comple90263536869209701_set_o(A) )
    <=> ! [X2: set_o] :
          ( member_set_o(X2,A)
         => ( X2 = bot_bot_set_o ) ) ) ).

% empty_Union_conv
tff(fact_3232_empty__Union__conv,axiom,
    ! [A: set_set_nat] :
      ( ( bot_bot_set_nat = comple7399068483239264473et_nat(A) )
    <=> ! [X2: set_nat] :
          ( member_set_nat(X2,A)
         => ( X2 = bot_bot_set_nat ) ) ) ).

% empty_Union_conv
tff(fact_3233_empty__Union__conv,axiom,
    ! [A: set_set_int] :
      ( ( bot_bot_set_int = comple3221217463730067765et_int(A) )
    <=> ! [X2: set_int] :
          ( member_set_int(X2,A)
         => ( X2 = bot_bot_set_int ) ) ) ).

% empty_Union_conv
tff(fact_3234_empty__Union__conv,axiom,
    ! [A: set_se6690607124073039791_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = comple4665849098253521980_nat_o(A) )
    <=> ! [X2: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(X2,A)
         => ( X2 = bot_bo8632255151191677115_nat_o ) ) ) ).

% empty_Union_conv
tff(fact_3235_Union__empty__conv,axiom,
    ! [A: set_set_Product_unit] :
      ( ( comple3349607820504768346t_unit(A) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: set_Product_unit] :
          ( member5877623283571906838t_unit(X2,A)
         => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ).

% Union_empty_conv
tff(fact_3236_Union__empty__conv,axiom,
    ! [A: set_set_o] :
      ( ( comple90263536869209701_set_o(A) = bot_bot_set_o )
    <=> ! [X2: set_o] :
          ( member_set_o(X2,A)
         => ( X2 = bot_bot_set_o ) ) ) ).

% Union_empty_conv
tff(fact_3237_Union__empty__conv,axiom,
    ! [A: set_set_nat] :
      ( ( comple7399068483239264473et_nat(A) = bot_bot_set_nat )
    <=> ! [X2: set_nat] :
          ( member_set_nat(X2,A)
         => ( X2 = bot_bot_set_nat ) ) ) ).

% Union_empty_conv
tff(fact_3238_Union__empty__conv,axiom,
    ! [A: set_set_int] :
      ( ( comple3221217463730067765et_int(A) = bot_bot_set_int )
    <=> ! [X2: set_int] :
          ( member_set_int(X2,A)
         => ( X2 = bot_bot_set_int ) ) ) ).

% Union_empty_conv
tff(fact_3239_Union__empty__conv,axiom,
    ! [A: set_se6690607124073039791_nat_o] :
      ( ( comple4665849098253521980_nat_o(A) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(X2,A)
         => ( X2 = bot_bo8632255151191677115_nat_o ) ) ) ).

% Union_empty_conv
tff(fact_3240_less__eq__Sup,axiom,
    ! [A: set_o,U: $o] :
      ( ! [V3: $o] :
          ( aa_set_o_o(member_o((V3)),A)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(U)),(V3)) )
     => ( ( A != bot_bot_set_o )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(U)),aa_set_o_o(complete_Sup_Sup_o,A)) ) ) ).

% less_eq_Sup
tff(fact_3241_less__eq__Sup,axiom,
    ! [A: set_fu3555266933746557007_nat_o,U: fun_Pr6460944699146269935_nat_o] :
      ( ! [V3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(V3),A)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,U),V3) )
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,U),comple7878897588548607196_nat_o(A)) ) ) ).

% less_eq_Sup
tff(fact_3242_less__eq__Sup,axiom,
    ! [A: set_filter_nat,U: filter_nat] :
      ( ! [V3: filter_nat] :
          ( member_filter_nat(V3,A)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,U),V3) )
     => ( ( A != bot_bo498966703094740906er_nat )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,U),comple2454491259644266675er_nat(A)) ) ) ).

% less_eq_Sup
tff(fact_3243_less__eq__Sup,axiom,
    ! [A: set_set_nat,U: set_nat] :
      ( ! [V3: set_nat] :
          ( member_set_nat(V3,A)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,U),V3) )
     => ( ( A != bot_bot_set_set_nat )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,U),comple7399068483239264473et_nat(A)) ) ) ).

% less_eq_Sup
tff(fact_3244_less__eq__Sup,axiom,
    ! [A: set_Product_unit,U: product_unit] :
      ( ! [V3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(V3),A)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,U),V3) )
     => ( ( A != bot_bo3957492148770167129t_unit )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,U),comple4687483117567863418t_unit(A)) ) ) ).

% less_eq_Sup
tff(fact_3245_cSup__least,axiom,
    ! [X5: set_o,Z: $o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X5)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(Z)) )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Sup_Sup_o,X5)),(Z)) ) ) ).

% cSup_least
tff(fact_3246_cSup__least,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,Z: fun_Pr6460944699146269935_nat_o] :
      ( ( X5 != bot_bo8632255151191677115_nat_o )
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),X5)
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X3),Z) )
       => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple7878897588548607196_nat_o(X5)),Z) ) ) ).

% cSup_least
tff(fact_3247_cSup__least,axiom,
    ! [X5: set_filter_nat,Z: filter_nat] :
      ( ( X5 != bot_bo498966703094740906er_nat )
     => ( ! [X3: filter_nat] :
            ( member_filter_nat(X3,X5)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X3),Z) )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple2454491259644266675er_nat(X5)),Z) ) ) ).

% cSup_least
tff(fact_3248_cSup__least,axiom,
    ! [X5: set_set_nat,Z: set_nat] :
      ( ( X5 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,X5)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Z) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(X5)),Z) ) ) ).

% cSup_least
tff(fact_3249_cSup__least,axiom,
    ! [X5: set_int,Z: int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Z) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(X5)),Z) ) ) ).

% cSup_least
tff(fact_3250_cSup__least,axiom,
    ! [X5: set_nat,Z: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Z) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,X5)),Z) ) ) ).

% cSup_least
tff(fact_3251_cSup__least,axiom,
    ! [X5: set_Product_unit,Z: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X3),Z) )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(X5)),Z) ) ) ).

% cSup_least
tff(fact_3252_cSup__eq__non__empty,axiom,
    ! [X5: set_o,A2: $o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X5)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X3)),(A2)) )
       => ( ! [Y2: $o] :
              ( ! [X4: $o] :
                  ( aa_set_o_o(member_o((X4)),X5)
                 => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X4)),(Y2)) )
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A2)),(Y2)) )
         => ( aa_set_o_o(complete_Sup_Sup_o,X5)
          <=> (A2) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3253_cSup__eq__non__empty,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( X5 != bot_bo8632255151191677115_nat_o )
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),X5)
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X3),A2) )
       => ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
              ( ! [X4: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),X5)
                 => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X4),Y2) )
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A2),Y2) )
         => ( comple7878897588548607196_nat_o(X5) = A2 ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3254_cSup__eq__non__empty,axiom,
    ! [X5: set_filter_nat,A2: filter_nat] :
      ( ( X5 != bot_bo498966703094740906er_nat )
     => ( ! [X3: filter_nat] :
            ( member_filter_nat(X3,X5)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X3),A2) )
       => ( ! [Y2: filter_nat] :
              ( ! [X4: filter_nat] :
                  ( member_filter_nat(X4,X5)
                 => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X4),Y2) )
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),Y2) )
         => ( comple2454491259644266675er_nat(X5) = A2 ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3255_cSup__eq__non__empty,axiom,
    ! [X5: set_set_nat,A2: set_nat] :
      ( ( X5 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,X5)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),A2) )
       => ( ! [Y2: set_nat] :
              ( ! [X4: set_nat] :
                  ( member_set_nat(X4,X5)
                 => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),Y2) )
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),Y2) )
         => ( comple7399068483239264473et_nat(X5) = A2 ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3256_cSup__eq__non__empty,axiom,
    ! [X5: set_int,A2: int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A2) )
       => ( ! [Y2: int] :
              ( ! [X4: int] :
                  ( aa_set_int_o(member_int(X4),X5)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y2) )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),Y2) )
         => ( complete_Sup_Sup_int(X5) = A2 ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3257_cSup__eq__non__empty,axiom,
    ! [X5: set_nat,A2: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A2) )
       => ( ! [Y2: nat] :
              ( ! [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),X5)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y2) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),Y2) )
         => ( aa_set_nat_nat(complete_Sup_Sup_nat,X5) = A2 ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3258_cSup__eq__non__empty,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X3),A2) )
       => ( ! [Y2: product_unit] :
              ( ! [X4: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X4),X5)
                 => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X4),Y2) )
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A2),Y2) )
         => ( comple4687483117567863418t_unit(X5) = A2 ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3259_SUP__eq__const,axiom,
    ! [I: set_nat,F: fun_nat_set_nat,X: set_nat] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => ( aa_nat_set_nat(F,I3) = X ) )
       => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),I)) = X ) ) ) ).

% SUP_eq_const
tff(fact_3260_SUP__eq__const,axiom,
    ! [I: set_Product_unit,F: fun_Pr2191293291710260995t_unit,X: product_unit] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => ( aa_Pro8294289767544744956t_unit(F,I3) = X ) )
       => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,I)) = X ) ) ) ).

% SUP_eq_const
tff(fact_3261_SUP__eq__const,axiom,
    ! [I: set_o,F: fun_o_Product_unit,X: product_unit] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => ( aa_o_Product_unit(F,(I3)) = X ) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit2(F,I)) = X ) ) ) ).

% SUP_eq_const
tff(fact_3262_SUP__eq__const,axiom,
    ! [I: set_nat,F: fun_nat_Product_unit,X: product_unit] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => ( aa_nat_Product_unit(F,I3) = X ) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,I)) = X ) ) ) ).

% SUP_eq_const
tff(fact_3263_SUP__eq__const,axiom,
    ! [I: set_int,F: fun_int_Product_unit,X: product_unit] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => ( aa_int_Product_unit(F,I3) = X ) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,I)) = X ) ) ) ).

% SUP_eq_const
tff(fact_3264_SUP__eq__const,axiom,
    ! [I: set_fu3555266933746557007_nat_o,F: fun_fu3096152755035260449t_unit,X: product_unit] :
      ( ( I != bot_bo8632255151191677115_nat_o )
     => ( ! [I3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I3),I)
           => ( aa_fun8210042732970293018t_unit(F,I3) = X ) )
       => ( comple4687483117567863418t_unit(image_5224052122049783779t_unit(F,I)) = X ) ) ) ).

% SUP_eq_const
tff(fact_3265_Union__disjoint,axiom,
    ! [C5: set_set_Product_unit,A: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(comple3349607820504768346t_unit(C5),A) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: set_Product_unit] :
          ( member5877623283571906838t_unit(X2,C5)
         => ( inf_in4660618365625256667t_unit(X2,A) = bot_bo3957492148770167129t_unit ) ) ) ).

% Union_disjoint
tff(fact_3266_Union__disjoint,axiom,
    ! [C5: set_set_o,A: set_o] :
      ( ( inf_inf_set_o(comple90263536869209701_set_o(C5),A) = bot_bot_set_o )
    <=> ! [X2: set_o] :
          ( member_set_o(X2,C5)
         => ( inf_inf_set_o(X2,A) = bot_bot_set_o ) ) ) ).

% Union_disjoint
tff(fact_3267_Union__disjoint,axiom,
    ! [C5: set_set_nat,A: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,comple7399068483239264473et_nat(C5)),A) = bot_bot_set_nat )
    <=> ! [X2: set_nat] :
          ( member_set_nat(X2,C5)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),A) = bot_bot_set_nat ) ) ) ).

% Union_disjoint
tff(fact_3268_Union__disjoint,axiom,
    ! [C5: set_set_int,A: set_int] :
      ( ( inf_inf_set_int(comple3221217463730067765et_int(C5),A) = bot_bot_set_int )
    <=> ! [X2: set_int] :
          ( member_set_int(X2,C5)
         => ( inf_inf_set_int(X2,A) = bot_bot_set_int ) ) ) ).

% Union_disjoint
tff(fact_3269_Union__disjoint,axiom,
    ! [C5: set_se6690607124073039791_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(comple4665849098253521980_nat_o(C5),A) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(X2,C5)
         => ( inf_in3376754634429691197_nat_o(X2,A) = bot_bo8632255151191677115_nat_o ) ) ) ).

% Union_disjoint
tff(fact_3270_Union__empty,axiom,
    comple3349607820504768346t_unit(bot_bo6284674557190688313t_unit) = bot_bo3957492148770167129t_unit ).

% Union_empty
tff(fact_3271_Union__empty,axiom,
    comple90263536869209701_set_o(bot_bot_set_set_o) = bot_bot_set_o ).

% Union_empty
tff(fact_3272_Union__empty,axiom,
    comple7399068483239264473et_nat(bot_bot_set_set_nat) = bot_bot_set_nat ).

% Union_empty
tff(fact_3273_Union__empty,axiom,
    comple3221217463730067765et_int(bot_bot_set_set_int) = bot_bot_set_int ).

% Union_empty
tff(fact_3274_Union__empty,axiom,
    comple4665849098253521980_nat_o(bot_bo4069457504818663451_nat_o) = bot_bo8632255151191677115_nat_o ).

% Union_empty
tff(fact_3275_SUP__eq__iff,axiom,
    ! [I: set_Product_unit,C: product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,C),aa_Pro8294289767544744956t_unit(F,I3)) )
       => ( ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,I)) = C )
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),I)
             => ( aa_Pro8294289767544744956t_unit(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3276_SUP__eq__iff,axiom,
    ! [I: set_o,C: product_unit,F: fun_o_Product_unit] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,C),aa_o_Product_unit(F,(I3))) )
       => ( ( comple4687483117567863418t_unit(image_o_Product_unit2(F,I)) = C )
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),I)
             => ( aa_o_Product_unit(F,(X2)) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3277_SUP__eq__iff,axiom,
    ! [I: set_nat,C: product_unit,F: fun_nat_Product_unit] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,C),aa_nat_Product_unit(F,I3)) )
       => ( ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,I)) = C )
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),I)
             => ( aa_nat_Product_unit(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3278_SUP__eq__iff,axiom,
    ! [I: set_int,C: product_unit,F: fun_int_Product_unit] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,C),aa_int_Product_unit(F,I3)) )
       => ( ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,I)) = C )
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),I)
             => ( aa_int_Product_unit(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3279_SUP__eq__iff,axiom,
    ! [I: set_Product_unit,C: filter_nat,F: fun_Pr566567543205111936er_nat] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),aa_Pro518797148967611313er_nat(F,I3)) )
       => ( ( comple2454491259644266675er_nat(image_1853478764106424488er_nat(F,I)) = C )
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),I)
             => ( aa_Pro518797148967611313er_nat(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3280_SUP__eq__iff,axiom,
    ! [I: set_o,C: filter_nat,F: fun_o_filter_nat] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),aa_o_filter_nat(F,(I3))) )
       => ( ( comple2454491259644266675er_nat(image_o_filter_nat(F,I)) = C )
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),I)
             => ( aa_o_filter_nat(F,(X2)) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3281_SUP__eq__iff,axiom,
    ! [I: set_nat,C: filter_nat,F: fun_nat_filter_nat] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),aa_nat_filter_nat(F,I3)) )
       => ( ( comple2454491259644266675er_nat(image_nat_filter_nat(F,I)) = C )
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),I)
             => ( aa_nat_filter_nat(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3282_SUP__eq__iff,axiom,
    ! [I: set_int,C: filter_nat,F: fun_int_filter_nat] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),aa_int_filter_nat(F,I3)) )
       => ( ( comple2454491259644266675er_nat(image_int_filter_nat(F,I)) = C )
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),I)
             => ( aa_int_filter_nat(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3283_SUP__eq__iff,axiom,
    ! [I: set_Product_unit,C: set_nat,F: fun_Pr1628945989756059814et_nat] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_Pro6064029160526250199et_nat(F,I3)) )
       => ( ( comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,I)) = C )
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),I)
             => ( aa_Pro6064029160526250199et_nat(F,X2) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3284_SUP__eq__iff,axiom,
    ! [I: set_o,C: set_nat,F: fun_o_set_nat] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_o_set_nat(F,(I3))) )
       => ( ( comple7399068483239264473et_nat(image_o_set_nat(F,I)) = C )
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),I)
             => ( aa_o_set_nat(F,(X2)) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3285_cSUP__least,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_int,M: int] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(F,X3)),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_873079544045703924it_int(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3286_cSUP__least,axiom,
    ! [A: set_o,F: fun_o_int,M: int] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X3))),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_o_int2(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3287_cSUP__least,axiom,
    ! [A: set_nat,F: fun_nat_int,M: int] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_nat_int2(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3288_cSUP__least,axiom,
    ! [A: set_int,F: fun_int_int,M: int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_int_int2(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3289_cSUP__least,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,M: nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,X3)),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3290_cSUP__least,axiom,
    ! [A: set_o,F: fun_o_nat,M: nat] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X3))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3291_cSUP__least,axiom,
    ! [A: set_nat,F: fun_nat_nat,M: nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),M) ) ) ).

% cSUP_least
tff(fact_3292_cSUP__least,axiom,
    ! [A: set_int,F: fun_int_nat,M: nat] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3293_cSUP__least,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,M: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(F,X3)),M) )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3294_cSUP__least,axiom,
    ! [A: set_o,F: fun_o_Product_unit,M: product_unit] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_o_Product_unit(F,(X3))),M) )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(image_o_Product_unit2(F,A))),M) ) ) ).

% cSUP_least
tff(fact_3295_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_set_Product_unit,A2: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(comple3349607820504768346t_unit(B4),A2) = bot_bo3957492148770167129t_unit )
    <=> ! [X2: set_Product_unit] :
          ( member5877623283571906838t_unit(X2,B4)
         => ( inf_in4660618365625256667t_unit(X2,A2) = bot_bo3957492148770167129t_unit ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3296_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_set_o,A2: set_o] :
      ( ( inf_inf_set_o(comple90263536869209701_set_o(B4),A2) = bot_bot_set_o )
    <=> ! [X2: set_o] :
          ( member_set_o(X2,B4)
         => ( inf_inf_set_o(X2,A2) = bot_bot_set_o ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3297_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_set_nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,comple7399068483239264473et_nat(B4)),A2) = bot_bot_set_nat )
    <=> ! [X2: set_nat] :
          ( member_set_nat(X2,B4)
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),A2) = bot_bot_set_nat ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3298_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_set_int,A2: set_int] :
      ( ( inf_inf_set_int(comple3221217463730067765et_int(B4),A2) = bot_bot_set_int )
    <=> ! [X2: set_int] :
          ( member_set_int(X2,B4)
         => ( inf_inf_set_int(X2,A2) = bot_bot_set_int ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3299_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_se6690607124073039791_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(comple4665849098253521980_nat_o(B4),A2) = bot_bo8632255151191677115_nat_o )
    <=> ! [X2: set_fu3555266933746557007_nat_o] :
          ( member361801372707117048_nat_o(X2,B4)
         => ( inf_in3376754634429691197_nat_o(X2,A2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3300_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_Product_unit,A2: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple4687483117567863418t_unit(B4)),A2) = bot_bot_Product_unit )
    <=> ! [X2: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X2),B4)
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X2),A2) = bot_bot_Product_unit ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3301_UNION__fun__upd,axiom,
    ! [A: fun_Pr3013102738158121955t_unit,I2: product_unit,B4: set_Product_unit,J2: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(fun_up488859312337516701t_unit(A,I2,B4),J2)) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,comple3349607820504768346t_unit(image_7778121751606633509t_unit(A,minus_6452836326544984404t_unit(J2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,I2),bot_bo3957492148770167129t_unit))))),
        $ite(aa_set8780210009671629607unit_o(member_Product_unit(I2),J2),B4,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3302_UNION__fun__upd,axiom,
    ! [A: fun_Pr6739565440344758894_set_o,I2: product_unit,B4: set_o,J2: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(fun_up1762901246852280360_set_o(A,I2,B4),J2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,comple90263536869209701_set_o(image_6930536745178679216_set_o(A,minus_6452836326544984404t_unit(J2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,I2),bot_bo3957492148770167129t_unit))))),
        $ite(aa_set8780210009671629607unit_o(member_Product_unit(I2),J2),B4,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3303_UNION__fun__upd,axiom,
    ! [A: fun_Pr1628945989756059814et_nat,I2: product_unit,B4: set_nat,J2: set_Product_unit] :
      comple7399068483239264473et_nat(image_8173849991827986510et_nat(fun_up4569834200710200790et_nat(A,I2,B4),J2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_8173849991827986510et_nat(A,minus_6452836326544984404t_unit(J2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,I2),bot_bo3957492148770167129t_unit))))),
        $ite(aa_set8780210009671629607unit_o(member_Product_unit(I2),J2),B4,bot_bot_set_nat)) ).

% UNION_fun_upd
tff(fact_3304_UNION__fun__upd,axiom,
    ! [A: fun_Pr8362234881679108866et_int,I2: product_unit,B4: set_int,J2: set_Product_unit] :
      comple3221217463730067765et_int(image_3995998972318789802et_int(fun_up391983181201004082et_int(A,I2,B4),J2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_3995998972318789802et_int(A,minus_6452836326544984404t_unit(J2,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,I2),bot_bo3957492148770167129t_unit))))),
        $ite(aa_set8780210009671629607unit_o(member_Product_unit(I2),J2),B4,bot_bot_set_int)) ).

% UNION_fun_upd
tff(fact_3305_UNION__fun__upd,axiom,
    ! [A: fun_o_1833192751255691672t_unit,I2: $o,B4: set_Product_unit,J2: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(fun_up4718931970374059986t_unit(A,(I2),B4),J2)) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,comple3349607820504768346t_unit(image_663195431845683034t_unit(A,minus_minus_set_o(J2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(I2)),bot_bot_set_o))))),
        $ite(aa_set_o_o(member_o((I2)),J2),B4,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3306_UNION__fun__upd,axiom,
    ! [A: fun_o_set_o,I2: $o,B4: set_o,J2: set_o] :
      comple90263536869209701_set_o(image_o_set_o(fun_upd_o_set_o(A,(I2),B4),J2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,comple90263536869209701_set_o(image_o_set_o(A,minus_minus_set_o(J2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(I2)),bot_bot_set_o))))),
        $ite(aa_set_o_o(member_o((I2)),J2),B4,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3307_UNION__fun__upd,axiom,
    ! [A: fun_o_set_nat,I2: $o,B4: set_nat,J2: set_o] :
      comple7399068483239264473et_nat(image_o_set_nat(fun_upd_o_set_nat(A,(I2),B4),J2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_o_set_nat(A,minus_minus_set_o(J2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(I2)),bot_bot_set_o))))),
        $ite(aa_set_o_o(member_o((I2)),J2),B4,bot_bot_set_nat)) ).

% UNION_fun_upd
tff(fact_3308_UNION__fun__upd,axiom,
    ! [A: fun_o_set_int,I2: $o,B4: set_int,J2: set_o] :
      comple3221217463730067765et_int(image_o_set_int(fun_upd_o_set_int(A,(I2),B4),J2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,comple3221217463730067765et_int(image_o_set_int(A,minus_minus_set_o(J2,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(I2)),bot_bot_set_o))))),
        $ite(aa_set_o_o(member_o((I2)),J2),B4,bot_bot_set_int)) ).

% UNION_fun_upd
tff(fact_3309_UNION__fun__upd,axiom,
    ! [A: fun_in4731230917697954842t_unit,I2: int,B4: set_Product_unit,J2: set_int] :
      comple3349607820504768346t_unit(image_8658272795163236474t_unit(fun_up5054257004045450754t_unit(A,I2,B4),J2)) = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,comple3349607820504768346t_unit(image_8658272795163236474t_unit(A,minus_minus_set_int(J2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,I2),bot_bot_set_int))))),
        $ite(aa_set_int_o(member_int(I2),J2),B4,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3310_UNION__fun__upd,axiom,
    ! [A: fun_int_set_o,I2: int,B4: set_o,J2: set_int] :
      comple90263536869209701_set_o(image_int_set_o(fun_upd_int_set_o(A,I2,B4),J2)) = aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,comple90263536869209701_set_o(image_int_set_o(A,minus_minus_set_int(J2,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,I2),bot_bot_set_int))))),
        $ite(aa_set_int_o(member_int(I2),J2),B4,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3311_cSUP__union,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(F,A))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(F,B4))
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3312_cSUP__union,axiom,
    ! [A: set_o,F: fun_o_nat,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat2(F,A))
       => ( ( B4 != bot_bot_set_o )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat2(F,B4))
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3313_cSUP__union,axiom,
    ! [A: set_nat,F: fun_nat_nat,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ( B4 != bot_bot_set_nat )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(F),B4))
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3314_cSUP__union,axiom,
    ! [A: set_int,F: fun_int_nat,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_int_nat2(F,A))
       => ( ( B4 != bot_bot_set_int )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_int_nat2(F,B4))
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3315_cSUP__union,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_405062704495631173t_unit(F,A))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_405062704495631173t_unit(F,B4))
           => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A))),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3316_cSUP__union,axiom,
    ! [A: set_o,F: fun_o_Product_unit,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_o_Product_unit2(F,A))
       => ( ( B4 != bot_bot_set_o )
         => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_o_Product_unit2(F,B4))
           => ( comple4687483117567863418t_unit(image_o_Product_unit2(F,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,comple4687483117567863418t_unit(image_o_Product_unit2(F,A))),comple4687483117567863418t_unit(image_o_Product_unit2(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3317_cSUP__union,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_8730104196221521654t_unit(F,A))
       => ( ( B4 != bot_bot_set_nat )
         => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_8730104196221521654t_unit(F,B4))
           => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A))),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3318_cSUP__union,axiom,
    ! [A: set_int,F: fun_int_Product_unit,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_5846526143423701914t_unit(F,A))
       => ( ( B4 != bot_bot_set_int )
         => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_5846526143423701914t_unit(F,B4))
           => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A))),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3319_cSUP__union,axiom,
    ! [A: set_Product_unit,F: fun_Pr1628945989756059814et_nat,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,A))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,B4))
           => ( comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,A))),comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3320_cSUP__union,axiom,
    ! [A: set_o,F: fun_o_set_nat,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( condit5477540289124974626et_nat(image_o_set_nat(F,A))
       => ( ( B4 != bot_bot_set_o )
         => ( condit5477540289124974626et_nat(image_o_set_nat(F,B4))
           => ( comple7399068483239264473et_nat(image_o_set_nat(F,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(image_o_set_nat(F,A))),comple7399068483239264473et_nat(image_o_set_nat(F,B4))) ) ) ) ) ) ).

% cSUP_union
tff(fact_3321_pair__in__swap__image,axiom,
    ! [Y: set_nat,X: heap_e7401611519738050253t_unit,A: set_Pr3948176798113811640et_nat] :
      ( member727419437372278867t_unit(produc2471974422597015918t_unit(Y,X),image_1893317993577548229t_unit(produc6300697153377358172et_nat,A))
    <=> member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y),A) ) ).

% pair_in_swap_image
tff(fact_3322_pair__in__swap__image,axiom,
    ! [Y: produc3658429121746597890et_nat,X: fun_Pr6460944699146269935_nat_o,A: set_Pr1312406793078187350et_nat] :
      ( member7677167966932745849_nat_o(produc6339119935501530516_nat_o(Y,X),image_1122556727974905353_nat_o(produc6102546028088763386et_nat,A))
    <=> member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y),A) ) ).

% pair_in_swap_image
tff(fact_3323_pair__in__swap__image,axiom,
    ! [Y: produc8799533983703829792et_nat,X: fun_Pr6460944699146269935_nat_o,A: set_Pr4040394263813967220et_nat] :
      ( member3696715088396795611_nat_o(produc6151749222514847990_nat_o(Y,X),image_2438217170757901833_nat_o(produc2593090504590949656et_nat,A))
    <=> member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y),A) ) ).

% pair_in_swap_image
tff(fact_3324_pair__in__swap__image,axiom,
    ! [Y: heap_e7401611519738050253t_unit,X: set_nat,A: set_Pr4071100604818519410t_unit] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y),X),image_5838825082199883205et_nat(produc1264744871843189710t_unit,A))
    <=> member727419437372278867t_unit(produc2471974422597015918t_unit(X,Y),A) ) ).

% pair_in_swap_image
tff(fact_3325_pair__in__swap__image,axiom,
    ! [Y: nat,X: nat,A: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X),image_5168914502847457605at_nat(product_swap_nat_nat,A))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),A) ) ).

% pair_in_swap_image
tff(fact_3326_pair__in__swap__image,axiom,
    ! [Y: int,X: int,A: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X),image_2653370878348428101nt_int(product_swap_int_int,A))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),A) ) ).

% pair_in_swap_image
tff(fact_3327_pair__in__swap__image,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o,X: produc3658429121746597890et_nat,A: set_Pr3013394129895136920_nat_o] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y),X),image_2538532745900505353et_nat(produc6424530459701215220_nat_o,A))
    <=> member7677167966932745849_nat_o(produc6339119935501530516_nat_o(X,Y),A) ) ).

% pair_in_swap_image
tff(fact_3328_pair__in__swap__image,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o,X: produc8799533983703829792et_nat,A: set_Pr7905283282039288826_nat_o] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y),X),image_660286381542540297et_nat(produc1452984123473447766_nat_o,A))
    <=> member3696715088396795611_nat_o(produc6151749222514847990_nat_o(X,Y),A) ) ).

% pair_in_swap_image
tff(fact_3329_cSUP__insert,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,A2: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(F,A))
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_Product_unit_nat(F,A2)),aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3330_cSUP__insert,axiom,
    ! [A: set_o,F: fun_o_nat,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat2(F,A))
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_o_nat(F,(A2))),aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3331_cSUP__insert,axiom,
    ! [A: set_nat,F: fun_nat_nat,A2: nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(F,A2)),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))) ) ) ) ).

% cSUP_insert
tff(fact_3332_cSUP__insert,axiom,
    ! [A: set_int,F: fun_int_nat,A2: int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_int_nat2(F,A))
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_int_nat(F,A2)),aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3333_cSUP__insert,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,A2: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_405062704495631173t_unit(F,A))
       => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_Pro8294289767544744956t_unit(F,A2)),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3334_cSUP__insert,axiom,
    ! [A: set_o,F: fun_o_Product_unit,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_o_Product_unit2(F,A))
       => ( comple4687483117567863418t_unit(image_o_Product_unit2(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_o_Product_unit(F,(A2))),comple4687483117567863418t_unit(image_o_Product_unit2(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3335_cSUP__insert,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit,A2: nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_8730104196221521654t_unit(F,A))
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_nat_Product_unit(F,A2)),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3336_cSUP__insert,axiom,
    ! [A: set_int,F: fun_int_Product_unit,A2: int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_5846526143423701914t_unit(F,A))
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,aa_int_Product_unit(F,A2)),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3337_cSUP__insert,axiom,
    ! [A: set_Product_unit,F: fun_Pr1628945989756059814et_nat,A2: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,A))
       => ( comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_Pro6064029160526250199et_nat(F,A2)),comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3338_cSUP__insert,axiom,
    ! [A: set_o,F: fun_o_set_nat,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( condit5477540289124974626et_nat(image_o_set_nat(F,A))
       => ( comple7399068483239264473et_nat(image_o_set_nat(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_o_set_nat(F,(A2))),comple7399068483239264473et_nat(image_o_set_nat(F,A))) ) ) ) ).

% cSUP_insert
tff(fact_3339_cSup__inter__less__eq,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(condit5488710616941104124bove_o,A)
     => ( aa_set_o_o(condit5488710616941104124bove_o,B4)
       => ( ( inf_inf_set_o(A,B4) != bot_bot_set_o )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Sup_Sup_o,inf_inf_set_o(A,B4))),aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(complete_Sup_Sup_o,A)),aa_set_o_o(complete_Sup_Sup_o,B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3340_cSup__inter__less__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,B4)
       => ( ( inf_in3376754634429691197_nat_o(A,B4) != bot_bo8632255151191677115_nat_o )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple7878897588548607196_nat_o(inf_in3376754634429691197_nat_o(A,B4))),aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(comple7878897588548607196_nat_o(A)),comple7878897588548607196_nat_o(B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3341_cSup__inter__less__eq,axiom,
    ! [A: set_filter_nat,B4: set_filter_nat] :
      ( condit4029664723990530684er_nat(A)
     => ( condit4029664723990530684er_nat(B4)
       => ( ( inf_in7395780696554941096er_nat(A,B4) != bot_bo498966703094740906er_nat )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple2454491259644266675er_nat(inf_in7395780696554941096er_nat(A,B4))),sup_sup_filter_nat(comple2454491259644266675er_nat(A),comple2454491259644266675er_nat(B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3342_cSup__inter__less__eq,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( condit5477540289124974626et_nat(A)
     => ( condit5477540289124974626et_nat(B4)
       => ( ( inf_inf_set_set_nat(A,B4) != bot_bot_set_set_nat )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(inf_inf_set_set_nat(A,B4))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(A)),comple7399068483239264473et_nat(B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3343_cSup__inter__less__eq,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,A)
     => ( aa_set_int_o(condit2212336002400062152ve_int,B4)
       => ( ( inf_inf_set_int(A,B4) != bot_bot_set_int )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(inf_inf_set_int(A,B4))),aa_int_int(aa_int_fun_int_int(sup_sup_int,complete_Sup_Sup_int(A)),complete_Sup_Sup_int(B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3344_cSup__inter__less__eq,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,A)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,A)),aa_set_nat_nat(complete_Sup_Sup_nat,B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3345_cSup__inter__less__eq,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,A)
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) != bot_bo3957492148770167129t_unit )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(inf_in4660618365625256667t_unit(A,B4))),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,comple4687483117567863418t_unit(A)),comple4687483117567863418t_unit(B4))) ) ) ) ).

% cSup_inter_less_eq
tff(fact_3346_cSUP__subset__mono,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_int,B4: set_Product_unit,F: fun_Product_unit_int] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(G,B4))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(F,X3)),aa_Product_unit_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_873079544045703924it_int(F,A))),complete_Sup_Sup_int(image_873079544045703924it_int(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3347_cSUP__subset__mono,axiom,
    ! [A: set_o,G: fun_o_int,B4: set_o,F: fun_o_int] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int2(G,B4))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X3))),aa_o_int(G,(X3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_o_int2(F,A))),complete_Sup_Sup_int(image_o_int2(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3348_cSUP__subset__mono,axiom,
    ! [A: set_int,G: fun_int_int,B4: set_int,F: fun_int_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_int_int2(G,B4))
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_int_int2(F,A))),complete_Sup_Sup_int(image_int_int2(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3349_cSUP__subset__mono,axiom,
    ! [A: set_nat,G: fun_nat_int,B4: set_nat,F: fun_nat_int] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_nat_int2(G,B4))
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_nat_int2(F,A))),complete_Sup_Sup_int(image_nat_int2(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3350_cSUP__subset__mono,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_nat,B4: set_Product_unit,F: fun_Product_unit_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(G,B4))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,X3)),aa_Product_unit_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3351_cSUP__subset__mono,axiom,
    ! [A: set_o,G: fun_o_nat,B4: set_o,F: fun_o_nat] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat2(G,B4))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X3))),aa_o_nat(G,(X3))) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3352_cSUP__subset__mono,axiom,
    ! [A: set_int,G: fun_int_nat,B4: set_int,F: fun_int_nat] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_int_nat2(G,B4))
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3353_cSUP__subset__mono,axiom,
    ! [A: set_nat,G: fun_nat_nat,B4: set_nat,F: fun_nat_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3354_cSUP__subset__mono,axiom,
    ! [A: set_Product_unit,G: fun_Pr2191293291710260995t_unit,B4: set_Product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_405062704495631173t_unit(G,B4))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(F,X3)),aa_Pro8294289767544744956t_unit(G,X3)) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A))),comple4687483117567863418t_unit(image_405062704495631173t_unit(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3355_cSUP__subset__mono,axiom,
    ! [A: set_o,G: fun_o_Product_unit,B4: set_o,F: fun_o_Product_unit] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_o_Product_unit2(G,B4))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_o_Product_unit(F,(X3))),aa_o_Product_unit(G,(X3))) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(image_o_Product_unit2(F,A))),comple4687483117567863418t_unit(image_o_Product_unit2(G,B4))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3356_Image__Int__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
      ( single5507084439971879973at_nat(converse_nat_nat(R))
     => ( aa_set_nat_set_nat(image_nat_nat(R),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(image_nat_nat(R),A)),aa_set_nat_set_nat(image_nat_nat(R),B4)) ) ) ).

% Image_Int_eq
tff(fact_3357_image__update,axiom,
    ! [X: code_integer,A: set_Code_integer,F: fun_Co8499238673658658103nteger,N2: code_integer] :
      ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
     => ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(fun_up8355007323143964473nteger(F,X),N2)),A) = aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A) ) ) ).

% image_update
tff(fact_3358_image__update,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_set_nat,N2: set_nat] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( aa_set7246812951530217472et_nat(image_nat_set_nat(fun_upd_nat_set_nat(F,X,N2)),A) = aa_set7246812951530217472et_nat(image_nat_set_nat(F),A) ) ) ).

% image_update
tff(fact_3359_image__update,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_nat,N2: nat] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_set_nat(image_nat_nat2(fun_upd_nat_nat(F,X,N2)),A) = aa_set_nat_set_nat(image_nat_nat2(F),A) ) ) ).

% image_update
tff(fact_3360_image__update,axiom,
    ! [X: nat,A: set_nat,F: fun_nat_Code_integer,N2: code_integer] :
      ( ~ aa_set_nat_o(member_nat(X),A)
     => ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(fun_up3904115110581602988nteger(F,X,N2)),A) = aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A) ) ) ).

% image_update
tff(fact_3361_bdd__above__empty,axiom,
    aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,bot_bo3957492148770167129t_unit) ).

% bdd_above_empty
tff(fact_3362_bdd__above__empty,axiom,
    aa_set_o_o(condit5488710616941104124bove_o,bot_bot_set_o) ).

% bdd_above_empty
tff(fact_3363_bdd__above__empty,axiom,
    aa_set_nat_o(condit2214826472909112428ve_nat,bot_bot_set_nat) ).

% bdd_above_empty
tff(fact_3364_bdd__above__empty,axiom,
    aa_set_int_o(condit2212336002400062152ve_int,bot_bot_set_int) ).

% bdd_above_empty
tff(fact_3365_bdd__above__empty,axiom,
    aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,bot_bo8632255151191677115_nat_o) ).

% bdd_above_empty
tff(fact_3366_bdd__above__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))
    <=> aa_set_nat_o(condit2214826472909112428ve_nat,A) ) ).

% bdd_above_insert
tff(fact_3367_bdd__above__insert,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))
    <=> aa_set_int_o(condit2212336002400062152ve_int,A) ) ).

% bdd_above_insert
tff(fact_3368_bdd__above__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(condit5488710616941104124bove_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> aa_set_o_o(condit5488710616941104124bove_o,A) ) ).

% bdd_above_insert
tff(fact_3369_bdd__above__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,A) ) ).

% bdd_above_insert
tff(fact_3370_swap__simp,axiom,
    ! [X: set_nat,Y: heap_e7401611519738050253t_unit] : aa_Pro2092192379717785596et_nat(produc1264744871843189710t_unit,produc2471974422597015918t_unit(X,Y)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y),X) ).

% swap_simp
tff(fact_3371_swap__simp,axiom,
    ! [X: produc3658429121746597890et_nat,Y: fun_Pr6460944699146269935_nat_o] : aa_Pro1287330115173305408et_nat(produc6424530459701215220_nat_o,produc6339119935501530516_nat_o(X,Y)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y),X) ).

% swap_simp
tff(fact_3372_swap__simp,axiom,
    ! [X: produc8799533983703829792et_nat,Y: fun_Pr6460944699146269935_nat_o] : aa_Pro7790764968675496000et_nat(produc1452984123473447766_nat_o,produc6151749222514847990_nat_o(X,Y)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y),X) ).

% swap_simp
tff(fact_3373_swap__simp,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat] : aa_Pro7370057327950226428t_unit(produc6300697153377358172et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = produc2471974422597015918t_unit(Y,X) ).

% swap_simp
tff(fact_3374_swap__simp,axiom,
    ! [X: nat,Y: nat] : aa_Pro262675715343573244at_nat(product_swap_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),X) ).

% swap_simp
tff(fact_3375_swap__simp,axiom,
    ! [X: int,Y: int] : aa_Pro6970504127699319548nt_int(product_swap_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = aa_int7462565939943104719nt_int(product_Pair_int_int(Y),X) ).

% swap_simp
tff(fact_3376_swap__simp,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : aa_Pro9094726134102481216_nat_o(produc6102546028088763386et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = produc6339119935501530516_nat_o(Y,X) ).

% swap_simp
tff(fact_3377_swap__simp,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : aa_Pro345323721036081728_nat_o(produc2593090504590949656et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = produc6151749222514847990_nat_o(Y,X) ).

% swap_simp
tff(fact_3378_single__valuedD,axiom,
    ! [R2: set_Pr3948176798113811640et_nat,X: heap_e7401611519738050253t_unit,Y: set_nat,Z: set_nat] :
      ( single6620584844202787856et_nat(R2)
     => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y),R2)
       => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Z),R2)
         => ( Y = Z ) ) ) ) ).

% single_valuedD
tff(fact_3379_single__valuedD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( single5507084439971879973at_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R2)
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),R2)
         => ( Y = Z ) ) ) ) ).

% single_valuedD
tff(fact_3380_single__valuedD,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
      ( single6503947681363884253nt_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),R2)
         => ( Y = Z ) ) ) ) ).

% single_valuedD
tff(fact_3381_single__valuedD,axiom,
    ! [R2: set_Pr1312406793078187350et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,Z: produc3658429121746597890et_nat] :
      ( single1421632596207000750et_nat(R2)
     => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y),R2)
       => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Z),R2)
         => ( Y = Z ) ) ) ) ).

% single_valuedD
tff(fact_3382_single__valuedD,axiom,
    ! [R2: set_Pr4040394263813967220et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,Z: produc8799533983703829792et_nat] :
      ( single3978381750310075340et_nat(R2)
     => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y),R2)
       => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Z),R2)
         => ( Y = Z ) ) ) ) ).

% single_valuedD
tff(fact_3383_single__valuedI,axiom,
    ! [R2: set_Pr3948176798113811640et_nat] :
      ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
          ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R2)
         => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Z3),R2)
           => ( Y2 = Z3 ) ) )
     => single6620584844202787856et_nat(R2) ) ).

% single_valuedI
tff(fact_3384_single__valuedI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y2: nat,Z3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R2)
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Z3),R2)
           => ( Y2 = Z3 ) ) )
     => single5507084439971879973at_nat(R2) ) ).

% single_valuedI
tff(fact_3385_single__valuedI,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y2: int,Z3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R2)
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Z3),R2)
           => ( Y2 = Z3 ) ) )
     => single6503947681363884253nt_int(R2) ) ).

% single_valuedI
tff(fact_3386_single__valuedI,axiom,
    ! [R2: set_Pr1312406793078187350et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),R2)
         => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Z3),R2)
           => ( Y2 = Z3 ) ) )
     => single1421632596207000750et_nat(R2) ) ).

% single_valuedI
tff(fact_3387_single__valuedI,axiom,
    ! [R2: set_Pr4040394263813967220et_nat] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat,Z3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2),R2)
         => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Z3),R2)
           => ( Y2 = Z3 ) ) )
     => single3978381750310075340et_nat(R2) ) ).

% single_valuedI
tff(fact_3388_single__valued__def,axiom,
    ! [R2: set_Pr3948176798113811640et_nat] :
      ( single6620584844202787856et_nat(R2)
    <=> ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat] :
          ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X2),Y3),R2)
         => ! [Z2: set_nat] :
              ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X2),Z2),R2)
             => ( Y3 = Z2 ) ) ) ) ).

% single_valued_def
tff(fact_3389_single__valued__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( single5507084439971879973at_nat(R2)
    <=> ! [X2: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y3),R2)
         => ! [Z2: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Z2),R2)
             => ( Y3 = Z2 ) ) ) ) ).

% single_valued_def
tff(fact_3390_single__valued__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( single6503947681363884253nt_int(R2)
    <=> ! [X2: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y3),R2)
         => ! [Z2: int] :
              ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Z2),R2)
             => ( Y3 = Z2 ) ) ) ) ).

% single_valued_def
tff(fact_3391_single__valued__def,axiom,
    ! [R2: set_Pr1312406793078187350et_nat] :
      ( single1421632596207000750et_nat(R2)
    <=> ! [X2: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Y3),R2)
         => ! [Z2: produc3658429121746597890et_nat] :
              ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X2),Z2),R2)
             => ( Y3 = Z2 ) ) ) ) ).

% single_valued_def
tff(fact_3392_single__valued__def,axiom,
    ! [R2: set_Pr4040394263813967220et_nat] :
      ( single3978381750310075340et_nat(R2)
    <=> ! [X2: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Y3),R2)
         => ! [Z2: produc8799533983703829792et_nat] :
              ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X2),Z2),R2)
             => ( Y3 = Z2 ) ) ) ) ).

% single_valued_def
tff(fact_3393_single__valued__confluent,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( single5507084439971879973at_nat(R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),transi2905341329935302413cl_nat(R2))
       => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z),transi2905341329935302413cl_nat(R2))
         => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),transi2905341329935302413cl_nat(R2))
            | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z),Y),transi2905341329935302413cl_nat(R2)) ) ) ) ) ).

% single_valued_confluent
tff(fact_3394_single__valued__confluent,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
      ( single6503947681363884253nt_int(R2)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),transi2902850859426252137cl_int(R2))
       => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Z),transi2902850859426252137cl_int(R2))
         => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z),transi2902850859426252137cl_int(R2))
            | member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Z),Y),transi2902850859426252137cl_int(R2)) ) ) ) ) ).

% single_valued_confluent
tff(fact_3395_cSup__le__iff,axiom,
    ! [S: set_o,A2: $o] :
      ( ( S != bot_bot_set_o )
     => ( aa_set_o_o(condit5488710616941104124bove_o,S)
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Sup_Sup_o,S)),(A2))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),S)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(A2)) ) ) ) ) ).

% cSup_le_iff
tff(fact_3396_cSup__le__iff,axiom,
    ! [S: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( S != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,S)
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple7878897588548607196_nat_o(S)),A2)
        <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X2),A2) ) ) ) ) ).

% cSup_le_iff
tff(fact_3397_cSup__le__iff,axiom,
    ! [S: set_filter_nat,A2: filter_nat] :
      ( ( S != bot_bo498966703094740906er_nat )
     => ( condit4029664723990530684er_nat(S)
       => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple2454491259644266675er_nat(S)),A2)
        <=> ! [X2: filter_nat] :
              ( member_filter_nat(X2,S)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X2),A2) ) ) ) ) ).

% cSup_le_iff
tff(fact_3398_cSup__le__iff,axiom,
    ! [S: set_set_nat,A2: set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ( condit5477540289124974626et_nat(S)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(S)),A2)
        <=> ! [X2: set_nat] :
              ( member_set_nat(X2,S)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),A2) ) ) ) ) ).

% cSup_le_iff
tff(fact_3399_cSup__le__iff,axiom,
    ! [S: set_int,A2: int] :
      ( ( S != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,S)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(S)),A2)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),S)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),A2) ) ) ) ) ).

% cSup_le_iff
tff(fact_3400_cSup__le__iff,axiom,
    ! [S: set_nat,A2: nat] :
      ( ( S != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,S)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,S)),A2)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),S)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),A2) ) ) ) ) ).

% cSup_le_iff
tff(fact_3401_cSup__le__iff,axiom,
    ! [S: set_Product_unit,A2: product_unit] :
      ( ( S != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,S)
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(S)),A2)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),S)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),A2) ) ) ) ) ).

% cSup_le_iff
tff(fact_3402_cSup__mono,axiom,
    ! [B4: set_o,A: set_o] :
      ( ( B4 != bot_bot_set_o )
     => ( aa_set_o_o(condit5488710616941104124bove_o,A)
       => ( ! [B2: $o] :
              ( aa_set_o_o(member_o((B2)),B4)
             => ? [X4: $o] :
                  ( aa_set_o_o(member_o((X4)),A)
                  & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B2)),(X4)) ) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Sup_Sup_o,B4)),aa_set_o_o(complete_Sup_Sup_o,A)) ) ) ) ).

% cSup_mono
tff(fact_3403_cSup__mono,axiom,
    ! [B4: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( B4 != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,A)
       => ( ! [B2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B2),B4)
             => ? [X4: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A)
                  & aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,B2),X4) ) )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple7878897588548607196_nat_o(B4)),comple7878897588548607196_nat_o(A)) ) ) ) ).

% cSup_mono
tff(fact_3404_cSup__mono,axiom,
    ! [B4: set_filter_nat,A: set_filter_nat] :
      ( ( B4 != bot_bo498966703094740906er_nat )
     => ( condit4029664723990530684er_nat(A)
       => ( ! [B2: filter_nat] :
              ( member_filter_nat(B2,B4)
             => ? [X4: filter_nat] :
                  ( member_filter_nat(X4,A)
                  & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B2),X4) ) )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple2454491259644266675er_nat(B4)),comple2454491259644266675er_nat(A)) ) ) ) ).

% cSup_mono
tff(fact_3405_cSup__mono,axiom,
    ! [B4: set_set_nat,A: set_set_nat] :
      ( ( B4 != bot_bot_set_set_nat )
     => ( condit5477540289124974626et_nat(A)
       => ( ! [B2: set_nat] :
              ( member_set_nat(B2,B4)
             => ? [X4: set_nat] :
                  ( member_set_nat(X4,A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),X4) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(B4)),comple7399068483239264473et_nat(A)) ) ) ) ).

% cSup_mono
tff(fact_3406_cSup__mono,axiom,
    ! [B4: set_int,A: set_int] :
      ( ( B4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,A)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),B4)
             => ? [X4: int] :
                  ( aa_set_int_o(member_int(X4),A)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),X4) ) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(B4)),complete_Sup_Sup_int(A)) ) ) ) ).

% cSup_mono
tff(fact_3407_cSup__mono,axiom,
    ! [B4: set_nat,A: set_nat] :
      ( ( B4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),X4) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,B4)),aa_set_nat_nat(complete_Sup_Sup_nat,A)) ) ) ) ).

% cSup_mono
tff(fact_3408_cSup__mono,axiom,
    ! [B4: set_Product_unit,A: set_Product_unit] :
      ( ( B4 != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,A)
       => ( ! [B2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(B2),B4)
             => ? [X4: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A)
                  & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B2),X4) ) )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(B4)),comple4687483117567863418t_unit(A)) ) ) ) ).

% cSup_mono
tff(fact_3409_cSUP__le__iff,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_int,U: int] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_873079544045703924it_int(F,A))),U)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3410_cSUP__le__iff,axiom,
    ! [A: set_o,F: fun_o_int,U: int] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int2(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_o_int2(F,A))),U)
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X2))),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3411_cSUP__le__iff,axiom,
    ! [A: set_nat,F: fun_nat_int,U: int] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_nat_int2(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_nat_int2(F,A))),U)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3412_cSUP__le__iff,axiom,
    ! [A: set_int,F: fun_int_int,U: int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_int_int2(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(image_int_int2(F,A))),U)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3413_cSUP__le__iff,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,U: nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(F,A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A))),U)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3414_cSUP__le__iff,axiom,
    ! [A: set_o,F: fun_o_nat,U: nat] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat2(F,A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,A))),U)
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X2))),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3415_cSUP__le__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat,U: nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),U)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3416_cSUP__le__iff,axiom,
    ! [A: set_int,F: fun_int_nat,U: nat] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_int_nat2(F,A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,A))),U)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3417_cSUP__le__iff,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,U: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_405062704495631173t_unit(F,A))
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A))),U)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(F,X2)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3418_cSUP__le__iff,axiom,
    ! [A: set_o,F: fun_o_Product_unit,U: product_unit] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,image_o_Product_unit2(F,A))
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(image_o_Product_unit2(F,A))),U)
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_o_Product_unit(F,(X2))),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3419_cSUP__mono,axiom,
    ! [A: set_Product_unit,G: fun_nat_set_nat,B4: set_nat,F: fun_Pr1628945989756059814et_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( condit5477540289124974626et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))
       => ( ! [N3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_Pro6064029160526250199et_nat(F,N3)),aa_nat_set_nat(G,X4)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,A))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3420_cSUP__mono,axiom,
    ! [A: set_o,G: fun_nat_set_nat,B4: set_nat,F: fun_o_set_nat] :
      ( ( A != bot_bot_set_o )
     => ( condit5477540289124974626et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))
       => ( ! [N3: $o] :
              ( aa_set_o_o(member_o((N3)),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_o_set_nat(F,(N3))),aa_nat_set_nat(G,X4)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_o_set_nat(F,A))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3421_cSUP__mono,axiom,
    ! [A: set_nat,G: fun_nat_set_nat,B4: set_nat,F: fun_nat_set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( condit5477540289124974626et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))
       => ( ! [N3: nat] :
              ( aa_set_nat_o(member_nat(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,N3)),aa_nat_set_nat(G,X4)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3422_cSUP__mono,axiom,
    ! [A: set_int,G: fun_nat_set_nat,B4: set_nat,F: fun_int_set_nat] :
      ( ( A != bot_bot_set_int )
     => ( condit5477540289124974626et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))
       => ( ! [N3: int] :
              ( aa_set_int_o(member_int(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_int_set_nat(F,N3)),aa_nat_set_nat(G,X4)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_int_set_nat(F,A))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3423_cSUP__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,G: fun_nat_set_nat,B4: set_nat,F: fun_fu2098174926252251144et_nat] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( condit5477540289124974626et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))
       => ( ! [N3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_fun7872747175131507769et_nat(F,N3)),aa_nat_set_nat(G,X4)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(image_7230913979079189040et_nat(F,A))),comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3424_cSUP__mono,axiom,
    ! [A: set_Product_unit,G: fun_nat_nat,B4: set_nat,F: fun_Product_unit_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( ! [N3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,N3)),aa_nat_nat(G,X4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3425_cSUP__mono,axiom,
    ! [A: set_o,G: fun_nat_nat,B4: set_nat,F: fun_o_nat] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( ! [N3: $o] :
              ( aa_set_o_o(member_o((N3)),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(N3))),aa_nat_nat(G,X4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat2(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3426_cSUP__mono,axiom,
    ! [A: set_nat,G: fun_nat_nat,B4: set_nat,F: fun_nat_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( ! [N3: nat] :
              ( aa_set_nat_o(member_nat(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(G,X4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3427_cSUP__mono,axiom,
    ! [A: set_int,G: fun_nat_nat,B4: set_nat,F: fun_int_nat] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( ! [N3: int] :
              ( aa_set_int_o(member_int(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,N3)),aa_nat_nat(G,X4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_int_nat2(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3428_cSUP__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,G: fun_nat_nat,B4: set_nat,F: fun_fu3504661237974713170_o_nat] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( ! [N3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(N3),A)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_fun767828674014743683_o_nat(F,N3)),aa_nat_nat(G,X4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_5746879016480857722_o_nat(F,A))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ).

% cSUP_mono
tff(fact_3429_cSup__subset__mono,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_o_o(condit5488710616941104124bove_o,B4)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Sup_Sup_o,A)),aa_set_o_o(complete_Sup_Sup_o,B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3430_cSup__subset__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,B4)
       => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple7878897588548607196_nat_o(A)),comple7878897588548607196_nat_o(B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3431_cSup__subset__mono,axiom,
    ! [A: set_filter_nat,B4: set_filter_nat] :
      ( ( A != bot_bo498966703094740906er_nat )
     => ( condit4029664723990530684er_nat(B4)
       => ( ord_le2426478655948331894er_nat(A,B4)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple2454491259644266675er_nat(A)),comple2454491259644266675er_nat(B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3432_cSup__subset__mono,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( condit5477540289124974626et_nat(B4)
       => ( ord_le6893508408891458716et_nat(A,B4)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7399068483239264473et_nat(A)),comple7399068483239264473et_nat(B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3433_cSup__subset__mono,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,B4)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Sup_Sup_int(A)),complete_Sup_Sup_int(B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3434_cSup__subset__mono,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,B4)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,A)),aa_set_nat_nat(complete_Sup_Sup_nat,B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3435_cSup__subset__mono,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,B4)
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple4687483117567863418t_unit(A)),comple4687483117567863418t_unit(B4)) ) ) ) ).

% cSup_subset_mono
tff(fact_3436_cSup__insert__If,axiom,
    ! [X5: set_set_nat,A2: set_nat] :
      ( condit5477540289124974626et_nat(X5)
     => ( comple7399068483239264473et_nat(insert_set_nat(A2,X5)) = $ite(X5 = bot_bot_set_set_nat,A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(X5))) ) ) ).

% cSup_insert_If
tff(fact_3437_cSup__insert__If,axiom,
    ! [X5: set_o,A2: $o] :
      ( aa_set_o_o(condit5488710616941104124bove_o,X5)
     => ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),X5))
      <=> $ite(X5 = bot_bot_set_o,(A2),aa_o_o(aa_o_fun_o_o(sup_sup_o,(A2)),aa_set_o_o(complete_Sup_Sup_o,X5))) ) ) ).

% cSup_insert_If
tff(fact_3438_cSup__insert__If,axiom,
    ! [X5: set_int,A2: int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,X5)
     => ( complete_Sup_Sup_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),X5)) = $ite(X5 = bot_bot_set_int,A2,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),complete_Sup_Sup_int(X5))) ) ) ).

% cSup_insert_If
tff(fact_3439_cSup__insert__If,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,X5)
     => ( comple7878897588548607196_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),X5)) = $ite(X5 = bot_bo8632255151191677115_nat_o,A2,aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(A2),comple7878897588548607196_nat_o(X5))) ) ) ).

% cSup_insert_If
tff(fact_3440_cSup__insert__If,axiom,
    ! [X5: set_nat,A2: nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,X5)
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),X5)) = $ite(X5 = bot_bot_set_nat,A2,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),aa_set_nat_nat(complete_Sup_Sup_nat,X5))) ) ) ).

% cSup_insert_If
tff(fact_3441_cSup__insert__If,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,X5)
     => ( comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),X5)) = $ite(X5 = bot_bo3957492148770167129t_unit,A2,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),comple4687483117567863418t_unit(X5))) ) ) ).

% cSup_insert_If
tff(fact_3442_cSup__insert,axiom,
    ! [X5: set_set_nat,A2: set_nat] :
      ( ( X5 != bot_bot_set_set_nat )
     => ( condit5477540289124974626et_nat(X5)
       => ( comple7399068483239264473et_nat(insert_set_nat(A2,X5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),comple7399068483239264473et_nat(X5)) ) ) ) ).

% cSup_insert
tff(fact_3443_cSup__insert,axiom,
    ! [X5: set_o,A2: $o] :
      ( ( X5 != bot_bot_set_o )
     => ( aa_set_o_o(condit5488710616941104124bove_o,X5)
       => ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),X5))
        <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,(A2)),aa_set_o_o(complete_Sup_Sup_o,X5)) ) ) ) ).

% cSup_insert
tff(fact_3444_cSup__insert,axiom,
    ! [X5: set_int,A2: int] :
      ( ( X5 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,X5)
       => ( complete_Sup_Sup_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),X5)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),complete_Sup_Sup_int(X5)) ) ) ) ).

% cSup_insert
tff(fact_3445_cSup__insert,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( X5 != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,X5)
       => ( comple7878897588548607196_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),X5)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(A2),comple7878897588548607196_nat_o(X5)) ) ) ) ).

% cSup_insert
tff(fact_3446_cSup__insert,axiom,
    ! [X5: set_nat,A2: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,X5)
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),X5)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),aa_set_nat_nat(complete_Sup_Sup_nat,X5)) ) ) ) ).

% cSup_insert
tff(fact_3447_cSup__insert,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,X5)
       => ( comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),X5)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),comple4687483117567863418t_unit(X5)) ) ) ) ).

% cSup_insert
tff(fact_3448_cSup__union__distrib,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( condit5477540289124974626et_nat(A)
       => ( ( B4 != bot_bot_set_set_nat )
         => ( condit5477540289124974626et_nat(B4)
           => ( comple7399068483239264473et_nat(sup_sup_set_set_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,comple7399068483239264473et_nat(A)),comple7399068483239264473et_nat(B4)) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_3449_cSup__union__distrib,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_o_o(condit5488710616941104124bove_o,A)
       => ( ( B4 != bot_bot_set_o )
         => ( aa_set_o_o(condit5488710616941104124bove_o,B4)
           => ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
            <=> aa_o_o(aa_o_fun_o_o(sup_sup_o,aa_set_o_o(complete_Sup_Sup_o,A)),aa_set_o_o(complete_Sup_Sup_o,B4)) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_3450_cSup__union__distrib,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,A)
       => ( ( B4 != bot_bot_set_int )
         => ( aa_set_int_o(condit2212336002400062152ve_int,B4)
           => ( complete_Sup_Sup_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,complete_Sup_Sup_int(A)),complete_Sup_Sup_int(B4)) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_3451_cSup__union__distrib,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,A)
       => ( ( B4 != bot_bo8632255151191677115_nat_o )
         => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,B4)
           => ( comple7878897588548607196_nat_o(aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) = aa_fun2638059832795814268_nat_o(sup_su6210301778384944323_nat_o(comple7878897588548607196_nat_o(A)),comple7878897588548607196_nat_o(B4)) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_3452_cSup__union__distrib,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A)
       => ( ( B4 != bot_bot_set_nat )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,B4)
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,A)),aa_set_nat_nat(complete_Sup_Sup_nat,B4)) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_3453_cSup__union__distrib,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,A)
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,B4)
           => ( comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,comple4687483117567863418t_unit(A)),comple4687483117567863418t_unit(B4)) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_3454_fun__upd__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: code_integer,Y: code_integer,A: set_Code_integer] :
      aa_set8622560600277679972nteger(image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(fun_up8355007323143964473nteger(F,X),Y)),A) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A)) ).

% fun_upd_image
tff(fact_3455_fun__upd__image,axiom,
    ! [F: fun_Product_unit_nat,X: product_unit,Y: nat,A: set_Product_unit] :
      image_875570014554754200it_nat(fun_up6473050886840373792it_nat(F,X,Y),A) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),image_875570014554754200it_nat(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),image_875570014554754200it_nat(F,A)) ).

% fun_upd_image
tff(fact_3456_fun__upd__image,axiom,
    ! [F: fun_Product_unit_int,X: product_unit,Y: int,A: set_Product_unit] :
      image_873079544045703924it_int(fun_up6470560416331323516it_int(F,X,Y),A) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),image_873079544045703924it_int(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),image_873079544045703924it_int(F,A)) ).

% fun_upd_image
tff(fact_3457_fun__upd__image,axiom,
    ! [F: fun_Product_unit_o,X: product_unit,Y: $o,A: set_Product_unit] :
      image_Product_unit_o2(fun_up5432024876597553480unit_o(F,X,(Y)),A) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),image_Product_unit_o2(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),image_Product_unit_o2(F,A)) ).

% fun_upd_image
tff(fact_3458_fun__upd__image,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,X: product_unit,Y: product_unit,A: set_Product_unit] :
      image_405062704495631173t_unit(fun_up6875000483049295805t_unit(F,X,Y),A) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),image_405062704495631173t_unit(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),image_405062704495631173t_unit(F,A)) ).

% fun_upd_image
tff(fact_3459_fun__upd__image,axiom,
    ! [F: fun_o_nat,X: $o,Y: nat,A: set_o] :
      image_o_nat2(fun_upd_o_nat(F,(X),Y),A) = $ite(aa_set_o_o(member_o((X)),A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),image_o_nat2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),image_o_nat2(F,A)) ).

% fun_upd_image
tff(fact_3460_fun__upd__image,axiom,
    ! [F: fun_o_int,X: $o,Y: int,A: set_o] :
      image_o_int2(fun_upd_o_int(F,(X),Y),A) = $ite(aa_set_o_o(member_o((X)),A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),image_o_int2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),image_o_int2(F,A)) ).

% fun_upd_image
tff(fact_3461_fun__upd__image,axiom,
    ! [F: fun_o_o,X: $o,Y: $o,A: set_o] :
      image_o_o2(aa_o_fun_o_o(fun_upd_o_o(F,(X)),(Y)),A) = $ite(aa_set_o_o(member_o((X)),A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),image_o_o2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),image_o_o2(F,A)) ).

% fun_upd_image
tff(fact_3462_fun__upd__image,axiom,
    ! [F: fun_o_Product_unit,X: $o,Y: product_unit,A: set_o] :
      image_o_Product_unit2(fun_up846054379331416818t_unit(F,(X),Y),A) = $ite(aa_set_o_o(member_o((X)),A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),image_o_Product_unit2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),image_o_Product_unit2(F,A)) ).

% fun_upd_image
tff(fact_3463_fun__upd__image,axiom,
    ! [F: fun_int_nat,X: int,Y: nat,A: set_int] :
      image_int_nat2(fun_upd_int_nat(F,X,Y),A) = $ite(aa_set_int_o(member_int(X),A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),image_int_nat2(F,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))),image_int_nat2(F,A)) ).

% fun_upd_image
tff(fact_3464_Sup__fin_Oeq__fold,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( lattic1091506334969745077in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = finite_fold_int_int(sup_sup_int,X,A) ) ) ).

% Sup_fin.eq_fold
tff(fact_3465_Sup__fin_Oeq__fold,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(lattic1508158080041050831_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
      <=> aa_set_o_o(finite_fold_o_o(sup_sup_o,(X)),A) ) ) ).

% Sup_fin.eq_fold
tff(fact_3466_Sup__fin_Oeq__fold,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(lattic1093996805478795353in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_set_nat_nat(finite_fold_nat_nat(sup_sup_nat,X),A) ) ) ).

% Sup_fin.eq_fold
tff(fact_3467_Sup__fin_Oeq__fold,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( lattic3835124923745554447et_nat(insert_set_nat(X,A)) = finite677925301803182934et_nat(sup_sup_set_nat,X,A) ) ) ).

% Sup_fin.eq_fold
tff(fact_3468_Sup__fin_Oeq__fold,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( lattic2150320897289308081n_assn(insert_assn(X,A)) = finite1440822074540989594n_assn(sup_sup_assn,X,A) ) ) ).

% Sup_fin.eq_fold
tff(fact_3469_Sup__fin_Oeq__fold,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( lattic5294303975357428420t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = finite8834431295148555784t_unit(sup_sup_Product_unit,X,A) ) ) ).

% Sup_fin.eq_fold
tff(fact_3470_Inf__fin_Oeq__fold,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( lattic5235898064620869839in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = finite_fold_int_int(inf_inf_int,X,A) ) ) ).

% Inf_fin.eq_fold
tff(fact_3471_Inf__fin_Oeq__fold,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(lattic4107685809792843317_fin_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
      <=> aa_set_o_o(finite_fold_o_o(inf_inf_o,(X)),A) ) ) ).

% Inf_fin.eq_fold
tff(fact_3472_Inf__fin_Oeq__fold,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(lattic5238388535129920115in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_set_nat_nat(finite_fold_nat_nat(inf_inf_nat,X),A) ) ) ).

% Inf_fin.eq_fold
tff(fact_3473_Inf__fin_Oeq__fold,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( lattic3014633134055518761et_nat(insert_set_nat(X,A)) = finite677925301803182934et_nat(inf_inf_set_nat,X,A) ) ) ).

% Inf_fin.eq_fold
tff(fact_3474_Inf__fin_Oeq__fold,axiom,
    ! [A: set_assn,X: assn] :
      ( finite_finite_assn(A)
     => ( lattic47131356835913163n_assn(insert_assn(X,A)) = finite1440822074540989594n_assn(inf_inf_assn,X,A) ) ) ).

% Inf_fin.eq_fold
tff(fact_3475_Inf__fin_Oeq__fold,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( lattic1263872656861969706t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = finite8834431295148555784t_unit(inf_inf_Product_unit,X,A) ) ) ).

% Inf_fin.eq_fold
tff(fact_3476_wf__bounded__set,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Ub: fun_na9118744110877893344nteger,F: fun_na9118744110877893344nteger] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),A3),R2)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_nat8922193080109606305nteger(Ub,A3))
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_nat8922193080109606305nteger(Ub,B2)),aa_nat8922193080109606305nteger(Ub,A3))
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_nat8922193080109606305nteger(F,B2)),aa_nat8922193080109606305nteger(Ub,A3))
            & aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(ord_le1307284697595431911nteger,aa_nat8922193080109606305nteger(F,A3)),aa_nat8922193080109606305nteger(F,B2)) ) )
     => wf_nat(R2) ) ).

% wf_bounded_set
tff(fact_3477_wf__bounded__set,axiom,
    ! [R2: set_Pr958786334691620121nt_int,Ub: fun_in6808926178784942396nteger,F: fun_in6808926178784942396nteger] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),A3),R2)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_int6612375148016655357nteger(Ub,A3))
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_int6612375148016655357nteger(Ub,B2)),aa_int6612375148016655357nteger(Ub,A3))
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_int6612375148016655357nteger(F,B2)),aa_int6612375148016655357nteger(Ub,A3))
            & aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(ord_le1307284697595431911nteger,aa_int6612375148016655357nteger(F,A3)),aa_int6612375148016655357nteger(F,B2)) ) )
     => wf_int(R2) ) ).

% wf_bounded_set
tff(fact_3478_wf__bounded__set,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Ub: fun_nat_set_nat,F: fun_nat_set_nat] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),A3),R2)
         => ( aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(Ub,A3))
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(Ub,B2)),aa_nat_set_nat(Ub,A3))
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,B2)),aa_nat_set_nat(Ub,A3))
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(F,A3)),aa_nat_set_nat(F,B2)) ) )
     => wf_nat(R2) ) ).

% wf_bounded_set
tff(fact_3479_wf__bounded__set,axiom,
    ! [R2: set_Pr958786334691620121nt_int,Ub: fun_int_set_nat,F: fun_int_set_nat] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),A3),R2)
         => ( aa_set_nat_o(finite_finite_nat,aa_int_set_nat(Ub,A3))
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_int_set_nat(Ub,B2)),aa_int_set_nat(Ub,A3))
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_int_set_nat(F,B2)),aa_int_set_nat(Ub,A3))
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_int_set_nat(F,A3)),aa_int_set_nat(F,B2)) ) )
     => wf_int(R2) ) ).

% wf_bounded_set
tff(fact_3480_psubset__insert__iff,axiom,
    ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),B4))
    <=> $ite(
          aa_set8780210009671629607unit_o(member_Product_unit(X),B4),
          aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A),B4),
          $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))),B4),aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3481_psubset__insert__iff,axiom,
    ! [A: set_o,X: $o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),B4))
    <=> $ite(
          aa_set_o_o(member_o((X)),B4),
          aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),B4),
          $ite(aa_set_o_o(member_o((X)),A),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))),B4),aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3482_psubset__insert__iff,axiom,
    ! [A: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(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,A),B4),
          $ite(aa_set_int_o(member_int(X),A),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))),B4),aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3483_psubset__insert__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),B4))
    <=> $ite(
          aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4),
          aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A),B4),
          $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A),aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),B4),aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3484_psubset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(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,A),B4),
          $ite(aa_set_nat_o(member_nat(X),A),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))),B4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3485_finite__induct__select,axiom,
    ! [S: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [T3: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(ord_le1307284697595431911nteger,T3),S)
             => ( aa_set5643387332196575111eger_o(P,T3)
               => ? [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),minus_2355218937544613996nteger(S,T3))
                    & aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X4),T3)) ) ) )
         => aa_set5643387332196575111eger_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_3486_finite__induct__select,axiom,
    ! [S: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [T3: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,T3),S)
             => ( aa_set8780210009671629607unit_o(P,T3)
               => ? [X4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(X4),minus_6452836326544984404t_unit(S,T3))
                    & aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X4),T3)) ) ) )
         => aa_set8780210009671629607unit_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_3487_finite__induct__select,axiom,
    ! [S: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [T3: set_o] :
              ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,T3),S)
             => ( aa_set_o_o(P,T3)
               => ? [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),minus_minus_set_o(S,T3))
                    & aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X4)),T3)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_3488_finite__induct__select,axiom,
    ! [S: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T3: set_int] :
              ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,T3),S)
             => ( aa_set_int_o(P,T3)
               => ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),minus_minus_set_int(S,T3))
                    & aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X4),T3)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_3489_finite__induct__select,axiom,
    ! [S: set_fu3555266933746557007_nat_o,P: fun_se7459991829846211660at_o_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( aa_set5091322549519198405at_o_o(P,bot_bo8632255151191677115_nat_o)
       => ( ! [T3: set_fu3555266933746557007_nat_o] :
              ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(T3),S)
             => ( aa_set5091322549519198405at_o_o(P,T3)
               => ? [X4: fun_Pr6460944699146269935_nat_o] :
                    ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),minus_6456595905716665654_nat_o(S,T3))
                    & aa_set5091322549519198405at_o_o(P,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X4),T3)) ) ) )
         => aa_set5091322549519198405at_o_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_3490_finite__induct__select,axiom,
    ! [S: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T3),S)
             => ( aa_set_nat_o(P,T3)
               => ? [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),aa_set_nat_set_nat(minus_minus_set_nat(S),T3))
                    & aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X4),T3)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_3491_Max_Osubset__imp,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B4)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A)),lattic4901227151466704046nteger(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_3492_Max_Osubset__imp,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1210392890313158908t_unit(A)),lattic1210392890313158908t_unit(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_3493_Max_Osubset__imp,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A)),aa_set_o_o(lattic1921953407002678535_Max_o,B4)) ) ) ) ).

% Max.subset_imp
tff(fact_3494_Max_Osubset__imp,axiom,
    ! [A: set_rat,B4: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A),B4)
     => ( ( A != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A)),lattic7630753665789217321ax_rat(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_3495_Max_Osubset__imp,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A)),lattic8263393255366662781ax_int(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_3496_Max_Osubset__imp,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),aa_set_nat_nat(lattic8265883725875713057ax_nat,B4)) ) ) ) ).

% Max.subset_imp
tff(fact_3497_Max__mono,axiom,
    ! [M: set_Code_integer,N: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(M),N)
     => ( ( M != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(M)),lattic4901227151466704046nteger(N)) ) ) ) ).

% Max_mono
tff(fact_3498_Max__mono,axiom,
    ! [M: set_Product_unit,N: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,M),N)
     => ( ( M != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,N)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1210392890313158908t_unit(M)),lattic1210392890313158908t_unit(N)) ) ) ) ).

% Max_mono
tff(fact_3499_Max__mono,axiom,
    ! [M: set_o,N: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,M),N)
     => ( ( M != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,N)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,M)),aa_set_o_o(lattic1921953407002678535_Max_o,N)) ) ) ) ).

% Max_mono
tff(fact_3500_Max__mono,axiom,
    ! [M: set_rat,N: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(M),N)
     => ( ( M != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(M)),lattic7630753665789217321ax_rat(N)) ) ) ) ).

% Max_mono
tff(fact_3501_Max__mono,axiom,
    ! [M: set_int,N: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,M),N)
     => ( ( M != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(M)),lattic8263393255366662781ax_int(N)) ) ) ) ).

% Max_mono
tff(fact_3502_Max__mono,axiom,
    ! [M: set_nat,N: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M),N)
     => ( ( M != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,M)),aa_set_nat_nat(lattic8265883725875713057ax_nat,N)) ) ) ) ).

% Max_mono
tff(fact_3503_image__vimage__eq,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),vimage1318793397690208119nteger(F,A)) = inf_in1364745209274528805nteger(A,aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger)) ).

% image_vimage_eq
tff(fact_3504_image__vimage__eq,axiom,
    ! [F: fun_Product_unit_nat,A: set_nat] : image_875570014554754200it_nat(F,vimage6253328473476588386it_nat(F,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),image_875570014554754200it_nat(F,top_to1996260823553986621t_unit)) ).

% image_vimage_eq
tff(fact_3505_image__vimage__eq,axiom,
    ! [F: fun_o_nat,A: set_nat] : image_o_nat2(F,vimage_o_nat(F,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),image_o_nat2(F,top_top_set_o)) ).

% image_vimage_eq
tff(fact_3506_image__vimage__eq,axiom,
    ! [F: fun_nat_set_nat,A: set_set_nat] : aa_set7246812951530217472et_nat(image_nat_set_nat(F),vimage_nat_set_nat(F,A)) = inf_inf_set_set_nat(A,aa_set7246812951530217472et_nat(image_nat_set_nat(F),top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3507_image__vimage__eq,axiom,
    ! [F: fun_nat_Code_integer,A: set_Code_integer] : aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),vimage6104394758324456170nteger(F,A)) = inf_in1364745209274528805nteger(A,aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3508_image__vimage__eq,axiom,
    ! [F: fun_nat_nat,A: set_nat] : aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(vimage_nat_nat(F),A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(image_nat_nat2(F),top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3509_image__vimage__eq,axiom,
    ! [F: fun_int_nat,A: set_nat] : image_int_nat2(F,vimage_int_nat(F,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),image_int_nat2(F,top_top_set_int)) ).

% image_vimage_eq
tff(fact_3510_UNIV__I,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),top_to9044565752620676511_nat_o) ).

% UNIV_I
tff(fact_3511_UNIV__I,axiom,
    ! [X: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(X),top_to1996260823553986621t_unit) ).

% UNIV_I
tff(fact_3512_UNIV__I,axiom,
    ! [X: $o] : aa_set_o_o(member_o((X)),top_top_set_o) ).

% UNIV_I
tff(fact_3513_UNIV__I,axiom,
    ! [X: nat] : aa_set_nat_o(member_nat(X),top_top_set_nat) ).

% UNIV_I
tff(fact_3514_UNIV__I,axiom,
    ! [X: int] : aa_set_int_o(member_int(X),top_top_set_int) ).

% UNIV_I
tff(fact_3515_Int__UNIV,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(A,B4) = top_to1996260823553986621t_unit )
    <=> ( ( A = top_to1996260823553986621t_unit )
        & ( B4 = top_to1996260823553986621t_unit ) ) ) ).

% Int_UNIV
tff(fact_3516_Int__UNIV,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A,B4) = top_top_set_o )
    <=> ( ( A = top_top_set_o )
        & ( B4 = top_top_set_o ) ) ) ).

% Int_UNIV
tff(fact_3517_Int__UNIV,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = top_top_set_nat )
    <=> ( ( A = top_top_set_nat )
        & ( B4 = top_top_set_nat ) ) ) ).

% Int_UNIV
tff(fact_3518_Int__UNIV,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A,B4) = top_top_set_int )
    <=> ( ( A = top_top_set_int )
        & ( B4 = top_top_set_int ) ) ) ).

% Int_UNIV
tff(fact_3519_psubsetI,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( ( A != B4 )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4) ) ) ).

% psubsetI
tff(fact_3520_vimage__UNIV,axiom,
    ! [F: fun_Pr2191293291710260995t_unit] : vimage7995052115951654139t_unit(F,top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).

% vimage_UNIV
tff(fact_3521_vimage__UNIV,axiom,
    ! [F: fun_o_Product_unit] : vimage4222377557387472176t_unit(F,top_to1996260823553986621t_unit) = top_top_set_o ).

% vimage_UNIV
tff(fact_3522_vimage__UNIV,axiom,
    ! [F: fun_nat_Product_unit] : vimage4884490618288580032t_unit(F,top_to1996260823553986621t_unit) = top_top_set_nat ).

% vimage_UNIV
tff(fact_3523_vimage__UNIV,axiom,
    ! [F: fun_int_Product_unit] : vimage2000912565490760292t_unit(F,top_to1996260823553986621t_unit) = top_top_set_int ).

% vimage_UNIV
tff(fact_3524_vimage__UNIV,axiom,
    ! [F: fun_Product_unit_o] : vimage8808348054653608838unit_o(F,top_top_set_o) = top_to1996260823553986621t_unit ).

% vimage_UNIV
tff(fact_3525_vimage__UNIV,axiom,
    ! [F: fun_o_o] : vimage_o_o(F,top_top_set_o) = top_top_set_o ).

% vimage_UNIV
tff(fact_3526_vimage__UNIV,axiom,
    ! [F: fun_nat_o] : vimage_nat_o(F,top_top_set_o) = top_top_set_nat ).

% vimage_UNIV
tff(fact_3527_vimage__UNIV,axiom,
    ! [F: fun_int_o] : vimage_int_o(F,top_top_set_o) = top_top_set_int ).

% vimage_UNIV
tff(fact_3528_vimage__UNIV,axiom,
    ! [F: fun_Product_unit_nat] : vimage6253328473476588386it_nat(F,top_top_set_nat) = top_to1996260823553986621t_unit ).

% vimage_UNIV
tff(fact_3529_vimage__UNIV,axiom,
    ! [F: fun_o_nat] : vimage_o_nat(F,top_top_set_nat) = top_top_set_o ).

% vimage_UNIV
tff(fact_3530_Pow__UNIV,axiom,
    pow_Product_unit(top_to1996260823553986621t_unit) = top_to1767297665138865437t_unit ).

% Pow_UNIV
tff(fact_3531_Pow__UNIV,axiom,
    pow_o(top_top_set_o) = top_top_set_set_o ).

% Pow_UNIV
tff(fact_3532_Pow__UNIV,axiom,
    pow_nat(top_top_set_nat) = top_top_set_set_nat ).

% Pow_UNIV
tff(fact_3533_Pow__UNIV,axiom,
    pow_int(top_top_set_int) = top_top_set_set_int ).

% Pow_UNIV
tff(fact_3534_Diff__UNIV,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : minus_6456595905716665654_nat_o(A,top_to9044565752620676511_nat_o) = bot_bo8632255151191677115_nat_o ).

% Diff_UNIV
tff(fact_3535_Diff__UNIV,axiom,
    ! [A: set_Product_unit] : minus_6452836326544984404t_unit(A,top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).

% Diff_UNIV
tff(fact_3536_Diff__UNIV,axiom,
    ! [A: set_o] : minus_minus_set_o(A,top_top_set_o) = bot_bot_set_o ).

% Diff_UNIV
tff(fact_3537_Diff__UNIV,axiom,
    ! [A: set_int] : minus_minus_set_int(A,top_top_set_int) = bot_bot_set_int ).

% Diff_UNIV
tff(fact_3538_Diff__UNIV,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),top_top_set_nat) = bot_bot_set_nat ).

% Diff_UNIV
tff(fact_3539_Max__singleton,axiom,
    ! [X: product_unit] : lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% Max_singleton
tff(fact_3540_Max__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% Max_singleton
tff(fact_3541_Max__singleton,axiom,
    ! [X: int] : lattic8263393255366662781ax_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% Max_singleton
tff(fact_3542_Max__singleton,axiom,
    ! [X: nat] : aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% Max_singleton
tff(fact_3543_range__snd,axiom,
    image_6088997261112373647et_nat(produc8586169260539613262et_nat,top_to2524957550908328808et_nat) = top_top_set_set_nat ).

% range_snd
tff(fact_3544_range__snd,axiom,
    image_5042161079198086560nt_int(product_snd_int_int,top_to4366644338036079209nt_int) = top_top_set_int ).

% range_snd
tff(fact_3545_Domain__Id,axiom,
    domain4292024440186670924t_unit(id_Product_unit) = top_to1996260823553986621t_unit ).

% Domain_Id
tff(fact_3546_Domain__Id,axiom,
    domain_o_o(id_o2) = top_top_set_o ).

% Domain_Id
tff(fact_3547_Domain__Id,axiom,
    domain_nat_nat(id_nat) = top_top_set_nat ).

% Domain_Id
tff(fact_3548_Domain__Id,axiom,
    domain_int_int(id_int) = top_top_set_int ).

% Domain_Id
tff(fact_3549_Range__Id,axiom,
    range_6617724647562260917t_unit(id_Product_unit) = top_to1996260823553986621t_unit ).

% Range_Id
tff(fact_3550_Range__Id,axiom,
    range_o_o(id_o2) = top_top_set_o ).

% Range_Id
tff(fact_3551_Range__Id,axiom,
    range_nat_nat(id_nat) = top_top_set_nat ).

% Range_Id
tff(fact_3552_Range__Id,axiom,
    range_int_int(id_int) = top_top_set_int ).

% Range_Id
tff(fact_3553_Max_Obounded__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A)),X)
        <=> ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3554_Max_Obounded__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1210392890313158908t_unit(A)),X)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3555_Max_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A)),(X))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(X)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3556_Max_Obounded__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A)),X)
        <=> ! [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3557_Max_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A)),X)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3558_Max_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),X)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),X) ) ) ) ) ).

% Max.bounded_iff
tff(fact_3559_Max__less__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,lattic1210392890313158908t_unit(A)),X)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X2),X) ) ) ) ) ).

% Max_less_iff
tff(fact_3560_Max__less__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_set_o_o(lattic1921953407002678535_Max_o,A)),(X))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_o,(X2)),(X)) ) ) ) ) ).

% Max_less_iff
tff(fact_3561_Max__less__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8263393255366662781ax_int(A)),X)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,X2),X) ) ) ) ) ).

% Max_less_iff
tff(fact_3562_Max__less__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,lattic4901227151466704046nteger(A)),X)
        <=> ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),X) ) ) ) ) ).

% Max_less_iff
tff(fact_3563_Max__less__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,lattic7630753665789217321ax_rat(A)),X)
        <=> ! [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),X) ) ) ) ) ).

% Max_less_iff
tff(fact_3564_Max__less__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),X)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),X) ) ) ) ) ).

% Max_less_iff
tff(fact_3565_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ).

% lt_ex
tff(fact_3566_lt__ex,axiom,
    ! [X: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ).

% lt_ex
tff(fact_3567_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_3568_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_3569_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_3570_dense,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ? [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y) ) ) ).

% dense
tff(fact_3571_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_3572_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_3573_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_3574_less__imp__neq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_3575_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_3576_order_Oasym,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2) ) ).

% order.asym
tff(fact_3577_order_Oasym,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2) ) ).

% order.asym
tff(fact_3578_order_Oasym,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ).

% order.asym
tff(fact_3579_order_Oasym,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ).

% order.asym
tff(fact_3580_order_Oasym,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ).

% order.asym
tff(fact_3581_ord__eq__less__trans,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( ( A2 = B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),C) ) ) ).

% ord_eq_less_trans
tff(fact_3582_ord__eq__less__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C) ) ) ).

% ord_eq_less_trans
tff(fact_3583_ord__eq__less__trans,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( A2 = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C) ) ) ).

% ord_eq_less_trans
tff(fact_3584_ord__eq__less__trans,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( A2 = B )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),C) ) ) ).

% ord_eq_less_trans
tff(fact_3585_ord__eq__less__trans,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( ( A2 = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C) ) ) ).

% ord_eq_less_trans
tff(fact_3586_ord__less__eq__trans,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( ( B = C )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),C) ) ) ).

% ord_less_eq_trans
tff(fact_3587_ord__less__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C) ) ) ).

% ord_less_eq_trans
tff(fact_3588_ord__less__eq__trans,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C) ) ) ).

% ord_less_eq_trans
tff(fact_3589_ord__less__eq__trans,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( ( B = C )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),C) ) ) ).

% ord_less_eq_trans
tff(fact_3590_ord__less__eq__trans,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C) ) ) ).

% ord_less_eq_trans
tff(fact_3591_less__induct,axiom,
    ! [P: fun_nat_o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X3)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X3) )
     => aa_nat_o(P,A2) ) ).

% less_induct
tff(fact_3592_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_3593_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_3594_antisym__conv3,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_3595_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_3596_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_3597_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_3598_linorder__cases,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( ( X != Y )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ) ).

% linorder_cases
tff(fact_3599_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_3600_dual__order_Oasym,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B) ) ).

% dual_order.asym
tff(fact_3601_dual__order_Oasym,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ).

% dual_order.asym
tff(fact_3602_dual__order_Oasym,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% dual_order.asym
tff(fact_3603_dual__order_Oasym,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ).

% dual_order.asym
tff(fact_3604_dual__order_Oasym,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% dual_order.asym
tff(fact_3605_dual__order_Oirrefl,axiom,
    ! [A2: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),A2) ).

% dual_order.irrefl
tff(fact_3606_dual__order_Oirrefl,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A2) ).

% dual_order.irrefl
tff(fact_3607_dual__order_Oirrefl,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),A2) ).

% dual_order.irrefl
tff(fact_3608_dual__order_Oirrefl,axiom,
    ! [A2: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),A2) ).

% dual_order.irrefl
tff(fact_3609_dual__order_Oirrefl,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),A2) ).

% dual_order.irrefl
tff(fact_3610_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N4)
             => ~ aa_nat_o(P,M3) ) ) ) ).

% exists_least_iff
tff(fact_3611_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A2: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
       => ( ! [A3: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3612_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A2: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B2)
         => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
       => ( ! [A3: int,B2: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B2),A3)
             => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
         => aa_int_o(aa_int_fun_int_o(P,A2),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3613_linorder__less__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,A2: code_integer,B: code_integer] :
      ( ! [A3: code_integer,B2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A3),B2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B2) )
     => ( ! [A3: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),A3)
       => ( ! [A3: code_integer,B2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,B2),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B2) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A2),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3614_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A2: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A3),A3)
       => ( ! [A3: rat,B2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A2),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3615_order_Ostrict__trans,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),C) ) ) ).

% order.strict_trans
tff(fact_3616_order_Ostrict__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C) ) ) ).

% order.strict_trans
tff(fact_3617_order_Ostrict__trans,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C) ) ) ).

% order.strict_trans
tff(fact_3618_order_Ostrict__trans,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),C) ) ) ).

% order.strict_trans
tff(fact_3619_order_Ostrict__trans,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C) ) ) ).

% order.strict_trans
tff(fact_3620_top_Oextremum__strict,axiom,
    ! [A2: set_Product_unit] : ~ aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,top_to1996260823553986621t_unit),A2) ).

% top.extremum_strict
tff(fact_3621_top_Oextremum__strict,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,top_top_set_o),A2) ).

% top.extremum_strict
tff(fact_3622_top_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,top_top_set_nat),A2) ).

% top.extremum_strict
tff(fact_3623_top_Oextremum__strict,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,top_top_set_int),A2) ).

% top.extremum_strict
tff(fact_3624_top_Oextremum__strict,axiom,
    ! [A2: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,top_top_assn),A2) ).

% top.extremum_strict
tff(fact_3625_top_Onot__eq__extremum,axiom,
    ! [A2: set_Product_unit] :
      ( ( A2 != top_to1996260823553986621t_unit )
    <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A2),top_to1996260823553986621t_unit) ) ).

% top.not_eq_extremum
tff(fact_3626_top_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != top_top_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A2),top_top_set_o) ) ).

% top.not_eq_extremum
tff(fact_3627_top_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != top_top_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),top_top_set_nat) ) ).

% top.not_eq_extremum
tff(fact_3628_top_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != top_top_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A2),top_top_set_int) ) ).

% top.not_eq_extremum
tff(fact_3629_top_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != top_top_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),top_top_assn) ) ).

% top.not_eq_extremum
tff(fact_3630_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_3631_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_3632_not__less__iff__gr__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X)
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_3633_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_3634_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A2: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A2) ) ) ).

% dual_order.strict_trans
tff(fact_3635_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A2) ) ) ).

% dual_order.strict_trans
tff(fact_3636_dual__order_Ostrict__trans,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A2) ) ) ).

% dual_order.strict_trans
tff(fact_3637_dual__order_Ostrict__trans,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A2) ) ) ).

% dual_order.strict_trans
tff(fact_3638_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A2) ) ) ).

% dual_order.strict_trans
tff(fact_3639_order_Ostrict__implies__not__eq,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3640_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3641_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3642_order_Ostrict__implies__not__eq,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3643_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3644_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3645_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3646_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3647_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3648_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3649_psubsetD,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o,C: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A),B4)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(C),B4) ) ) ).

% psubsetD
tff(fact_3650_psubsetD,axiom,
    ! [A: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
     => ( aa_set_nat_o(member_nat(C),A)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% psubsetD
tff(fact_3651_psubsetD,axiom,
    ! [A: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B4)
     => ( aa_set_int_o(member_int(C),A)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% psubsetD
tff(fact_3652_psubsetD,axiom,
    ! [A: set_o,B4: set_o,C: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),B4)
     => ( aa_set_o_o(member_o((C)),A)
       => aa_set_o_o(member_o((C)),B4) ) ) ).

% psubsetD
tff(fact_3653_psubsetD,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A),B4)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(C),A)
       => aa_set8780210009671629607unit_o(member_Product_unit(C),B4) ) ) ).

% psubsetD
tff(fact_3654_UNIV__eq__I,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ! [X3: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
     => ( top_to9044565752620676511_nat_o = A ) ) ).

% UNIV_eq_I
tff(fact_3655_UNIV__eq__I,axiom,
    ! [A: set_Product_unit] :
      ( ! [X3: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
     => ( top_to1996260823553986621t_unit = A ) ) ).

% UNIV_eq_I
tff(fact_3656_UNIV__eq__I,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] : aa_set_o_o(member_o((X3)),A)
     => ( top_top_set_o = A ) ) ).

% UNIV_eq_I
tff(fact_3657_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : aa_set_nat_o(member_nat(X3),A)
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
tff(fact_3658_UNIV__eq__I,axiom,
    ! [A: set_int] :
      ( ! [X3: int] : aa_set_int_o(member_int(X3),A)
     => ( top_top_set_int = A ) ) ).

% UNIV_eq_I
tff(fact_3659_UNIV__witness,axiom,
    ? [X3: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),top_to9044565752620676511_nat_o) ).

% UNIV_witness
tff(fact_3660_UNIV__witness,axiom,
    ? [X3: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(X3),top_to1996260823553986621t_unit) ).

% UNIV_witness
tff(fact_3661_UNIV__witness,axiom,
    ? [X3: $o] : aa_set_o_o(member_o((X3)),top_top_set_o) ).

% UNIV_witness
tff(fact_3662_UNIV__witness,axiom,
    ? [X3: nat] : aa_set_nat_o(member_nat(X3),top_top_set_nat) ).

% UNIV_witness
tff(fact_3663_UNIV__witness,axiom,
    ? [X3: int] : aa_set_int_o(member_int(X3),top_top_set_int) ).

% UNIV_witness
tff(fact_3664_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_3665_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_3666_linorder__neqE,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ) ).

% linorder_neqE
tff(fact_3667_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_3668_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_3669_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_3670_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_3671_order__less__asym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).

% order_less_asym
tff(fact_3672_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_3673_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_3674_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_3675_linorder__neq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ) ).

% linorder_neq_iff
tff(fact_3676_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_3677_order__less__asym_H,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2) ) ).

% order_less_asym'
tff(fact_3678_order__less__asym_H,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2) ) ).

% order_less_asym'
tff(fact_3679_order__less__asym_H,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ).

% order_less_asym'
tff(fact_3680_order__less__asym_H,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ).

% order_less_asym'
tff(fact_3681_order__less__asym_H,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ).

% order_less_asym'
tff(fact_3682_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: 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),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_less_trans
tff(fact_3683_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: 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),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_less_trans
tff(fact_3684_order__less__trans,axiom,
    ! [X: int,Y: int,Z: 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),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_less_trans
tff(fact_3685_order__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z) ) ) ).

% order_less_trans
tff(fact_3686_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: 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),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z) ) ) ).

% order_less_trans
tff(fact_3687_ord__eq__less__subst,axiom,
    ! [A2: assn,F: fun_assn_assn,B: assn,C: assn] :
      ( ( A2 = aa_assn_assn(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_assn_assn(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3688_ord__eq__less__subst,axiom,
    ! [A2: nat,F: fun_assn_nat,B: assn,C: assn] :
      ( ( A2 = aa_assn_nat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_assn_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3689_ord__eq__less__subst,axiom,
    ! [A2: int,F: fun_assn_int,B: assn,C: assn] :
      ( ( A2 = aa_assn_int(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X3)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_assn_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3690_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
      ( ( A2 = aa_assn_Code_integer(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X3)),aa_assn_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3691_ord__eq__less__subst,axiom,
    ! [A2: rat,F: fun_assn_rat,B: assn,C: assn] :
      ( ( A2 = aa_assn_rat(F,B) )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_assn_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3692_ord__eq__less__subst,axiom,
    ! [A2: assn,F: fun_nat_assn,B: nat,C: nat] :
      ( ( A2 = aa_nat_assn(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_nat_assn(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3693_ord__eq__less__subst,axiom,
    ! [A2: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( ( A2 = aa_nat_nat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3694_ord__eq__less__subst,axiom,
    ! [A2: int,F: fun_nat_int,B: nat,C: nat] :
      ( ( A2 = aa_nat_int(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3695_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
      ( ( A2 = aa_nat_Code_integer(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3696_ord__eq__less__subst,axiom,
    ! [A2: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A2 = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3697_ord__less__eq__subst,axiom,
    ! [A2: assn,B: assn,F: fun_assn_assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( ( aa_assn_assn(F,B) = C )
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3698_ord__less__eq__subst,axiom,
    ! [A2: assn,B: assn,F: fun_assn_nat,C: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( ( aa_assn_nat(F,B) = C )
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3699_ord__less__eq__subst,axiom,
    ! [A2: assn,B: assn,F: fun_assn_int,C: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( ( aa_assn_int(F,B) = C )
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X3)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3700_ord__less__eq__subst,axiom,
    ! [A2: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( ( aa_assn_Code_integer(F,B) = C )
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X3)),aa_assn_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3701_ord__less__eq__subst,axiom,
    ! [A2: assn,B: assn,F: fun_assn_rat,C: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( ( aa_assn_rat(F,B) = C )
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3702_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_assn,C: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( ( aa_nat_assn(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3703_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( ( aa_nat_nat(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3704_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( ( aa_nat_int(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3705_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( ( aa_nat_Code_integer(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3706_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A2)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_3707_order__less__irrefl,axiom,
    ! [X: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),X) ).

% order_less_irrefl
tff(fact_3708_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_3709_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_3710_order__less__irrefl,axiom,
    ! [X: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),X) ).

% order_less_irrefl
tff(fact_3711_order__less__irrefl,axiom,
    ! [X: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X) ).

% order_less_irrefl
tff(fact_3712_order__less__subst1,axiom,
    ! [A2: assn,F: fun_assn_assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_assn_assn(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3713_order__less__subst1,axiom,
    ! [A2: assn,F: fun_nat_assn,B: nat,C: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_nat_assn(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3714_order__less__subst1,axiom,
    ! [A2: assn,F: fun_int_assn,B: int,C: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X3)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_int_assn(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3715_order__less__subst1,axiom,
    ! [A2: assn,F: fun_Co4490034186337852034r_assn,B: code_integer,C: code_integer] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_Code_integer_assn(F,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,X3)),aa_Code_integer_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_Code_integer_assn(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3716_order__less__subst1,axiom,
    ! [A2: assn,F: fun_rat_assn,B: rat,C: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_rat_assn(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_rat_assn(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3717_order__less__subst1,axiom,
    ! [A2: nat,F: fun_assn_nat,B: assn,C: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_assn_nat(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_assn_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3718_order__less__subst1,axiom,
    ! [A2: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3719_order__less__subst1,axiom,
    ! [A2: nat,F: fun_int_nat,B: int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_int_nat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_int_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3720_order__less__subst1,axiom,
    ! [A2: nat,F: fun_Code_integer_nat,B: code_integer,C: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_Code_integer_nat(F,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X3)),aa_Code_integer_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_Code_integer_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3721_order__less__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_3722_order__less__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_assn_assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3723_order__less__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_assn_nat,C: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,X3)),aa_assn_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_assn_nat(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3724_order__less__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_assn_int,C: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,X3)),aa_assn_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_assn_int(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3725_order__less__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X3)),aa_assn_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3726_order__less__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_assn_rat,C: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,X3)),aa_assn_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_assn_rat(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3727_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_assn,C: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3728_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3729_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3730_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3731_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A2)),C) ) ) ) ).

% order_less_subst2
tff(fact_3732_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_3733_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_3734_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_3735_order__less__not__sym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).

% order_less_not_sym
tff(fact_3736_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_3737_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,P: $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)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3738_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $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)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3739_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $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)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3740_order__less__imp__triv,axiom,
    ! [X: code_integer,Y: code_integer,P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3741_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $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)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3742_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_3743_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_3744_linorder__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
      | ( X = Y )
      | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).

% linorder_less_linear
tff(fact_3745_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_3746_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_3747_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_3748_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_3749_order__less__imp__not__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_3750_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_3751_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_3752_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_3753_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_3754_order__less__imp__not__eq2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_3755_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_3756_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_3757_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_3758_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_3759_order__less__imp__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).

% order_less_imp_not_less
tff(fact_3760_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_3761_Max__gr__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X),lattic1210392890313158908t_unit(A))
        <=> ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X),X2) ) ) ) ) ).

% Max_gr_iff
tff(fact_3762_Max__gr__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A))
        <=> ? [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
              & aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(X2)) ) ) ) ) ).

% Max_gr_iff
tff(fact_3763_Max__gr__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),lattic8263393255366662781ax_int(A))
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X),X2) ) ) ) ) ).

% Max_gr_iff
tff(fact_3764_Max__gr__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),lattic4901227151466704046nteger(A))
        <=> ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),X2) ) ) ) ) ).

% Max_gr_iff
tff(fact_3765_Max__gr__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),lattic7630753665789217321ax_rat(A))
        <=> ? [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X2) ) ) ) ) ).

% Max_gr_iff
tff(fact_3766_Max__gr__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_set_nat_nat(lattic8265883725875713057ax_nat,A))
        <=> ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X2) ) ) ) ) ).

% Max_gr_iff
tff(fact_3767_top__greatest,axiom,
    ! [A2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),top_top_assn) ).

% top_greatest
tff(fact_3768_top__greatest,axiom,
    ! [A2: set_Product_unit] : aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A2),top_to1996260823553986621t_unit) ).

% top_greatest
tff(fact_3769_top__greatest,axiom,
    ! [A2: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A2),top_top_set_o) ).

% top_greatest
tff(fact_3770_top__greatest,axiom,
    ! [A2: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A2),top_top_set_int) ).

% top_greatest
tff(fact_3771_top__greatest,axiom,
    ! [A2: filter_nat] : aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),top_top_filter_nat) ).

% top_greatest
tff(fact_3772_top__greatest,axiom,
    ! [A2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),top_top_set_nat) ).

% top_greatest
tff(fact_3773_top_Oextremum__unique,axiom,
    ! [A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,top_top_assn),A2)
    <=> ( A2 = top_top_assn ) ) ).

% top.extremum_unique
tff(fact_3774_top_Oextremum__unique,axiom,
    ! [A2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,top_to1996260823553986621t_unit),A2)
    <=> ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_unique
tff(fact_3775_top_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,top_top_set_o),A2)
    <=> ( A2 = top_top_set_o ) ) ).

% top.extremum_unique
tff(fact_3776_top_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,top_top_set_int),A2)
    <=> ( A2 = top_top_set_int ) ) ).

% top.extremum_unique
tff(fact_3777_top_Oextremum__unique,axiom,
    ! [A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,top_top_filter_nat),A2)
    <=> ( A2 = top_top_filter_nat ) ) ).

% top.extremum_unique
tff(fact_3778_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),A2)
    <=> ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
tff(fact_3779_top_Oextremum__uniqueI,axiom,
    ! [A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,top_top_assn),A2)
     => ( A2 = top_top_assn ) ) ).

% top.extremum_uniqueI
tff(fact_3780_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,top_to1996260823553986621t_unit),A2)
     => ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_uniqueI
tff(fact_3781_top_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,top_top_set_o),A2)
     => ( A2 = top_top_set_o ) ) ).

% top.extremum_uniqueI
tff(fact_3782_top_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,top_top_set_int),A2)
     => ( A2 = top_top_set_int ) ) ).

% top.extremum_uniqueI
tff(fact_3783_top_Oextremum__uniqueI,axiom,
    ! [A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,top_top_filter_nat),A2)
     => ( A2 = top_top_filter_nat ) ) ).

% top.extremum_uniqueI
tff(fact_3784_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,top_top_set_nat),A2)
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
tff(fact_3785_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_3786_leD,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
     => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ).

% leD
tff(fact_3787_leD,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y) ) ).

% leD
tff(fact_3788_leD,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_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y) ) ).

% leD
tff(fact_3789_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_3790_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_3791_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_3792_leI,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ).

% leI
tff(fact_3793_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_3794_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_3795_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_3796_nless__le,axiom,
    ! [A2: assn,B: assn] :
      ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
    <=> ( ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3797_nless__le,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3798_nless__le,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B)
    <=> ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3799_nless__le,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
    <=> ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3800_nless__le,axiom,
    ! [A2: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3801_nless__le,axiom,
    ! [A2: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3802_nless__le,axiom,
    ! [A2: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
        | ( A2 = B ) ) ) ).

% nless_le
tff(fact_3803_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_3804_antisym__conv1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_3805_antisym__conv1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_3806_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_3807_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_3808_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_3809_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_3810_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_3811_antisym__conv2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_3812_antisym__conv2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_3813_antisym__conv2,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_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_3814_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_3815_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_3816_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_3817_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),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),Z) ) ).

% dense_ge
tff(fact_3818_dense__le,axiom,
    ! [Y: rat,Z: 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),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_3819_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_3820_less__le__not__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ) ).

% less_le_not_le
tff(fact_3821_less__le__not__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X) ) ) ).

% less_le_not_le
tff(fact_3822_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
    <=> ( 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,Y),X) ) ) ).

% less_le_not_le
tff(fact_3823_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_3824_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_3825_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_3826_not__le__imp__less,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ).

% not_le_imp_less
tff(fact_3827_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_3828_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_3829_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_3830_order_Oorder__iff__strict,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3831_order_Oorder__iff__strict,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3832_order_Oorder__iff__strict,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3833_order_Oorder__iff__strict,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3834_order_Oorder__iff__strict,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3835_order_Oorder__iff__strict,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3836_order_Oorder__iff__strict,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
        | ( A2 = B ) ) ) ).

% order.order_iff_strict
tff(fact_3837_order_Ostrict__iff__order,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3838_order_Ostrict__iff__order,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3839_order_Ostrict__iff__order,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3840_order_Ostrict__iff__order,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3841_order_Ostrict__iff__order,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3842_order_Ostrict__iff__order,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3843_order_Ostrict__iff__order,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
        & ( A2 != B ) ) ) ).

% order.strict_iff_order
tff(fact_3844_order_Ostrict__trans1,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3845_order_Ostrict__trans1,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3846_order_Ostrict__trans1,axiom,
    ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3847_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3848_order_Ostrict__trans1,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3849_order_Ostrict__trans1,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3850_order_Ostrict__trans1,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C) ) ) ).

% order.strict_trans1
tff(fact_3851_order_Ostrict__trans2,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3852_order_Ostrict__trans2,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3853_order_Ostrict__trans2,axiom,
    ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),C)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3854_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3855_order_Ostrict__trans2,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3856_order_Ostrict__trans2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3857_order_Ostrict__trans2,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C) ) ) ).

% order.strict_trans2
tff(fact_3858_order_Ostrict__iff__not,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3859_order_Ostrict__iff__not,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3860_order_Ostrict__iff__not,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
        & ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3861_order_Ostrict__iff__not,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3862_order_Ostrict__iff__not,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3863_order_Ostrict__iff__not,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3864_order_Ostrict__iff__not,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ) ).

% order.strict_iff_not
tff(fact_3865_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),X)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_3866_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_3867_dual__order_Oorder__iff__strict,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A2)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3868_dual__order_Oorder__iff__strict,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3869_dual__order_Oorder__iff__strict,axiom,
    ! [B: filter_nat,A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3870_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3871_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3872_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3873_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
        | ( A2 = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3874_dual__order_Ostrict__iff__order,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3875_dual__order_Ostrict__iff__order,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3876_dual__order_Ostrict__iff__order,axiom,
    ! [B: filter_nat,A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A2)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3877_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3878_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3879_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3880_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
        & ( A2 != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3881_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A2: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A2)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3882_dual__order_Ostrict__trans1,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3883_dual__order_Ostrict__trans1,axiom,
    ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,C),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3884_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A2: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3885_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3886_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3887_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_3888_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A2: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3889_dual__order_Ostrict__trans2,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3890_dual__order_Ostrict__trans2,axiom,
    ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A2)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,C),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3891_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A2: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3892_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3893_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3894_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_3895_dual__order_Ostrict__iff__not,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A2)
        & ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3896_dual__order_Ostrict__iff__not,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3897_dual__order_Ostrict__iff__not,axiom,
    ! [B: filter_nat,A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A2)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2)
        & ~ aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3898_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A2)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3899_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3900_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3901_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3902_order_Ostrict__implies__order,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B) ) ).

% order.strict_implies_order
tff(fact_3903_order_Ostrict__implies__order,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ).

% order.strict_implies_order
tff(fact_3904_order_Ostrict__implies__order,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B) ) ).

% order.strict_implies_order
tff(fact_3905_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B) ) ).

% order.strict_implies_order
tff(fact_3906_order_Ostrict__implies__order,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ).

% order.strict_implies_order
tff(fact_3907_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ).

% order.strict_implies_order
tff(fact_3908_order_Ostrict__implies__order,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ).

% order.strict_implies_order
tff(fact_3909_dual__order_Ostrict__implies__order,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A2)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3910_dual__order_Ostrict__implies__order,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3911_dual__order_Ostrict__implies__order,axiom,
    ! [B: filter_nat,A2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,B),A2)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3912_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3913_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3914_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3915_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ).

% dual_order.strict_implies_order
tff(fact_3916_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_3917_order__le__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_3918_order__le__less,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_3919_order__le__less,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_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_3920_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_3921_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_3922_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_3923_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_3924_order__less__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_3925_order__less__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
    <=> ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_3926_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_3927_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_3928_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_3929_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_3930_linorder__not__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).

% linorder_not_le
tff(fact_3931_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_3932_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_3933_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_3934_linorder__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),X) ) ).

% linorder_not_less
tff(fact_3935_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_3936_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_3937_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_3938_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_3939_order__less__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ).

% order_less_imp_le
tff(fact_3940_order__less__imp__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_3941_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y) ) ).

% order_less_imp_le
tff(fact_3942_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_3943_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_3944_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_3945_order__le__neq__trans,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
     => ( ( A2 != B )
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3946_order__le__neq__trans,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( ( A2 != B )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3947_order__le__neq__trans,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
     => ( ( A2 != B )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3948_order__le__neq__trans,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
     => ( ( A2 != B )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3949_order__le__neq__trans,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( ( A2 != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3950_order__le__neq__trans,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( ( A2 != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3951_order__le__neq__trans,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( ( A2 != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ) ).

% order_le_neq_trans
tff(fact_3952_order__neq__le__trans,axiom,
    ! [A2: assn,B: assn] :
      ( ( A2 != B )
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3953_order__neq__le__trans,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 != B )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3954_order__neq__le__trans,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ( ( A2 != B )
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3955_order__neq__le__trans,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( A2 != B )
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3956_order__neq__le__trans,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3957_order__neq__le__trans,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3958_order__neq__le__trans,axiom,
    ! [A2: int,B: int] :
      ( ( A2 != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ) ).

% order_neq_le_trans
tff(fact_3959_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: 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),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3960_order__le__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3961_order__le__less__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,Y),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3962_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: 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_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3963_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: 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),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3964_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: 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),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3965_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: 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),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_le_less_trans
tff(fact_3966_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: 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),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3967_order__less__le__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3968_order__less__le__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),Z)
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3969_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3970_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: 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),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3971_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: 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),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3972_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: 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),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z) ) ) ).

% order_less_le_trans
tff(fact_3973_order__le__less__subst1,axiom,
    ! [A2: assn,F: fun_assn_assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),aa_assn_assn(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_assn_assn(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3974_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: fun_as9138348466139350018nteger,B: assn,C: assn] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_assn_Code_integer(F,B))
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X3)),aa_assn_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3975_order__le__less__subst1,axiom,
    ! [A2: assn,F: fun_nat_assn,B: nat,C: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_nat_assn(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3976_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_nat_Code_integer(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3977_order__le__less__subst1,axiom,
    ! [A2: assn,F: fun_int_assn,B: int,C: int] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),aa_int_assn(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X3)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_int_assn(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3978_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: fun_int_Code_integer,B: int,C: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_int_Code_integer(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X3)),aa_int_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_int_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3979_order__le__less__subst1,axiom,
    ! [A2: assn,F: fun_Co4490034186337852034r_assn,B: code_integer,C: code_integer] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),aa_Code_integer_assn(F,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,X3)),aa_Code_integer_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_Code_integer_assn(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3980_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: fun_Co8499238673658658103nteger,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(F,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X3)),aa_Cod1783974652533396728nteger(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3981_order__le__less__subst1,axiom,
    ! [A2: assn,F: fun_rat_assn,B: rat,C: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),aa_rat_assn(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_rat_assn(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3982_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_rat_Code_integer(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X3)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_rat_Code_integer(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3983_order__le__less__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_assn,C: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3984_order__le__less__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,X3)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3985_order__le__less__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3986_order__le__less__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3987_order__le__less__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3988_order__le__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_assn,C: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3989_order__le__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3990_order__le__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3991_order__le__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3992_order__le__less__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_int,C: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,A2)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3993_order__less__le__subst1,axiom,
    ! [A2: assn,F: fun_rat_assn,B: rat,C: rat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_rat_assn(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_rat_assn(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3994_order__less__le__subst1,axiom,
    ! [A2: code_integer,F: fun_rat_Code_integer,B: rat,C: rat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_rat_Code_integer(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,X3)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_rat_Code_integer(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3995_order__less__le__subst1,axiom,
    ! [A2: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3996_order__less__le__subst1,axiom,
    ! [A2: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3997_order__less__le__subst1,axiom,
    ! [A2: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3998_order__less__le__subst1,axiom,
    ! [A2: assn,F: fun_nat_assn,B: nat,C: nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_nat_assn(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),aa_nat_assn(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3999_order__less__le__subst1,axiom,
    ! [A2: code_integer,F: fun_nat_Code_integer,B: nat,C: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4000_order__less__le__subst1,axiom,
    ! [A2: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_nat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4001_order__less__le__subst1,axiom,
    ! [A2: nat,F: fun_nat_nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X3)),aa_nat_nat(F,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4002_order__less__le__subst1,axiom,
    ! [A2: int,F: fun_nat_int,B: nat,C: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(F,Y2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4003_order__less__le__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_assn_assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,X3)),aa_assn_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4004_order__less__le__subst2,axiom,
    ! [A2: assn,B: assn,F: fun_as9138348466139350018nteger,C: code_integer] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_assn_Code_integer(F,B)),C)
       => ( ! [X3: assn,Y2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,X3)),aa_assn_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_assn_Code_integer(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4005_order__less__le__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_assn,C: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_nat_assn(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X3)),aa_nat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4006_order__less__le__subst2,axiom,
    ! [A2: nat,B: nat,F: fun_nat_Code_integer,C: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,B)),C)
       => ( ! [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,X3)),aa_nat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4007_order__less__le__subst2,axiom,
    ! [A2: int,B: int,F: fun_int_assn,C: assn] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_int_assn(F,B)),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X3)),aa_int_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4008_order__less__le__subst2,axiom,
    ! [A2: int,B: int,F: fun_int_Code_integer,C: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,B)),C)
       => ( ! [X3: int,Y2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,X3)),aa_int_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4009_order__less__le__subst2,axiom,
    ! [A2: code_integer,B: code_integer,F: fun_Co4490034186337852034r_assn,C: assn] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_Code_integer_assn(F,B)),C)
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,X3)),aa_Code_integer_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4010_order__less__le__subst2,axiom,
    ! [A2: code_integer,B: code_integer,F: fun_Co8499238673658658103nteger,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,B)),C)
       => ( ! [X3: code_integer,Y2: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X3)),aa_Cod1783974652533396728nteger(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4011_order__less__le__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_assn,C: assn] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_rat_assn(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y2)
             => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,X3)),aa_rat_assn(F,Y2)) )
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_rat_assn(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4012_order__less__le__subst2,axiom,
    ! [A2: rat,B: rat,F: fun_rat_Code_integer,C: code_integer] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_rat_Code_integer(F,B)),C)
       => ( ! [X3: rat,Y2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y2)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,X3)),aa_rat_Code_integer(F,Y2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_rat_Code_integer(F,A2)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4013_linorder__le__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
      | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),X) ) ).

% linorder_le_less_linear
tff(fact_4014_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_4015_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_4016_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_4017_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_4018_order__le__imp__less__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4019_order__le__imp__less__or__eq,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_less_filter_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4020_order__le__imp__less__or__eq,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_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4021_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_4022_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_4023_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_4024_less__numeral__extra_I4_J,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),one_one_Code_natural) ).

% less_numeral_extra(4)
tff(fact_4025_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_4026_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_4027_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_4028_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_4029_diff__strict__mono,axiom,
    ! [A2: int,B: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),C)),aa_int_int(minus_minus_int(B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4030_diff__strict__mono,axiom,
    ! [A2: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,D2),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4031_diff__strict__mono,axiom,
    ! [A2: rat,B: rat,D2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),C)),aa_rat_rat(minus_minus_rat(B),D2)) ) ) ).

% diff_strict_mono
tff(fact_4032_diff__eq__diff__less,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(minus_minus_int(C),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4033_diff__eq__diff__less,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4034_diff__eq__diff__less,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(minus_minus_rat(C),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_4035_diff__strict__left__mono,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(C),A2)),aa_int_int(minus_minus_int(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4036_diff__strict__left__mono,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B)) ) ).

% diff_strict_left_mono
tff(fact_4037_diff__strict__left__mono,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A2)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4038_diff__strict__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),C)),aa_int_int(minus_minus_int(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4039_diff__strict__right__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) ) ).

% diff_strict_right_mono
tff(fact_4040_diff__strict__right__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4041_bot_Oextremum__strict,axiom,
    ! [A2: set_Product_unit] : ~ aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A2),bot_bo3957492148770167129t_unit) ).

% bot.extremum_strict
tff(fact_4042_bot_Oextremum__strict,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A2),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_4043_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A2),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_4044_bot_Oextremum__strict,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A2),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_4045_bot_Oextremum__strict,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : ~ aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A2),bot_bo8632255151191677115_nat_o) ).

% bot.extremum_strict
tff(fact_4046_bot_Oextremum__strict,axiom,
    ! [A2: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),bot_bot_assn) ).

% bot.extremum_strict
tff(fact_4047_bot_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_4048_bot_Onot__eq__extremum,axiom,
    ! [A2: set_Product_unit] :
      ( ( A2 != bot_bo3957492148770167129t_unit )
    <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,bot_bo3957492148770167129t_unit),A2) ) ).

% bot.not_eq_extremum
tff(fact_4049_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,bot_bot_set_o),A2) ) ).

% bot.not_eq_extremum
tff(fact_4050_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A2) ) ).

% bot.not_eq_extremum
tff(fact_4051_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,bot_bot_set_int),A2) ) ).

% bot.not_eq_extremum
tff(fact_4052_bot_Onot__eq__extremum,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] :
      ( ( A2 != bot_bo8632255151191677115_nat_o )
    <=> aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(bot_bo8632255151191677115_nat_o),A2) ) ).

% bot.not_eq_extremum
tff(fact_4053_bot_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,bot_bot_assn),A2) ) ).

% bot.not_eq_extremum
tff(fact_4054_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A2) ) ).

% bot.not_eq_extremum
tff(fact_4055_not__psubset__empty,axiom,
    ! [A: set_Product_unit] : ~ aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A),bot_bo3957492148770167129t_unit) ).

% not_psubset_empty
tff(fact_4056_not__psubset__empty,axiom,
    ! [A: set_o] : ~ aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_4057_not__psubset__empty,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_4058_not__psubset__empty,axiom,
    ! [A: set_int] : ~ aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_4059_not__psubset__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : ~ aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A),bot_bo8632255151191677115_nat_o) ).

% not_psubset_empty
tff(fact_4060_psubsetE,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A) ) ) ).

% psubsetE
tff(fact_4061_psubset__eq,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
        & ( A != B4 ) ) ) ).

% psubset_eq
tff(fact_4062_psubset__imp__subset,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4) ) ).

% psubset_imp_subset
tff(fact_4063_psubset__subset__trans,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C5)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C5) ) ) ).

% psubset_subset_trans
tff(fact_4064_subset__not__subset__eq,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
        & ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A) ) ) ).

% subset_not_subset_eq
tff(fact_4065_subset__psubset__trans,axiom,
    ! [A: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B4),C5)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C5) ) ) ).

% subset_psubset_trans
tff(fact_4066_subset__iff__psubset__eq,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
        | ( A = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_4067_rangeI,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: code_integer] : aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(F,X)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger)) ).

% rangeI
tff(fact_4068_rangeI,axiom,
    ! [F: fun_Product_unit_nat,X: product_unit] : aa_set_nat_o(member_nat(aa_Product_unit_nat(F,X)),image_875570014554754200it_nat(F,top_to1996260823553986621t_unit)) ).

% rangeI
tff(fact_4069_rangeI,axiom,
    ! [F: fun_Product_unit_int,X: product_unit] : aa_set_int_o(member_int(aa_Product_unit_int(F,X)),image_873079544045703924it_int(F,top_to1996260823553986621t_unit)) ).

% rangeI
tff(fact_4070_rangeI,axiom,
    ! [F: fun_Product_unit_o,X: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(F,X)),image_Product_unit_o2(F,top_to1996260823553986621t_unit)) ).

% rangeI
tff(fact_4071_rangeI,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,X: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(F,X)),image_405062704495631173t_unit(F,top_to1996260823553986621t_unit)) ).

% rangeI
tff(fact_4072_rangeI,axiom,
    ! [F: fun_o_nat,X: $o] : aa_set_nat_o(member_nat(aa_o_nat(F,(X))),image_o_nat2(F,top_top_set_o)) ).

% rangeI
tff(fact_4073_rangeI,axiom,
    ! [F: fun_o_int,X: $o] : aa_set_int_o(member_int(aa_o_int(F,(X))),image_o_int2(F,top_top_set_o)) ).

% rangeI
tff(fact_4074_rangeI,axiom,
    ! [F: fun_o_o,X: $o] : aa_set_o_o(member_o(aa_o_o(F,(X))),image_o_o2(F,top_top_set_o)) ).

% rangeI
tff(fact_4075_rangeI,axiom,
    ! [F: fun_o_Product_unit,X: $o] : aa_set8780210009671629607unit_o(member_Product_unit(aa_o_Product_unit(F,(X))),image_o_Product_unit2(F,top_top_set_o)) ).

% rangeI
tff(fact_4076_rangeI,axiom,
    ! [F: fun_nat_Code_integer,X: nat] : aa_set5643387332196575111eger_o(member_Code_integer(aa_nat_Code_integer(F,X)),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),top_top_set_nat)) ).

% rangeI
tff(fact_4077_range__eqI,axiom,
    ! [B: code_integer,F: fun_Co8499238673658658103nteger,X: code_integer] :
      ( ( B = aa_Cod1783974652533396728nteger(F,X) )
     => aa_set5643387332196575111eger_o(member_Code_integer(B),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger)) ) ).

% range_eqI
tff(fact_4078_range__eqI,axiom,
    ! [B: nat,F: fun_Product_unit_nat,X: product_unit] :
      ( ( B = aa_Product_unit_nat(F,X) )
     => aa_set_nat_o(member_nat(B),image_875570014554754200it_nat(F,top_to1996260823553986621t_unit)) ) ).

% range_eqI
tff(fact_4079_range__eqI,axiom,
    ! [B: int,F: fun_Product_unit_int,X: product_unit] :
      ( ( B = aa_Product_unit_int(F,X) )
     => aa_set_int_o(member_int(B),image_873079544045703924it_int(F,top_to1996260823553986621t_unit)) ) ).

% range_eqI
tff(fact_4080_range__eqI,axiom,
    ! [B: $o,F: fun_Product_unit_o,X: product_unit] :
      ( ( (B)
      <=> aa_Product_unit_o(F,X) )
     => aa_set_o_o(member_o((B)),image_Product_unit_o2(F,top_to1996260823553986621t_unit)) ) ).

% range_eqI
tff(fact_4081_range__eqI,axiom,
    ! [B: product_unit,F: fun_Pr2191293291710260995t_unit,X: product_unit] :
      ( ( B = aa_Pro8294289767544744956t_unit(F,X) )
     => aa_set8780210009671629607unit_o(member_Product_unit(B),image_405062704495631173t_unit(F,top_to1996260823553986621t_unit)) ) ).

% range_eqI
tff(fact_4082_range__eqI,axiom,
    ! [B: nat,F: fun_o_nat,X: $o] :
      ( ( B = aa_o_nat(F,(X)) )
     => aa_set_nat_o(member_nat(B),image_o_nat2(F,top_top_set_o)) ) ).

% range_eqI
tff(fact_4083_range__eqI,axiom,
    ! [B: int,F: fun_o_int,X: $o] :
      ( ( B = aa_o_int(F,(X)) )
     => aa_set_int_o(member_int(B),image_o_int2(F,top_top_set_o)) ) ).

% range_eqI
tff(fact_4084_range__eqI,axiom,
    ! [B: $o,F: fun_o_o,X: $o] :
      ( ( (B)
      <=> aa_o_o(F,(X)) )
     => aa_set_o_o(member_o((B)),image_o_o2(F,top_top_set_o)) ) ).

% range_eqI
tff(fact_4085_range__eqI,axiom,
    ! [B: product_unit,F: fun_o_Product_unit,X: $o] :
      ( ( B = aa_o_Product_unit(F,(X)) )
     => aa_set8780210009671629607unit_o(member_Product_unit(B),image_o_Product_unit2(F,top_top_set_o)) ) ).

% range_eqI
tff(fact_4086_range__eqI,axiom,
    ! [B: code_integer,F: fun_nat_Code_integer,X: nat] :
      ( ( B = aa_nat_Code_integer(F,X) )
     => aa_set5643387332196575111eger_o(member_Code_integer(B),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),top_top_set_nat)) ) ).

% range_eqI
tff(fact_4087_empty__not__UNIV,axiom,
    bot_bo8632255151191677115_nat_o != top_to9044565752620676511_nat_o ).

% empty_not_UNIV
tff(fact_4088_empty__not__UNIV,axiom,
    bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).

% empty_not_UNIV
tff(fact_4089_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
tff(fact_4090_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
tff(fact_4091_empty__not__UNIV,axiom,
    bot_bot_set_int != top_top_set_int ).

% empty_not_UNIV
tff(fact_4092_subset__UNIV,axiom,
    ! [A: set_Product_unit] : aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),top_to1996260823553986621t_unit) ).

% subset_UNIV
tff(fact_4093_subset__UNIV,axiom,
    ! [A: set_o] : aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),top_top_set_o) ).

% subset_UNIV
tff(fact_4094_subset__UNIV,axiom,
    ! [A: set_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),top_top_set_int) ).

% subset_UNIV
tff(fact_4095_subset__UNIV,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),top_top_set_nat) ).

% subset_UNIV
tff(fact_4096_psubset__imp__ex__mem,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(A),B4)
     => ? [B2: fun_Pr6460944699146269935_nat_o] : aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B2),minus_6456595905716665654_nat_o(B4,A)) ) ).

% psubset_imp_ex_mem
tff(fact_4097_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,A),B4)
     => ? [B2: int] : aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A)) ) ).

% psubset_imp_ex_mem
tff(fact_4098_psubset__imp__ex__mem,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,A),B4)
     => ? [B2: $o] : aa_set_o_o(member_o((B2)),minus_minus_set_o(B4,A)) ) ).

% psubset_imp_ex_mem
tff(fact_4099_psubset__imp__ex__mem,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,A),B4)
     => ? [B2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(B4,A)) ) ).

% psubset_imp_ex_mem
tff(fact_4100_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B4)
     => ? [B2: nat] : aa_set_nat_o(member_nat(B2),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)) ) ).

% psubset_imp_ex_mem
tff(fact_4101_insert__UNIV,axiom,
    ! [X: product_unit] : aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).

% insert_UNIV
tff(fact_4102_insert__UNIV,axiom,
    ! [X: $o] : aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),top_top_set_o) = top_top_set_o ).

% insert_UNIV
tff(fact_4103_insert__UNIV,axiom,
    ! [X: nat] : aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),top_top_set_nat) = top_top_set_nat ).

% insert_UNIV
tff(fact_4104_insert__UNIV,axiom,
    ! [X: int] : aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),top_top_set_int) = top_top_set_int ).

% insert_UNIV
tff(fact_4105_Int__UNIV__left,axiom,
    ! [B4: set_Product_unit] : inf_in4660618365625256667t_unit(top_to1996260823553986621t_unit,B4) = B4 ).

% Int_UNIV_left
tff(fact_4106_Int__UNIV__left,axiom,
    ! [B4: set_o] : inf_inf_set_o(top_top_set_o,B4) = B4 ).

% Int_UNIV_left
tff(fact_4107_Int__UNIV__left,axiom,
    ! [B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),B4) = B4 ).

% Int_UNIV_left
tff(fact_4108_Int__UNIV__left,axiom,
    ! [B4: set_int] : inf_inf_set_int(top_top_set_int,B4) = B4 ).

% Int_UNIV_left
tff(fact_4109_Int__UNIV__right,axiom,
    ! [A: set_Product_unit] : inf_in4660618365625256667t_unit(A,top_to1996260823553986621t_unit) = A ).

% Int_UNIV_right
tff(fact_4110_Int__UNIV__right,axiom,
    ! [A: set_o] : inf_inf_set_o(A,top_top_set_o) = A ).

% Int_UNIV_right
tff(fact_4111_Int__UNIV__right,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),top_top_set_nat) = A ).

% Int_UNIV_right
tff(fact_4112_Int__UNIV__right,axiom,
    ! [A: set_int] : inf_inf_set_int(A,top_top_set_int) = A ).

% Int_UNIV_right
tff(fact_4113_Un__UNIV__left,axiom,
    ! [B4: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,top_to1996260823553986621t_unit),B4) = top_to1996260823553986621t_unit ).

% Un_UNIV_left
tff(fact_4114_Un__UNIV__left,axiom,
    ! [B4: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,top_top_set_o),B4) = top_top_set_o ).

% Un_UNIV_left
tff(fact_4115_Un__UNIV__left,axiom,
    ! [B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),B4) = top_top_set_nat ).

% Un_UNIV_left
tff(fact_4116_Un__UNIV__left,axiom,
    ! [B4: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,top_top_set_int),B4) = top_top_set_int ).

% Un_UNIV_left
tff(fact_4117_Un__UNIV__right,axiom,
    ! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).

% Un_UNIV_right
tff(fact_4118_Un__UNIV__right,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),top_top_set_o) = top_top_set_o ).

% Un_UNIV_right
tff(fact_4119_Un__UNIV__right,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),top_top_set_nat) = top_top_set_nat ).

% Un_UNIV_right
tff(fact_4120_Un__UNIV__right,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),top_top_set_int) = top_top_set_int ).

% Un_UNIV_right
tff(fact_4121_type__copy__ex__RepI,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,F2: fun_fu5135969500543783852at_o_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),top_to9044565752620676511_nat_o)
     => ( ? [X_12: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(F2,X_12)
      <=> ? [B6: assn] : aa_fun1675921005893230629at_o_o(F2,aa_ass4243947775541419017_nat_o(Rep,B6)) ) ) ).

% type_copy_ex_RepI
tff(fact_4122_type__copy__ex__RepI,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,F2: fun_int_o] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( ? [X_12: int] : aa_int_o(F2,X_12)
      <=> ? [B6: code_integer] : aa_int_o(F2,aa_Code_integer_int(Rep,B6)) ) ) ).

% type_copy_ex_RepI
tff(fact_4123_type__copy__ex__RepI,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,F2: fun_o_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( ? [X_12: $o] : aa_o_o(F2,(X_12))
      <=> ? [B6: product_unit] : aa_o_o(F2,aa_Product_unit_o(Rep,B6)) ) ) ).

% type_copy_ex_RepI
tff(fact_4124_type__copy__ex__RepI,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,F2: fun_nat_o] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),top_top_set_nat)
     => ( ? [X_12: nat] : aa_nat_o(F2,X_12)
      <=> ? [B6: code_natural] : aa_nat_o(F2,aa_Code_natural_nat(Rep,B6)) ) ) ).

% type_copy_ex_RepI
tff(fact_4125_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,S2: assn] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),top_to9044565752620676511_nat_o)
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] : S2 != aa_fun5526454687433275355o_assn(Abs,X3) ) ).

% type_copy_obj_one_point_absE
tff(fact_4126_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,S2: code_integer] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ~ ! [X3: int] : S2 != aa_int_Code_integer(Abs,X3) ) ).

% type_copy_obj_one_point_absE
tff(fact_4127_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,S2: product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ~ ! [X3: $o] : S2 != aa_o_Product_unit(Abs,(X3)) ) ).

% type_copy_obj_one_point_absE
tff(fact_4128_type__copy__obj__one__point__absE,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,S2: code_natural] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),top_top_set_nat)
     => ~ ! [X3: nat] : S2 != aa_nat_Code_natural(Abs,X3) ) ).

% type_copy_obj_one_point_absE
tff(fact_4129_less__1__mult,axiom,
    ! [M4: code_natural,N2: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),M4)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),N2)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,M4),N2)) ) ) ).

% less_1_mult
tff(fact_4130_less__1__mult,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M4)
     => ( 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,M4),N2)) ) ) ).

% less_1_mult
tff(fact_4131_less__1__mult,axiom,
    ! [M4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M4)
     => ( 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,M4),N2)) ) ) ).

% less_1_mult
tff(fact_4132_less__1__mult,axiom,
    ! [M4: code_integer,N2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),M4)
     => ( 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,M4),N2)) ) ) ).

% less_1_mult
tff(fact_4133_less__1__mult,axiom,
    ! [M4: rat,N2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M4)
     => ( 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,M4),N2)) ) ) ).

% less_1_mult
tff(fact_4134_infinite__growing,axiom,
    ! [X5: set_Product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
           => ? [Xa3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(Xa3),X5)
                & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X3),Xa3) ) )
       => ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5) ) ) ).

% infinite_growing
tff(fact_4135_infinite__growing,axiom,
    ! [X5: set_o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X5)
           => ? [Xa3: $o] :
                ( aa_set_o_o(member_o((Xa3)),X5)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(Xa3)) ) )
       => ~ aa_set_o_o(finite_finite_o,X5) ) ) ).

% infinite_growing
tff(fact_4136_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => ? [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa3) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_4137_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => ? [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa3) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_4138_infinite__growing,axiom,
    ! [X5: set_Code_integer] :
      ( ( X5 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X5)
           => ? [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),X5)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Xa3) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X5) ) ) ).

% infinite_growing
tff(fact_4139_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),X5)
           => ? [Xa3: rat] :
                ( aa_set_rat_o(member_rat(Xa3),X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa3) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X5) ) ) ).

% infinite_growing
tff(fact_4140_ex__min__if__finite,axiom,
    ! [S: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( ( S != bot_bo3957492148770167129t_unit )
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),S)
            & ~ ? [Xa3: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(Xa3),S)
                  & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4141_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)
            & ~ ? [Xa3: $o] :
                  ( aa_set_o_o(member_o((Xa3)),S)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa3)),(X3)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4142_ex__min__if__finite,axiom,
    ! [S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( ( S != bot_bo8632255151191677115_nat_o )
       => ? [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),S)
            & ~ ? [Xa3: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Xa3),S)
                  & aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le6684905062368609691_nat_o,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4143_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( finite_finite_assn(S)
     => ( ( S != bot_bot_set_assn )
       => ? [X3: assn] :
            ( member_assn(X3,S)
            & ~ ? [Xa3: assn] :
                  ( member_assn(Xa3,S)
                  & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4144_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)
            & ~ ? [Xa3: nat] :
                  ( aa_set_nat_o(member_nat(Xa3),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4145_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)
            & ~ ? [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4146_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)
            & ~ ? [Xa3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),S)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4147_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)
            & ~ ? [Xa3: rat] :
                  ( aa_set_rat_o(member_rat(Xa3),S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa3),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4148_less__cSupD,axiom,
    ! [X5: set_int,Z: int] :
      ( ( X5 != bot_bot_set_int )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),complete_Sup_Sup_int(X5))
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X3) ) ) ) ).

% less_cSupD
tff(fact_4149_less__cSupD,axiom,
    ! [X5: set_nat,Z: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_set_nat_nat(complete_Sup_Sup_nat,X5))
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X3) ) ) ) ).

% less_cSupD
tff(fact_4150_less__cSupD,axiom,
    ! [X5: set_Product_unit,Z: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Z),comple4687483117567863418t_unit(X5))
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
            & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Z),X3) ) ) ) ).

% less_cSupD
tff(fact_4151_less__cSupE,axiom,
    ! [Y: int,X5: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),complete_Sup_Sup_int(X5))
     => ( ( X5 != bot_bot_set_int )
       => ~ ! [X3: int] :
              ( aa_set_int_o(member_int(X3),X5)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X3) ) ) ) ).

% less_cSupE
tff(fact_4152_less__cSupE,axiom,
    ! [Y: nat,X5: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_set_nat_nat(complete_Sup_Sup_nat,X5))
     => ( ( X5 != bot_bot_set_nat )
       => ~ ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),X5)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X3) ) ) ) ).

% less_cSupE
tff(fact_4153_less__cSupE,axiom,
    ! [Y: product_unit,X5: set_Product_unit] :
      ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Y),comple4687483117567863418t_unit(X5))
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ~ ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
             => ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Y),X3) ) ) ) ).

% less_cSupE
tff(fact_4154_Max__in,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => aa_set5643387332196575111eger_o(member_Code_integer(lattic4901227151466704046nteger(A)),A) ) ) ).

% Max_in
tff(fact_4155_Max__in,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => aa_set8780210009671629607unit_o(member_Product_unit(lattic1210392890313158908t_unit(A)),A) ) ) ).

% Max_in
tff(fact_4156_Max__in,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => aa_set_o_o(member_o(aa_set_o_o(lattic1921953407002678535_Max_o,A)),A) ) ) ).

% Max_in
tff(fact_4157_Max__in,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => aa_set_int_o(member_int(lattic8263393255366662781ax_int(A)),A) ) ) ).

% Max_in
tff(fact_4158_Max__in,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),A) ) ) ).

% Max_in
tff(fact_4159_range__subsetD,axiom,
    ! [F: fun_Co8499238673658658103nteger,B4: set_Code_integer,I2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger)),B4)
     => aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4160_range__subsetD,axiom,
    ! [F: fun_Product_unit_int,B4: set_int,I2: product_unit] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_873079544045703924it_int(F,top_to1996260823553986621t_unit)),B4)
     => aa_set_int_o(member_int(aa_Product_unit_int(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4161_range__subsetD,axiom,
    ! [F: fun_Product_unit_o,B4: set_o,I2: product_unit] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_Product_unit_o2(F,top_to1996260823553986621t_unit)),B4)
     => aa_set_o_o(member_o(aa_Product_unit_o(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4162_range__subsetD,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,B4: set_Product_unit,I2: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_405062704495631173t_unit(F,top_to1996260823553986621t_unit)),B4)
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4163_range__subsetD,axiom,
    ! [F: fun_o_int,B4: set_int,I2: $o] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_o_int2(F,top_top_set_o)),B4)
     => aa_set_int_o(member_int(aa_o_int(F,(I2))),B4) ) ).

% range_subsetD
tff(fact_4164_range__subsetD,axiom,
    ! [F: fun_o_o,B4: set_o,I2: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_o_o2(F,top_top_set_o)),B4)
     => aa_set_o_o(member_o(aa_o_o(F,(I2))),B4) ) ).

% range_subsetD
tff(fact_4165_range__subsetD,axiom,
    ! [F: fun_o_Product_unit,B4: set_Product_unit,I2: $o] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,image_o_Product_unit2(F,top_top_set_o)),B4)
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_o_Product_unit(F,(I2))),B4) ) ).

% range_subsetD
tff(fact_4166_range__subsetD,axiom,
    ! [F: fun_nat_Code_integer,B4: set_Code_integer,I2: nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),top_top_set_nat)),B4)
     => aa_set5643387332196575111eger_o(member_Code_integer(aa_nat_Code_integer(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4167_range__subsetD,axiom,
    ! [F: fun_nat_int,B4: set_int,I2: nat] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_nat_int2(F,top_top_set_nat)),B4)
     => aa_set_int_o(member_int(aa_nat_int(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4168_range__subsetD,axiom,
    ! [F: fun_nat_o,B4: set_o,I2: nat] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,image_nat_o2(F,top_top_set_nat)),B4)
     => aa_set_o_o(member_o(aa_nat_o(F,I2)),B4) ) ).

% range_subsetD
tff(fact_4169_surj__vimage__empty,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( vimage1318793397690208119nteger(F,A) = bot_bo3990330152332043303nteger )
      <=> ( A = bot_bo3990330152332043303nteger ) ) ) ).

% surj_vimage_empty
tff(fact_4170_surj__vimage__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A: set_Product_unit] :
      ( ( image_405062704495631173t_unit(F,top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit )
     => ( ( vimage7995052115951654139t_unit(F,A) = bot_bo3957492148770167129t_unit )
      <=> ( A = bot_bo3957492148770167129t_unit ) ) ) ).

% surj_vimage_empty
tff(fact_4171_surj__vimage__empty,axiom,
    ! [F: fun_Product_unit_o,A: set_o] :
      ( ( image_Product_unit_o2(F,top_to1996260823553986621t_unit) = top_top_set_o )
     => ( ( vimage8808348054653608838unit_o(F,A) = bot_bo3957492148770167129t_unit )
      <=> ( A = bot_bot_set_o ) ) ) ).

% surj_vimage_empty
tff(fact_4172_surj__vimage__empty,axiom,
    ! [F: fun_Product_unit_nat,A: set_nat] :
      ( ( image_875570014554754200it_nat(F,top_to1996260823553986621t_unit) = top_top_set_nat )
     => ( ( vimage6253328473476588386it_nat(F,A) = bot_bo3957492148770167129t_unit )
      <=> ( A = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
tff(fact_4173_surj__vimage__empty,axiom,
    ! [F: fun_Product_unit_int,A: set_int] :
      ( ( image_873079544045703924it_int(F,top_to1996260823553986621t_unit) = top_top_set_int )
     => ( ( vimage6250838002967538110it_int(F,A) = bot_bo3957492148770167129t_unit )
      <=> ( A = bot_bot_set_int ) ) ) ).

% surj_vimage_empty
tff(fact_4174_surj__vimage__empty,axiom,
    ! [F: fun_o_Product_unit,A: set_Product_unit] :
      ( ( image_o_Product_unit2(F,top_top_set_o) = top_to1996260823553986621t_unit )
     => ( ( vimage4222377557387472176t_unit(F,A) = bot_bot_set_o )
      <=> ( A = bot_bo3957492148770167129t_unit ) ) ) ).

% surj_vimage_empty
tff(fact_4175_surj__vimage__empty,axiom,
    ! [F: fun_o_o,A: set_o] :
      ( ( image_o_o2(F,top_top_set_o) = top_top_set_o )
     => ( ( vimage_o_o(F,A) = bot_bot_set_o )
      <=> ( A = bot_bot_set_o ) ) ) ).

% surj_vimage_empty
tff(fact_4176_surj__vimage__empty,axiom,
    ! [F: fun_o_nat,A: set_nat] :
      ( ( image_o_nat2(F,top_top_set_o) = top_top_set_nat )
     => ( ( vimage_o_nat(F,A) = bot_bot_set_o )
      <=> ( A = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
tff(fact_4177_surj__vimage__empty,axiom,
    ! [F: fun_o_int,A: set_int] :
      ( ( image_o_int2(F,top_top_set_o) = top_top_set_int )
     => ( ( vimage_o_int(F,A) = bot_bot_set_o )
      <=> ( A = bot_bot_set_int ) ) ) ).

% surj_vimage_empty
tff(fact_4178_surj__vimage__empty,axiom,
    ! [F: fun_nat_Code_integer,A: set_Code_integer] :
      ( ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),top_top_set_nat) = top_to4645266643341252675nteger )
     => ( ( vimage6104394758324456170nteger(F,A) = bot_bot_set_nat )
      <=> ( A = bot_bo3990330152332043303nteger ) ) ) ).

% surj_vimage_empty
tff(fact_4179_acyclicI__order,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: fun_nat_assn] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),R2)
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,B2)),aa_nat_assn(F,A3)) )
     => transi3139058337892597825ic_nat(R2) ) ).

% acyclicI_order
tff(fact_4180_acyclicI__order,axiom,
    ! [R2: set_Pr958786334691620121nt_int,F: fun_int_assn] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),R2)
         => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,B2)),aa_int_assn(F,A3)) )
     => transi3136567867383547549ic_int(R2) ) ).

% acyclicI_order
tff(fact_4181_acyclicI__order,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: fun_nat_nat] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),R2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,B2)),aa_nat_nat(F,A3)) )
     => transi3139058337892597825ic_nat(R2) ) ).

% acyclicI_order
tff(fact_4182_acyclicI__order,axiom,
    ! [R2: set_Pr958786334691620121nt_int,F: fun_int_nat] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),R2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,B2)),aa_int_nat(F,A3)) )
     => transi3136567867383547549ic_int(R2) ) ).

% acyclicI_order
tff(fact_4183_acyclicI__order,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: fun_nat_int] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),R2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,B2)),aa_nat_int(F,A3)) )
     => transi3139058337892597825ic_nat(R2) ) ).

% acyclicI_order
tff(fact_4184_acyclicI__order,axiom,
    ! [R2: set_Pr958786334691620121nt_int,F: fun_int_int] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),R2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,B2)),aa_int_int(F,A3)) )
     => transi3136567867383547549ic_int(R2) ) ).

% acyclicI_order
tff(fact_4185_acyclicI__order,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: fun_nat_Code_integer] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),R2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,B2)),aa_nat_Code_integer(F,A3)) )
     => transi3139058337892597825ic_nat(R2) ) ).

% acyclicI_order
tff(fact_4186_acyclicI__order,axiom,
    ! [R2: set_Pr958786334691620121nt_int,F: fun_int_Code_integer] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),R2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,B2)),aa_int_Code_integer(F,A3)) )
     => transi3136567867383547549ic_int(R2) ) ).

% acyclicI_order
tff(fact_4187_acyclicI__order,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: fun_nat_rat] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2),R2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,B2)),aa_nat_rat(F,A3)) )
     => transi3139058337892597825ic_nat(R2) ) ).

% acyclicI_order
tff(fact_4188_acyclicI__order,axiom,
    ! [R2: set_Pr958786334691620121nt_int,F: fun_int_rat] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B2),R2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,B2)),aa_int_rat(F,A3)) )
     => transi3136567867383547549ic_int(R2) ) ).

% acyclicI_order
tff(fact_4189_type__definition_OAbs__image,axiom,
    ! [Rep: fun_set_nat_nat,Abs: fun_nat_set_nat,A: set_nat] :
      ( aa_set_nat_o(type_d202864840109488260at_nat(Rep,Abs),A)
     => ( aa_set7246812951530217472et_nat(image_nat_set_nat(Abs),A) = top_top_set_set_nat ) ) ).

% type_definition.Abs_image
tff(fact_4190_type__definition_OAbs__image,axiom,
    ! [Rep: fun_Co8499238673658658103nteger,Abs: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(type_d4796611474492816872nteger(Rep,Abs),A)
     => ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(Abs),A) = top_to4645266643341252675nteger ) ) ).

% type_definition.Abs_image
tff(fact_4191_type__definition_OAbs__image,axiom,
    ! [Rep: fun_Code_integer_nat,Abs: fun_nat_Code_integer,A: set_nat] :
      ( aa_set_nat_o(type_d8368584451094728027er_nat(Rep,Abs),A)
     => ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(Abs),A) = top_to4645266643341252675nteger ) ) ).

% type_definition.Abs_image
tff(fact_4192_type__definition_OAbs__image,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),A)
     => ( aa_set_nat_set_nat(image_nat_nat2(Abs),A) = top_top_set_nat ) ) ).

% type_definition.Abs_image
tff(fact_4193_type__definition_OAbs__image,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( image_3838674425481255890o_assn(Abs,A) = top_top_set_assn ) ) ).

% type_definition.Abs_image
tff(fact_4194_type__definition_OAbs__image,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( image_1587234942943678608nteger(Abs,A) = top_to4645266643341252675nteger ) ) ).

% type_definition.Abs_image
tff(fact_4195_type__definition_OAbs__image,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( image_o_Product_unit2(Abs,A) = top_to1996260823553986621t_unit ) ) ).

% type_definition.Abs_image
tff(fact_4196_type__definition_OAbs__image,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( image_39176534463551487atural(Abs,A) = top_to9103716463852441742atural ) ) ).

% type_definition.Abs_image
tff(fact_4197_type__definition_ORep__range,axiom,
    ! [Rep: fun_Co8499238673658658103nteger,Abs: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(type_d4796611474492816872nteger(Rep,Abs),A)
     => ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(Rep),top_to4645266643341252675nteger) = A ) ) ).

% type_definition.Rep_range
tff(fact_4198_type__definition_ORep__range,axiom,
    ! [Rep: fun_nat_set_nat,Abs: fun_set_nat_nat,A: set_set_nat] :
      ( type_d3768499744764857476et_nat(Rep,Abs,A)
     => ( aa_set7246812951530217472et_nat(image_nat_set_nat(Rep),top_top_set_nat) = A ) ) ).

% type_definition.Rep_range
tff(fact_4199_type__definition_ORep__range,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),A)
     => ( aa_set_nat_set_nat(image_nat_nat2(Rep),top_top_set_nat) = A ) ) ).

% type_definition.Rep_range
tff(fact_4200_type__definition_ORep__range,axiom,
    ! [Rep: fun_nat_Code_integer,Abs: fun_Code_integer_nat,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(type_d8633139899873770843nteger(Rep,Abs),A)
     => ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(Rep),top_top_set_nat) = A ) ) ).

% type_definition.Rep_range
tff(fact_4201_type__definition_ORep__range,axiom,
    ! [Rep: fun_as3747249873083764768_nat_o,Abs: fun_fu3790787410453625514o_assn,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(type_d5492019815722432281_nat_o(Rep,Abs),A)
     => ( image_2556167513589399552_nat_o(Rep,top_top_set_assn) = A ) ) ).

% type_definition.Rep_range
tff(fact_4202_type__definition_ORep__range,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,A: set_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),A)
     => ( image_948535463418740880er_int(Rep,top_to4645266643341252675nteger) = A ) ) ).

% type_definition.Rep_range
tff(fact_4203_type__definition_ORep__range,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A: set_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A)
     => ( image_Product_unit_o2(Rep,top_to1996260823553986621t_unit) = A ) ) ).

% type_definition.Rep_range
tff(fact_4204_type__definition_ORep__range,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,A: set_nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),A)
     => ( image_6215854944615398399al_nat(Rep,top_to9103716463852441742atural) = A ) ) ).

% type_definition.Rep_range
tff(fact_4205_refl__Id,axiom,
    refl_on_Product_unit(top_to1996260823553986621t_unit,id_Product_unit) ).

% refl_Id
tff(fact_4206_refl__Id,axiom,
    refl_on_o(top_top_set_o,id_o2) ).

% refl_Id
tff(fact_4207_refl__Id,axiom,
    refl_on_nat(top_top_set_nat,id_nat) ).

% refl_Id
tff(fact_4208_refl__Id,axiom,
    refl_on_int(top_top_set_int,id_int) ).

% refl_Id
tff(fact_4209_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: product_unit,X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A2),X) = bot_bot_Product_unit )
     => ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),X) = top_top_Product_unit )
       => ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A2),Y) = bot_bot_Product_unit )
         => ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,A2),Y) = top_top_Product_unit )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4210_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A2,X) = bot_bo8632255151191677115_nat_o )
     => ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A2),X) = top_to9044565752620676511_nat_o )
       => ( ( inf_in3376754634429691197_nat_o(A2,Y) = bot_bo8632255151191677115_nat_o )
         => ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A2),Y) = top_to9044565752620676511_nat_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4211_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: assn,X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),X) = bot_bot_assn )
     => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),X) = top_top_assn )
       => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),Y) = bot_bot_assn )
         => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),Y) = top_top_assn )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4212_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(A2,X) = bot_bo3957492148770167129t_unit )
     => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A2),X) = top_to1996260823553986621t_unit )
       => ( ( inf_in4660618365625256667t_unit(A2,Y) = bot_bo3957492148770167129t_unit )
         => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A2),Y) = top_to1996260823553986621t_unit )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4213_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_o,X: set_o,Y: set_o] :
      ( ( inf_inf_set_o(A2,X) = bot_bot_set_o )
     => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),X) = top_top_set_o )
       => ( ( inf_inf_set_o(A2,Y) = bot_bot_set_o )
         => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A2),Y) = top_top_set_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4214_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_nat,X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),X) = bot_bot_set_nat )
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),X) = top_top_set_nat )
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),Y) = bot_bot_set_nat )
         => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),Y) = top_top_set_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4215_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_int,X: set_int,Y: set_int] :
      ( ( inf_inf_set_int(A2,X) = bot_bot_set_int )
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),X) = top_top_set_int )
       => ( ( inf_inf_set_int(A2,Y) = bot_bot_set_int )
         => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),Y) = top_top_set_int )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4216_union__fold__insert,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4) = finite1723285575846220376et_int(insert_int,B4,A) ) ) ).

% union_fold_insert
tff(fact_4217_union__fold__insert,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4) = finite_fold_o_set_o(insert_o,B4,A) ) ) ).

% union_fold_insert
tff(fact_4218_union__fold__insert,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4) = finite2740360166580943592t_unit(insert_Product_unit,B4,A) ) ) ).

% union_fold_insert
tff(fact_4219_union__fold__insert,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( sup_su848401254843788991nteger(A,B4) = finite7486036041278456378nteger(insert_Code_integer,B4,A) ) ) ).

% union_fold_insert
tff(fact_4220_union__fold__insert,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4) = aa_set_nat_set_nat(finite5529483035118572448et_nat(insert_nat,B4),A) ) ) ).

% union_fold_insert
tff(fact_4221_finite__linorder__min__induct,axiom,
    ! [A: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [B2: product_unit,A8: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8)
             => ( ! [X4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A8)
                   => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,B2),X4) )
               => ( aa_set8780210009671629607unit_o(P,A8)
                 => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B2),A8)) ) ) )
         => aa_set8780210009671629607unit_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4222_finite__linorder__min__induct,axiom,
    ! [A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B2: $o,A8: set_o] :
              ( aa_set_o_o(finite_finite_o,A8)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A8)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B2)),(X4)) )
               => ( aa_set_o_o(P,A8)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B2)),A8)) ) ) )
         => aa_set_o_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4223_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B2: nat,A8: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A8)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A8)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),X4) )
               => ( aa_set_nat_o(P,A8)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B2),A8)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4224_finite__linorder__min__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B2: int,A8: set_int] :
              ( aa_set_int_o(finite_finite_int,A8)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A8)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),X4) )
               => ( aa_set_int_o(P,A8)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B2),A8)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4225_finite__linorder__min__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B2: code_integer,A8: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8)
             => ( ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A8)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),X4) )
               => ( aa_set5643387332196575111eger_o(P,A8)
                 => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,B2),A8)) ) ) )
         => aa_set5643387332196575111eger_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4226_finite__linorder__min__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B2: rat,A8: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A8)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A8)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),X4) )
               => ( aa_set_rat_o(P,A8)
                 => aa_set_rat_o(P,insert_rat(B2,A8)) ) ) )
         => aa_set_rat_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4227_finite__linorder__max__induct,axiom,
    ! [A: set_Product_unit,P: fun_se4342725320470837934unit_o] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(P,bot_bo3957492148770167129t_unit)
       => ( ! [B2: product_unit,A8: set_Product_unit] :
              ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A8)
             => ( ! [X4: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A8)
                   => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X4),B2) )
               => ( aa_set8780210009671629607unit_o(P,A8)
                 => aa_set8780210009671629607unit_o(P,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B2),A8)) ) ) )
         => aa_set8780210009671629607unit_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4228_finite__linorder__max__induct,axiom,
    ! [A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B2: $o,A8: set_o] :
              ( aa_set_o_o(finite_finite_o,A8)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A8)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X4)),(B2)) )
               => ( aa_set_o_o(P,A8)
                 => aa_set_o_o(P,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B2)),A8)) ) ) )
         => aa_set_o_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4229_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B2: nat,A8: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A8)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A8)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),B2) )
               => ( aa_set_nat_o(P,A8)
                 => aa_set_nat_o(P,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B2),A8)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4230_finite__linorder__max__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B2: int,A8: set_int] :
              ( aa_set_int_o(finite_finite_int,A8)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A8)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),B2) )
               => ( aa_set_int_o(P,A8)
                 => aa_set_int_o(P,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B2),A8)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4231_finite__linorder__max__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B2: code_integer,A8: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8)
             => ( ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A8)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),B2) )
               => ( aa_set5643387332196575111eger_o(P,A8)
                 => aa_set5643387332196575111eger_o(P,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,B2),A8)) ) ) )
         => aa_set5643387332196575111eger_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4232_finite__linorder__max__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B2: rat,A8: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A8)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A8)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),B2) )
               => ( aa_set_rat_o(P,A8)
                 => aa_set_rat_o(P,insert_rat(B2,A8)) ) ) )
         => aa_set_rat_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4233_finite__Sup__less__iff,axiom,
    ! [X5: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( ( X5 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,complete_Sup_Sup_int(X5)),A2)
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,X2),A2) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4234_finite__Sup__less__iff,axiom,
    ! [X5: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( ( X5 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(complete_Sup_Sup_nat,X5)),A2)
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),X5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),A2) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4235_finite__Sup__less__iff,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,comple4687483117567863418t_unit(X5)),A2)
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),X5)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X2),A2) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4236_Max__eq__iff,axiom,
    ! [A: set_Code_integer,M4: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ( lattic4901227151466704046nteger(A) = M4 )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M4),A)
            & ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),M4) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4237_Max__eq__iff,axiom,
    ! [A: set_Product_unit,M4: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ( lattic1210392890313158908t_unit(A) = M4 )
        <=> ( aa_set8780210009671629607unit_o(member_Product_unit(M4),A)
            & ! [X2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),M4) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4238_Max__eq__iff,axiom,
    ! [A: set_o,M4: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ( aa_set_o_o(lattic1921953407002678535_Max_o,A)
          <=> (M4) )
        <=> ( aa_set_o_o(member_o((M4)),A)
            & ! [X2: $o] :
                ( aa_set_o_o(member_o((X2)),A)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(M4)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4239_Max__eq__iff,axiom,
    ! [A: set_rat,M4: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ( lattic7630753665789217321ax_rat(A) = M4 )
        <=> ( aa_set_rat_o(member_rat(M4),A)
            & ! [X2: rat] :
                ( aa_set_rat_o(member_rat(X2),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),M4) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4240_Max__eq__iff,axiom,
    ! [A: set_int,M4: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ( lattic8263393255366662781ax_int(A) = M4 )
        <=> ( aa_set_int_o(member_int(M4),A)
            & ! [X2: int] :
                ( aa_set_int_o(member_int(X2),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),M4) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4241_Max__eq__iff,axiom,
    ! [A: set_nat,M4: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A) = M4 )
        <=> ( aa_set_nat_o(member_nat(M4),A)
            & ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M4) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4242_Max__ge__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),lattic4901227151466704046nteger(A))
        <=> ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),X2) ) ) ) ) ).

% Max_ge_iff
tff(fact_4243_Max__ge__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic1210392890313158908t_unit(A))
        <=> ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),X2) ) ) ) ) ).

% Max_ge_iff
tff(fact_4244_Max__ge__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A))
        <=> ? [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
              & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X2)) ) ) ) ) ).

% Max_ge_iff
tff(fact_4245_Max__ge__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic7630753665789217321ax_rat(A))
        <=> ? [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X2) ) ) ) ) ).

% Max_ge_iff
tff(fact_4246_Max__ge__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8263393255366662781ax_int(A))
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X2) ) ) ) ) ).

% Max_ge_iff
tff(fact_4247_Max__ge__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic8265883725875713057ax_nat,A))
        <=> ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X2) ) ) ) ) ).

% Max_ge_iff
tff(fact_4248_eq__Max__iff,axiom,
    ! [A: set_Code_integer,M4: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ( M4 = lattic4901227151466704046nteger(A) )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M4),A)
            & ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),M4) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4249_eq__Max__iff,axiom,
    ! [A: set_Product_unit,M4: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ( M4 = lattic1210392890313158908t_unit(A) )
        <=> ( aa_set8780210009671629607unit_o(member_Product_unit(M4),A)
            & ! [X2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),M4) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4250_eq__Max__iff,axiom,
    ! [A: set_o,M4: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ( (M4)
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,A) )
        <=> ( aa_set_o_o(member_o((M4)),A)
            & ! [X2: $o] :
                ( aa_set_o_o(member_o((X2)),A)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(M4)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4251_eq__Max__iff,axiom,
    ! [A: set_rat,M4: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ( M4 = lattic7630753665789217321ax_rat(A) )
        <=> ( aa_set_rat_o(member_rat(M4),A)
            & ! [X2: rat] :
                ( aa_set_rat_o(member_rat(X2),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),M4) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4252_eq__Max__iff,axiom,
    ! [A: set_int,M4: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ( M4 = lattic8263393255366662781ax_int(A) )
        <=> ( aa_set_int_o(member_int(M4),A)
            & ! [X2: int] :
                ( aa_set_int_o(member_int(X2),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),M4) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4253_eq__Max__iff,axiom,
    ! [A: set_nat,M4: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ( M4 = aa_set_nat_nat(lattic8265883725875713057ax_nat,A) )
        <=> ( aa_set_nat_o(member_nat(M4),A)
            & ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M4) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4254_Max_OboundedE,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A)),X)
         => ! [A10: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A10),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A10),X) ) ) ) ) ).

% Max.boundedE
tff(fact_4255_Max_OboundedE,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1210392890313158908t_unit(A)),X)
         => ! [A10: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A10),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A10),X) ) ) ) ) ).

% Max.boundedE
tff(fact_4256_Max_OboundedE,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A)),(X))
         => ! [A10: $o] :
              ( aa_set_o_o(member_o((A10)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A10)),(X)) ) ) ) ) ).

% Max.boundedE
tff(fact_4257_Max_OboundedE,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A)),X)
         => ! [A10: rat] :
              ( aa_set_rat_o(member_rat(A10),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A10),X) ) ) ) ) ).

% Max.boundedE
tff(fact_4258_Max_OboundedE,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A)),X)
         => ! [A10: int] :
              ( aa_set_int_o(member_int(A10),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A10),X) ) ) ) ) ).

% Max.boundedE
tff(fact_4259_Max_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),X)
         => ! [A10: nat] :
              ( aa_set_nat_o(member_nat(A10),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A10),X) ) ) ) ) ).

% Max.boundedE
tff(fact_4260_Max_OboundedI,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A3),X) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A)),X) ) ) ) ).

% Max.boundedI
tff(fact_4261_Max_OboundedI,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A3),X) )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic1210392890313158908t_unit(A)),X) ) ) ) ).

% Max.boundedI
tff(fact_4262_Max_OboundedI,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(X)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A)),(X)) ) ) ) ).

% Max.boundedI
tff(fact_4263_Max_OboundedI,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),X) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A)),X) ) ) ) ).

% Max.boundedI
tff(fact_4264_Max_OboundedI,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),X) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A)),X) ) ) ) ).

% Max.boundedI
tff(fact_4265_Max_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),X) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),X) ) ) ) ).

% Max.boundedI
tff(fact_4266_Max__insert2,axiom,
    ! [A: set_o,A2: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ! [B2: $o] :
            ( aa_set_o_o(member_o((B2)),A)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B2)),(A2)) )
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
        <=> (A2) ) ) ) ).

% Max_insert2
tff(fact_4267_Max__insert2,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ! [B2: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(B2),A)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,B2),A2) )
       => ( lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) = A2 ) ) ) ).

% Max_insert2
tff(fact_4268_Max__insert2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [B2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(B2),A)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2) )
       => ( lattic4901227151466704046nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),A)) = A2 ) ) ) ).

% Max_insert2
tff(fact_4269_Max__insert2,axiom,
    ! [A: set_rat,A2: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ! [B2: rat] :
            ( aa_set_rat_o(member_rat(B2),A)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) )
       => ( lattic7630753665789217321ax_rat(insert_rat(A2,A)) = A2 ) ) ) ).

% Max_insert2
tff(fact_4270_Max__insert2,axiom,
    ! [A: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [B2: int] :
            ( aa_set_int_o(member_int(B2),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) )
       => ( lattic8263393255366662781ax_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A)) = A2 ) ) ) ).

% Max_insert2
tff(fact_4271_Max__insert2,axiom,
    ! [A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [B2: nat] :
            ( aa_set_nat_o(member_nat(B2),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)) = A2 ) ) ) ).

% Max_insert2
tff(fact_4272_remove__subset,axiom,
    ! [X: product_unit,S: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),S)
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,minus_6452836326544984404t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))),S) ) ).

% remove_subset
tff(fact_4273_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( aa_set_o_o(member_o((X)),S)
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(S,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))),S) ) ).

% remove_subset
tff(fact_4274_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,minus_minus_set_int(S,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))),S) ) ).

% remove_subset
tff(fact_4275_remove__subset,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
     => aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(S,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))),S) ) ).

% remove_subset
tff(fact_4276_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(minus_minus_set_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))),S) ) ).

% remove_subset
tff(fact_4277_cSup__eq__Max,axiom,
    ! [X5: set_int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( ( X5 != bot_bot_set_int )
       => ( complete_Sup_Sup_int(X5) = lattic8263393255366662781ax_int(X5) ) ) ) ).

% cSup_eq_Max
tff(fact_4278_cSup__eq__Max,axiom,
    ! [X5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ( comple4687483117567863418t_unit(X5) = lattic1210392890313158908t_unit(X5) ) ) ) ).

% cSup_eq_Max
tff(fact_4279_cSup__eq__Max,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( ( X5 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X5) ) ) ) ).

% cSup_eq_Max
tff(fact_4280_Max__Sup,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic1210392890313158908t_unit(A) = comple4687483117567863418t_unit(A) ) ) ) ).

% Max_Sup
tff(fact_4281_disjoint__alt__simp3,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,minus_6452836326544984404t_unit(A,B4)),A)
    <=> ( inf_in4660618365625256667t_unit(A,B4) != bot_bo3957492148770167129t_unit ) ) ).

% disjoint_alt_simp3
tff(fact_4282_disjoint__alt__simp3,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,minus_minus_set_o(A,B4)),A)
    <=> ( inf_inf_set_o(A,B4) != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
tff(fact_4283_disjoint__alt__simp3,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,minus_minus_set_int(A,B4)),A)
    <=> ( inf_inf_set_int(A,B4) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_4284_disjoint__alt__simp3,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(minus_6456595905716665654_nat_o(A,B4)),A)
    <=> ( inf_in3376754634429691197_nat_o(A,B4) != bot_bo8632255151191677115_nat_o ) ) ).

% disjoint_alt_simp3
tff(fact_4285_disjoint__alt__simp3,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_4286_range__eq__singletonD,axiom,
    ! [F: fun_Co8499238673658658103nteger,A2: code_integer,X: code_integer] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),bot_bo3990330152332043303nteger) )
     => ( aa_Cod1783974652533396728nteger(F,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4287_range__eq__singletonD,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A2: product_unit,X: product_unit] :
      ( ( image_405062704495631173t_unit(F,top_to1996260823553986621t_unit) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit) )
     => ( aa_Pro8294289767544744956t_unit(F,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4288_range__eq__singletonD,axiom,
    ! [F: fun_Product_unit_o,A2: $o,X: product_unit] :
      ( ( image_Product_unit_o2(F,top_to1996260823553986621t_unit) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o) )
     => ( aa_Product_unit_o(F,X)
      <=> (A2) ) ) ).

% range_eq_singletonD
tff(fact_4289_range__eq__singletonD,axiom,
    ! [F: fun_Product_unit_nat,A2: nat,X: product_unit] :
      ( ( image_875570014554754200it_nat(F,top_to1996260823553986621t_unit) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat) )
     => ( aa_Product_unit_nat(F,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4290_range__eq__singletonD,axiom,
    ! [F: fun_Product_unit_int,A2: int,X: product_unit] :
      ( ( image_873079544045703924it_int(F,top_to1996260823553986621t_unit) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int) )
     => ( aa_Product_unit_int(F,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4291_range__eq__singletonD,axiom,
    ! [F: fun_o_Product_unit,A2: product_unit,X: $o] :
      ( ( image_o_Product_unit2(F,top_top_set_o) = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit) )
     => ( aa_o_Product_unit(F,(X)) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4292_range__eq__singletonD,axiom,
    ! [F: fun_o_o,A2: $o,X: $o] :
      ( ( image_o_o2(F,top_top_set_o) = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o) )
     => ( aa_o_o(F,(X))
      <=> (A2) ) ) ).

% range_eq_singletonD
tff(fact_4293_range__eq__singletonD,axiom,
    ! [F: fun_o_nat,A2: nat,X: $o] :
      ( ( image_o_nat2(F,top_top_set_o) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat) )
     => ( aa_o_nat(F,(X)) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4294_range__eq__singletonD,axiom,
    ! [F: fun_o_int,A2: int,X: $o] :
      ( ( image_o_int2(F,top_top_set_o) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int) )
     => ( aa_o_int(F,(X)) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4295_range__eq__singletonD,axiom,
    ! [F: fun_nat_Code_integer,A2: code_integer,X: nat] :
      ( ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),top_top_set_nat) = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),bot_bo3990330152332043303nteger) )
     => ( aa_nat_Code_integer(F,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4296_less__cSup__iff,axiom,
    ! [X5: set_int,Y: int] :
      ( ( X5 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,X5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),complete_Sup_Sup_int(X5))
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),X5)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),X2) ) ) ) ) ).

% less_cSup_iff
tff(fact_4297_less__cSup__iff,axiom,
    ! [X5: set_nat,Y: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,X5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_set_nat_nat(complete_Sup_Sup_nat,X5))
        <=> ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),X5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X2) ) ) ) ) ).

% less_cSup_iff
tff(fact_4298_less__cSup__iff,axiom,
    ! [X5: set_Product_unit,Y: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,X5)
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Y),comple4687483117567863418t_unit(X5))
        <=> ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),X5)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Y),X2) ) ) ) ) ).

% less_cSup_iff
tff(fact_4299_underS__Field3,axiom,
    ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
      ( ( field_Product_unit(R2) != bot_bo3957492148770167129t_unit )
     => aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le8056459307392131481t_unit,aa_Pro234974305641502940t_unit(order_5809128134054296019t_unit(R2),A2)),field_Product_unit(R2)) ) ).

% underS_Field3
tff(fact_4300_underS__Field3,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o] :
      ( ( field_o(R2) != bot_bot_set_o )
     => aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_set_o,aa_o_set_o(order_underS_o(R2),(A2))),field_o(R2)) ) ).

% underS_Field3
tff(fact_4301_underS__Field3,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( ( field_nat(R2) != bot_bot_set_nat )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(order_underS_nat(R2),A2)),field_nat(R2)) ) ).

% underS_Field3
tff(fact_4302_underS__Field3,axiom,
    ! [R2: set_Pr958786334691620121nt_int,A2: int] :
      ( ( field_int(R2) != bot_bot_set_int )
     => aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_set_int,aa_int_set_int(order_underS_int(R2),A2)),field_int(R2)) ) ).

% underS_Field3
tff(fact_4303_underS__Field3,axiom,
    ! [R2: set_Pr2890355626087285447_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( field_48612661033931452_nat_o(R2) != bot_bo8632255151191677115_nat_o )
     => aa_set5091322549519198405at_o_o(ord_le1274849205035319547_nat_o(order_7714984474677380533_nat_o(R2,A2)),field_48612661033931452_nat_o(R2)) ) ).

% underS_Field3
tff(fact_4304_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Code_integer,F: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ~ ? [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Code_integer_assn(F,X4)),aa_Code_integer_assn(F,lattic4682378690128378784r_assn(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4305_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Product_unit,F: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( ( S != bot_bo3957492148770167129t_unit )
       => ~ ? [X4: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X4),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_Product_unit_assn(F,X4)),aa_Product_unit_assn(F,lattic4376901495776413404t_assn(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4306_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_o_assn(F,(X4))),aa_o_assn(F,aa_set_o_o(lattic7708394068118247271o_assn(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4307_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_nat_assn(F,X4)),aa_nat_assn(F,aa_set_nat_nat(lattic2187264146484958483t_assn(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4308_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( aa_set_int_o(member_int(X4),S)
              & aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_int_assn(F,X4)),aa_int_assn(F,lattic3162799611691135855t_assn(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4309_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ~ ? [X4: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X4),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X4)),aa_Code_integer_nat(F,lattic6074936555552774728er_nat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4310_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Product_unit,F: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( ( S != bot_bo3957492148770167129t_unit )
       => ~ ? [X4: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X4),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Product_unit_nat(F,X4)),aa_Product_unit_nat(F,lattic1360832392136924548it_nat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4311_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( aa_set_o_o(member_o((X4)),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X4))),aa_o_nat(F,aa_set_o_o(lattic2775856028456453135_o_nat(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4312_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X4)),aa_nat_nat(F,aa_set_nat_nat(lattic7446932960582359483at_nat(F),S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4313_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( aa_set_int_o(member_int(X4),S)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(F,lattic8446286672483414039nt_nat(F,S))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4314_Sup__fold__sup,axiom,
    ! [A: set_set_Product_unit] :
      ( finite1772178364199683094t_unit(A)
     => ( comple3349607820504768346t_unit(A) = finite4243028863286625800t_unit(sup_su793286257634532545t_unit,bot_bo3957492148770167129t_unit,A) ) ) ).

% Sup_fold_sup
tff(fact_4315_Sup__fold__sup,axiom,
    ! [A: set_set_o] :
      ( finite_finite_set_o(A)
     => ( comple90263536869209701_set_o(A) = finite4337638375924247368_set_o(sup_sup_set_o,bot_bot_set_o,A) ) ) ).

% Sup_fold_sup
tff(fact_4316_Sup__fold__sup,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( comple7399068483239264473et_nat(A) = finite677925301803182934et_nat(sup_sup_set_nat,bot_bot_set_nat,A) ) ) ).

% Sup_fold_sup
tff(fact_4317_Sup__fold__sup,axiom,
    ! [A: set_set_int] :
      ( finite6197958912794628473et_int(A)
     => ( comple3221217463730067765et_int(A) = finite2508883933289561102et_int(sup_sup_set_int,bot_bot_set_int,A) ) ) ).

% Sup_fold_sup
tff(fact_4318_Sup__fold__sup,axiom,
    ! [A: set_se6690607124073039791_nat_o] :
      ( finite9028524065462436600_nat_o(A)
     => ( comple4665849098253521980_nat_o(A) = finite4256660883244961672_nat_o(sup_su2896084399781877795_nat_o,bot_bo8632255151191677115_nat_o,A) ) ) ).

% Sup_fold_sup
tff(fact_4319_Sup__fold__sup,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( comple4687483117567863418t_unit(A) = finite8834431295148555784t_unit(sup_sup_Product_unit,bot_bot_Product_unit,A) ) ) ).

% Sup_fold_sup
tff(fact_4320_SUP__fold__sup,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A)) = finite6602901031542678131t_unit(comp_P7295696529757372863it_nat(sup_sup_Product_unit,F),bot_bot_Product_unit,A) ) ) ).

% SUP_fold_sup
tff(fact_4321_SUP__fold__sup,axiom,
    ! [A: set_Code_integer,F: fun_Co1365676809456677961t_unit] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( comple4687483117567863418t_unit(image_7182656606676592809t_unit(F,A)) = finite2281032298429728806t_unit(comp_P5585084804395868748nteger(sup_sup_Product_unit,F),bot_bot_Product_unit,A) ) ) ).

% SUP_fold_sup
tff(fact_4322_SUP__fold__sup,axiom,
    ! [A: set_nat,F: fun_na896991816813928310t_unit] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( comple3349607820504768346t_unit(image_4824033694279209942t_unit(F,A)) = finite9145781422887121747t_unit(comp_s10149624282781343it_nat(sup_su793286257634532545t_unit,F),bot_bo3957492148770167129t_unit,A) ) ) ).

% SUP_fold_sup
tff(fact_4323_SUP__fold__sup,axiom,
    ! [A: set_Code_integer,F: fun_Co7675133579868055849t_unit] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( comple3349607820504768346t_unit(image_5870443399019143561t_unit(F,A)) = finite3687622995246107910t_unit(comp_s8366014661416236844nteger(sup_su793286257634532545t_unit,F),bot_bo3957492148770167129t_unit,A) ) ) ).

% SUP_fold_sup
tff(fact_4324_SUP__fold__sup,axiom,
    ! [A: set_nat,F: fun_nat_set_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( comple90263536869209701_set_o(image_nat_set_o(F,A)) = finite3217087857726763998_set_o(comp_s8800259901120174058_o_nat(sup_sup_set_o,F),bot_bot_set_o,A) ) ) ).

% SUP_fold_sup
tff(fact_4325_SUP__fold__sup,axiom,
    ! [A: set_Code_integer,F: fun_Co3841156504242925748_set_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( comple90263536869209701_set_o(image_1867029960091478804_set_o(F,A)) = finite1928688952044918289_set_o(comp_s6642683956436225143nteger(sup_sup_set_o,F),bot_bot_set_o,A) ) ) ).

% SUP_fold_sup
tff(fact_4326_SUP__fold__sup,axiom,
    ! [A: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( comple7399068483239264473et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A)) = aa_set_nat_set_nat(finite5529483035118572448et_nat(comp_s5871518449905772612at_nat(sup_sup_set_nat,F),bot_bot_set_nat),A) ) ) ).

% SUP_fold_sup
tff(fact_4327_SUP__fold__sup,axiom,
    ! [A: set_Code_integer,F: fun_Co8307959106670069728et_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( comple7399068483239264473et_nat(image_8829277216472192490et_nat(F,A)) = finite1590314701948642349et_nat(comp_s9048578035094795473nteger(sup_sup_set_nat,F),bot_bot_set_nat,A) ) ) ).

% SUP_fold_sup
tff(fact_4328_SUP__fold__sup,axiom,
    ! [A: set_nat,F: fun_nat_set_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( comple3221217463730067765et_int(image_nat_set_int(F,A)) = finite1351632015609375740et_int(comp_s2690021802452441944nt_nat(sup_sup_set_int,F),bot_bot_set_int,A) ) ) ).

% SUP_fold_sup
tff(fact_4329_SUP__fold__sup,axiom,
    ! [A: set_Code_integer,F: fun_Co5817875961738342972et_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( comple3221217463730067765et_int(image_4651426196962995782et_int(F,A)) = finite6635835719294221449et_int(comp_s4085161575516116965nteger(sup_sup_set_int,F),bot_bot_set_int,A) ) ) ).

% SUP_fold_sup
tff(fact_4330_finite__subset__Union__chain,axiom,
    ! [A: set_Code_integer,B8: set_set_Code_integer,A11: set_set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),comple739944243200306918nteger(B8))
       => ( ( B8 != bot_bo9204850693367297629nteger )
         => ( pred_c5891848916688490280nteger(A11,ord_le1307284697595431911nteger,B8)
           => ~ ! [B5: set_Code_integer] :
                  ( member1834298987299537450nteger(B5,B8)
                 => ~ aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B5) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_4331_finite__subset__Union__chain,axiom,
    ! [A: set_nat,B8: set_set_nat,A11: set_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),comple7399068483239264473et_nat(B8))
       => ( ( B8 != bot_bot_set_set_nat )
         => ( pred_chain_set_nat(A11,ord_less_set_nat,B8)
           => ~ ! [B5: set_nat] :
                  ( member_set_nat(B5,B8)
                 => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B5) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_4332_merge__true__star,axiom,
    aa_assn_assn(times_times_assn(top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_4333_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_4334_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_4335_mod__true,axiom,
    ! [Ha: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),Ha)
    <=> aa_Pro653082453696054616_nat_o(in_range,Ha) ) ).

% mod_true
tff(fact_4336_fun__comp__eq__conv,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger,Fg: fun_Co1561183069634336733nteger] :
      ( ( comp_C7848664941760151970nteger(F,G) = Fg )
    <=> ! [X2: code_integer] : aa_Cod1978467581589928094nteger(F,aa_Cod1783974652533396728nteger(G,X2)) = aa_Cod1978467581589928094nteger(Fg,X2) ) ).

% fun_comp_eq_conv
tff(fact_4337_fun__comp__eq__conv,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Fg: fun_Co1561183069634336733nteger] :
      ( ( comp_f6739083541385432444nteger(F,G) = Fg )
    <=> ! [X2: code_integer] : aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X2)) = aa_Cod1978467581589928094nteger(Fg,X2) ) ).

% fun_comp_eq_conv
tff(fact_4338_comp__cong__right,axiom,
    ! [X: fun_Co8499238673658658103nteger,Y: fun_Co8499238673658658103nteger,F: fun_Co1561183069634336733nteger] :
      ( ( X = Y )
     => ( comp_C7848664941760151970nteger(F,X) = comp_C7848664941760151970nteger(F,Y) ) ) ).

% comp_cong_right
tff(fact_4339_comp__cong__right,axiom,
    ! [X: fun_Co1981900195452921617nteger,Y: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger] :
      ( ( X = Y )
     => ( comp_f6739083541385432444nteger(F,X) = comp_f6739083541385432444nteger(F,Y) ) ) ).

% comp_cong_right
tff(fact_4340_comp__cong__left,axiom,
    ! [X: fun_Co1561183069634336733nteger,Y: fun_Co1561183069634336733nteger,F: fun_Co8499238673658658103nteger] :
      ( ( X = Y )
     => ( comp_C7848664941760151970nteger(X,F) = comp_C7848664941760151970nteger(Y,F) ) ) ).

% comp_cong_left
tff(fact_4341_comp__cong__left,axiom,
    ! [X: fun_fu4800041728187187127nteger,Y: fun_fu4800041728187187127nteger,F: fun_Co1981900195452921617nteger] :
      ( ( X = Y )
     => ( comp_f6739083541385432444nteger(X,F) = comp_f6739083541385432444nteger(Y,F) ) ) ).

% comp_cong_left
tff(fact_4342_comp__apply__eq,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger,X: code_integer,Ha: fun_Co1561183069634336733nteger,K3: fun_Co8499238673658658103nteger] :
      ( ( aa_Cod1978467581589928094nteger(F,aa_Cod1783974652533396728nteger(G,X)) = aa_Cod1978467581589928094nteger(Ha,aa_Cod1783974652533396728nteger(K3,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(F,G),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(Ha,K3),X) ) ) ).

% comp_apply_eq
tff(fact_4343_comp__apply__eq,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger,X: code_integer,Ha: fun_fu4800041728187187127nteger,K3: fun_Co1981900195452921617nteger] :
      ( ( aa_Cod1978467581589928094nteger(F,aa_Cod1783974652533396728nteger(G,X)) = aa_fun2509385170507305080nteger(Ha,aa_Cod4161879620478718930nteger(K3,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(F,G),X) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(Ha,K3),X) ) ) ).

% comp_apply_eq
tff(fact_4344_comp__apply__eq,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,X: code_integer,Ha: fun_Co1561183069634336733nteger,K3: fun_Co8499238673658658103nteger] :
      ( ( aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X)) = aa_Cod1978467581589928094nteger(Ha,aa_Cod1783974652533396728nteger(K3,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),X) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(Ha,K3),X) ) ) ).

% comp_apply_eq
tff(fact_4345_comp__apply__eq,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,X: code_integer,Ha: fun_fu4800041728187187127nteger,K3: fun_Co1981900195452921617nteger] :
      ( ( aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X)) = aa_fun2509385170507305080nteger(Ha,aa_Cod4161879620478718930nteger(K3,X)) )
     => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(F,G),X) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(Ha,K3),X) ) ) ).

% comp_apply_eq
tff(fact_4346_type__copy__set__map0,axiom,
    ! [Rep: fun_fu9117715662874197469nteger,Abs: fun_Co1561183069634336733nteger,S: fun_Co8307959106670069728et_nat,M: fun_Co8499238673658658103nteger,F: fun_nat_nat,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(type_d8201345049721092750nteger(Rep,Abs),top_to4645266643341252675nteger)
     => ( ( comp_C6537567860429791781nteger(S,M) = comp_s4605850120322998606nteger(image_nat_nat2(F),S4) )
       => ( comp_f6733788873650792907nteger(comp_C958128367788360779nteger(S,Rep),comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Abs,M),G)) = comp_s4605850120322998606nteger(image_nat_nat2(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4347_type__copy__set__map0,axiom,
    ! [Rep: fun_fu9117715662874197469nteger,Abs: fun_Co1561183069634336733nteger,S: fun_Co744628173305925485nteger,M: fun_Co8499238673658658103nteger,F: fun_Co8499238673658658103nteger,S4: fun_Co744628173305925485nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(type_d8201345049721092750nteger(Rep,Abs),top_to4645266643341252675nteger)
     => ( ( comp_C8186931911283765298nteger(S,M) = comp_s956550612328592744nteger(image_4470545334726330049nteger(F),S4) )
       => ( comp_f8646512988615909336nteger(comp_C8996093786508476632nteger(S,Rep),comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Abs,M),G)) = comp_s956550612328592744nteger(image_4470545334726330049nteger(F),comp_C8186931911283765298nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4348_type__copy__set__map0,axiom,
    ! [Rep: fun_fu9117715662874197469nteger,Abs: fun_Co1561183069634336733nteger,S: fun_Co744628173305925485nteger,M: fun_Co8499238673658658103nteger,F: fun_nat_Code_integer,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(type_d8201345049721092750nteger(Rep,Abs),top_to4645266643341252675nteger)
     => ( ( comp_C8186931911283765298nteger(S,M) = comp_s8292494440407567195nteger(image_1215581382706833972nteger(F),S4) )
       => ( comp_f8646512988615909336nteger(comp_C8996093786508476632nteger(S,Rep),comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(Abs,M),G)) = comp_s8292494440407567195nteger(image_1215581382706833972nteger(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4349_type__copy__set__map0,axiom,
    ! [Rep: fun_fu4478038021464438563t_unit,Abs: fun_Pr3800342156844704675nteger,S: fun_Pr1628945989756059814et_nat,M: fun_Co1365676809456677961t_unit,F: fun_nat_nat,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set8780210009671629607unit_o(type_d3971210459120089884t_unit(Rep,Abs),top_to1996260823553986621t_unit)
     => ( ( comp_P5878433733084393551nteger(S,M) = comp_s4605850120322998606nteger(image_nat_nat2(F),S4) )
       => ( comp_f6733788873650792907nteger(comp_P7160832843261671285nteger(S,Rep),comp_C7848664941760151970nteger(comp_P4827997380378686668nteger(Abs,M),G)) = comp_s4605850120322998606nteger(image_nat_nat2(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4350_type__copy__set__map0,axiom,
    ! [Rep: fun_fu4478038021464438563t_unit,Abs: fun_Pr3800342156844704675nteger,S: fun_Pr1789781366282006579nteger,M: fun_Co1365676809456677961t_unit,F: fun_Co8499238673658658103nteger,S4: fun_Co744628173305925485nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set8780210009671629607unit_o(type_d3971210459120089884t_unit(Rep,Abs),top_to1996260823553986621t_unit)
     => ( ( comp_P4767411901762164316nteger(S,M) = comp_s956550612328592744nteger(image_4470545334726330049nteger(F),S4) )
       => ( comp_f8646512988615909336nteger(comp_P1748418806231731202nteger(S,Rep),comp_C7848664941760151970nteger(comp_P4827997380378686668nteger(Abs,M),G)) = comp_s956550612328592744nteger(image_4470545334726330049nteger(F),comp_C8186931911283765298nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4351_type__copy__set__map0,axiom,
    ! [Rep: fun_fu4478038021464438563t_unit,Abs: fun_Pr3800342156844704675nteger,S: fun_Pr1789781366282006579nteger,M: fun_Co1365676809456677961t_unit,F: fun_nat_Code_integer,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set8780210009671629607unit_o(type_d3971210459120089884t_unit(Rep,Abs),top_to1996260823553986621t_unit)
     => ( ( comp_P4767411901762164316nteger(S,M) = comp_s8292494440407567195nteger(image_1215581382706833972nteger(F),S4) )
       => ( comp_f8646512988615909336nteger(comp_P1748418806231731202nteger(S,Rep),comp_C7848664941760151970nteger(comp_P4827997380378686668nteger(Abs,M),G)) = comp_s8292494440407567195nteger(image_1215581382706833972nteger(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4352_type__copy__set__map0,axiom,
    ! [Rep: fun_fu2958438437739016110eger_o,Abs: fun_o_1027063593290716974nteger,S: fun_o_set_nat,M: fun_Code_integer_o,F: fun_nat_nat,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set_o_o(type_d7787643095087435559eger_o(Rep,Abs),top_top_set_o)
     => ( ( comp_o172754640513974618nteger(S,M) = comp_s4605850120322998606nteger(image_nat_nat2(F),S4) )
       => ( comp_f6733788873650792907nteger(comp_o3823523404893087232nteger(S,Rep),comp_C7848664941760151970nteger(comp_o1490687942010102615nteger(Abs,M),G)) = comp_s4605850120322998606nteger(image_nat_nat2(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4353_type__copy__set__map0,axiom,
    ! [Rep: fun_fu2958438437739016110eger_o,Abs: fun_o_1027063593290716974nteger,S: fun_o_5240169576699714110nteger,M: fun_Code_integer_o,F: fun_Co8499238673658658103nteger,S4: fun_Co744628173305925485nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set_o_o(type_d7787643095087435559eger_o(Rep,Abs),top_top_set_o)
     => ( ( comp_o4291842030903409511nteger(S,M) = comp_s956550612328592744nteger(image_4470545334726330049nteger(F),S4) )
       => ( comp_f8646512988615909336nteger(comp_o3033810377234304653nteger(S,Rep),comp_C7848664941760151970nteger(comp_o1490687942010102615nteger(Abs,M),G)) = comp_s956550612328592744nteger(image_4470545334726330049nteger(F),comp_C8186931911283765298nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4354_type__copy__set__map0,axiom,
    ! [Rep: fun_fu2958438437739016110eger_o,Abs: fun_o_1027063593290716974nteger,S: fun_o_5240169576699714110nteger,M: fun_Code_integer_o,F: fun_nat_Code_integer,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set_o_o(type_d7787643095087435559eger_o(Rep,Abs),top_top_set_o)
     => ( ( comp_o4291842030903409511nteger(S,M) = comp_s8292494440407567195nteger(image_1215581382706833972nteger(F),S4) )
       => ( comp_f8646512988615909336nteger(comp_o3033810377234304653nteger(S,Rep),comp_C7848664941760151970nteger(comp_o1490687942010102615nteger(Abs,M),G)) = comp_s8292494440407567195nteger(image_1215581382706833972nteger(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4355_type__copy__set__map0,axiom,
    ! [Rep: fun_fu7638885401485650896er_nat,Abs: fun_na7028214812545403088nteger,S: fun_nat_set_nat,M: fun_Code_integer_nat,F: fun_nat_nat,S4: fun_Co8307959106670069728et_nat,G: fun_Co8499238673658658103nteger] :
      ( aa_set_nat_o(type_d922709932879716865er_nat(Rep,Abs),top_top_set_nat)
     => ( ( comp_n5963683855531483544nteger(S,M) = comp_s4605850120322998606nteger(image_nat_nat2(F),S4) )
       => ( comp_f6733788873650792907nteger(comp_n2709255890054698302nteger(S,Rep),comp_C7848664941760151970nteger(comp_n376420427171713685nteger(Abs,M),G)) = comp_s4605850120322998606nteger(image_nat_nat2(F),comp_C6537567860429791781nteger(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4356_top__empty__eq,axiom,
    ! [X4: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(top_to79712303169546844at_o_o,X4)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),top_to9044565752620676511_nat_o) ) ).

% top_empty_eq
tff(fact_4357_top__empty__eq,axiom,
    ! [X4: product_unit] :
      ( aa_Product_unit_o(top_to7743374099656719934unit_o,X4)
    <=> aa_set8780210009671629607unit_o(member_Product_unit(X4),top_to1996260823553986621t_unit) ) ).

% top_empty_eq
tff(fact_4358_top__empty__eq,axiom,
    ! [X4: $o] :
      ( aa_o_o(top_top_fun_o_o,(X4))
    <=> aa_set_o_o(member_o((X4)),top_top_set_o) ) ).

% top_empty_eq
tff(fact_4359_top__empty__eq,axiom,
    ! [X4: nat] :
      ( aa_nat_o(top_top_fun_nat_o,X4)
    <=> aa_set_nat_o(member_nat(X4),top_top_set_nat) ) ).

% top_empty_eq
tff(fact_4360_top__empty__eq,axiom,
    ! [X4: int] :
      ( aa_int_o(top_top_fun_int_o,X4)
    <=> aa_set_int_o(member_int(X4),top_top_set_int) ) ).

% top_empty_eq
tff(fact_4361_top__set__def,axiom,
    top_to9044565752620676511_nat_o = aa_fun1518904406198085989_nat_o(collec6407654713642994138_nat_o,top_to79712303169546844at_o_o) ).

% top_set_def
tff(fact_4362_top__set__def,axiom,
    top_to1996260823553986621t_unit = aa_fun8337552879598298401t_unit(collect_Product_unit,top_to7743374099656719934unit_o) ).

% top_set_def
tff(fact_4363_top__set__def,axiom,
    top_top_set_o = aa_fun_o_o_set_o(collect_o,top_top_fun_o_o) ).

% top_set_def
tff(fact_4364_top__set__def,axiom,
    top_top_set_nat = aa_fun_nat_o_set_nat(collect_nat,top_top_fun_nat_o) ).

% top_set_def
tff(fact_4365_top__set__def,axiom,
    top_top_set_int = aa_fun_int_o_set_int(collect_int,top_top_fun_int_o) ).

% top_set_def
tff(fact_4366_pred__on_Ochain__empty,axiom,
    ! [A: set_Product_unit,P: fun_Pr1520749766685617582unit_o] : aa_set8780210009671629607unit_o(pred_c6915491785556283576t_unit(A,P),bot_bo3957492148770167129t_unit) ).

% pred_on.chain_empty
tff(fact_4367_pred__on_Ochain__empty,axiom,
    ! [A: set_o,P: fun_o_fun_o_o] : aa_set_o_o(pred_chain_o(A,P),bot_bot_set_o) ).

% pred_on.chain_empty
tff(fact_4368_pred__on_Ochain__empty,axiom,
    ! [A: set_nat,P: fun_nat_fun_nat_o] : aa_set_nat_o(pred_chain_nat(A,P),bot_bot_set_nat) ).

% pred_on.chain_empty
tff(fact_4369_pred__on_Ochain__empty,axiom,
    ! [A: set_int,P: fun_int_fun_int_o] : aa_set_int_o(pred_chain_int(A,P),bot_bot_set_int) ).

% pred_on.chain_empty
tff(fact_4370_pred__on_Ochain__empty,axiom,
    ! [A: set_fu3555266933746557007_nat_o,P: fun_fu4164567611050918574at_o_o] : aa_set5091322549519198405at_o_o(pred_c5982856694712694810_nat_o(A,P),bot_bo8632255151191677115_nat_o) ).

% pred_on.chain_empty
tff(fact_4371_surj__fun__eq,axiom,
    ! [F: fun_Co1981900195452921617nteger,X5: set_Code_integer,G1: fun_fu4800041728187187127nteger,G2: fun_fu4800041728187187127nteger] :
      ( ( image_2716445779319712667nteger(F,X5) = top_to805621317456759325nteger )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X5)
           => ( aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(G1,F),X3) = aa_Cod1978467581589928094nteger(comp_f6739083541385432444nteger(G2,F),X3) ) )
       => ( G1 = G2 ) ) ) ).

% surj_fun_eq
tff(fact_4372_surj__fun__eq,axiom,
    ! [F: fun_Co8499238673658658103nteger,X5: set_Code_integer,G1: fun_Co1561183069634336733nteger,G2: fun_Co1561183069634336733nteger] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),X5) = top_to4645266643341252675nteger )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X5)
           => ( aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(G1,F),X3) = aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(G2,F),X3) ) )
       => ( G1 = G2 ) ) ) ).

% surj_fun_eq
tff(fact_4373_type__copy__map__comp0,axiom,
    ! [Rep: fun_Co8499238673658658103nteger,Abs: fun_Co8499238673658658103nteger,M: fun_Co8499238673658658103nteger,M1: fun_Co8499238673658658103nteger,M22: fun_Co8499238673658658103nteger,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(type_d4796611474492816872nteger(Rep,Abs),top_to4645266643341252675nteger)
     => ( ( M = comp_C7449957260575251196nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M1),Rep),comp_C7449957260575251196nteger(comp_C7449957260575251196nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4374_type__copy__map__comp0,axiom,
    ! [Rep: fun_Co1365676809456677961t_unit,Abs: fun_Pr1820225556744648189nteger,M: fun_Co8499238673658658103nteger,M1: fun_Pr1820225556744648189nteger,M22: fun_Co1365676809456677961t_unit,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set8780210009671629607unit_o(type_d9006111508436661314t_unit(Rep,Abs),top_to1996260823553986621t_unit)
     => ( ( M = comp_P6790823133229852966nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7848664941760151970nteger(comp_P4827997380378686668nteger(comp_C5159084050497807624t_unit(F,M1),Rep),comp_C7449957260575251196nteger(comp_P6790823133229852966nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4375_type__copy__map__comp0,axiom,
    ! [Rep: fun_Code_integer_o,Abs: fun_o_Code_integer,M: fun_Co8499238673658658103nteger,M1: fun_o_Code_integer,M22: fun_Code_integer_o,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set_o_o(type_d6812736842639505741eger_o(Rep,Abs),top_top_set_o)
     => ( ( M = comp_o1085144040659434033nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7848664941760151970nteger(comp_o1490687942010102615nteger(comp_C4761632023083944723eger_o(F,M1),Rep),comp_C7449957260575251196nteger(comp_o1085144040659434033nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4376_type__copy__map__comp0,axiom,
    ! [Rep: fun_Code_integer_nat,Abs: fun_nat_Code_integer,M: fun_Co8499238673658658103nteger,M1: fun_nat_Code_integer,M22: fun_Code_integer_nat,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set_nat_o(type_d8368584451094728027er_nat(Rep,Abs),top_top_set_nat)
     => ( ( M = comp_n6876073255676942959nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7848664941760151970nteger(comp_n376420427171713685nteger(comp_C5593919495898904853er_nat(F,M1),Rep),comp_C7449957260575251196nteger(comp_n6876073255676942959nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4377_type__copy__map__comp0,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,M: fun_Co8499238673658658103nteger,M1: fun_int_Code_integer,M22: fun_Code_integer_int,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( ( M = comp_i6014793192326455499nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7848664941760151970nteger(comp_i6210154785576167153nteger(comp_C5591429025389854577er_int(F,M1),Rep),comp_C7449957260575251196nteger(comp_i6014793192326455499nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4378_type__copy__map__comp0,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,M: fun_Co8499238673658658103nteger,M1: fun_o_Code_integer,M22: fun_Code_integer_o,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( ( M = comp_o1085144040659434033nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_P4827997380378686668nteger(comp_o8595797847074315155t_unit(comp_C4761632023083944723eger_o(F,M1),Rep),comp_C8146212659816229034nteger(comp_o6635994240230588853nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4379_type__copy__map__comp0,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,M: fun_Co8499238673658658103nteger,M1: fun_nat_Code_integer,M22: fun_Code_integer_nat,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),top_top_set_nat)
     => ( ( M = comp_n6876073255676942959nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7951028950906698989nteger(comp_n8423387615783207008atural(comp_C5593919495898904853er_nat(F,M1),Rep),comp_C1891057325546811591nteger(comp_n1317173320648503354nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4380_type__copy__map__comp0,axiom,
    ! [Rep: fun_Co1981900195452921617nteger,Abs: fun_fu5689865433563736081nteger,M: fun_Co8499238673658658103nteger,M1: fun_fu5689865433563736081nteger,M22: fun_Co1981900195452921617nteger,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( type_d7606044679298358466nteger(Rep,Abs,top_to805621317456759325nteger)
     => ( ( M = comp_f2642788907534428630nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(comp_C7827331287018003580nteger(F,M1),Rep),comp_C7449957260575251196nteger(comp_f2642788907534428630nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4381_type__copy__map__comp0,axiom,
    ! [Rep: fun_fu5689865433563736081nteger,Abs: fun_Co1981900195452921617nteger,M: fun_Co8499238673658658103nteger,M1: fun_Co8499238673658658103nteger,M22: fun_Co8499238673658658103nteger,F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(type_d6382125070450342082nteger(Rep,Abs),top_to4645266643341252675nteger)
     => ( ( M = comp_C7449957260575251196nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(F,M),G) = comp_f6739083541385432444nteger(comp_C7827331287018003580nteger(comp_C7848664941760151970nteger(F,M1),Rep),comp_C7233611567195229654nteger(comp_C7233611567195229654nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4382_type__copy__map__comp0,axiom,
    ! [Rep: fun_Co8499238673658658103nteger,Abs: fun_Co8499238673658658103nteger,M: fun_Co1981900195452921617nteger,M1: fun_Co1981900195452921617nteger,M22: fun_Co8499238673658658103nteger,F: fun_fu4800041728187187127nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(type_d4796611474492816872nteger(Rep,Abs),top_to4645266643341252675nteger)
     => ( ( M = comp_C7233611567195229654nteger(M1,M22) )
       => ( comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(F,M),G) = comp_C7848664941760151970nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(F,M1),Rep),comp_C7449957260575251196nteger(comp_C7449957260575251196nteger(Abs,M22),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4383_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Code_integer_int,Abs2: fun_int_Code_integer,Rep3: fun_Code_integer_int,Abs3: fun_int_Code_integer,M: fun_int_int,M1: fun_int_int,M22: fun_int_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_int_o(type_d8366093980585677751er_int(Rep2,Abs2),top_top_set_int)
       => ( aa_set_int_o(type_d8366093980585677751er_int(Rep3,Abs3),top_top_set_int)
         => ( ( comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs2,M),Rep3) = comp_C7449957260575251196nteger(comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs2,M1),Rep),comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs,M22),Rep3)) )
           => ( comp_int_int_int(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4384_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Code_integer_int,Abs2: fun_int_Code_integer,Rep3: fun_Product_unit_o,Abs3: fun_o_Product_unit,M: fun_o_int,M1: fun_int_int,M22: fun_o_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_int_o(type_d8366093980585677751er_int(Rep2,Abs2),top_top_set_int)
       => ( aa_set_o_o(type_d6188575255521822967unit_o(Rep3,Abs3),top_top_set_o)
         => ( ( comp_o780160699970281785t_unit(comp_i5286552343147518890eger_o(Abs2,M),Rep3) = comp_C2290379119555921966t_unit(comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs2,M1),Rep),comp_o780160699970281785t_unit(comp_i5286552343147518890eger_o(Abs,M22),Rep3)) )
           => ( comp_int_int_o(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4385_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Code_integer_int,Abs2: fun_int_Code_integer,Rep3: fun_Code_natural_nat,Abs3: fun_nat_Code_natural,M: fun_nat_int,M1: fun_int_int,M22: fun_nat_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_int_o(type_d8366093980585677751er_int(Rep2,Abs2),top_top_set_int)
       => ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( comp_n5699668407433660474atural(comp_i1588355021709917246er_nat(Abs2,M),Rep3) = comp_C6273552412331968711atural(comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs2,M1),Rep),comp_n5699668407433660474atural(comp_i1588355021709917246er_nat(Abs,M22),Rep3)) )
           => ( comp_int_int_nat(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4386_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Product_unit_o,Abs2: fun_o_Product_unit,Rep3: fun_Code_integer_int,Abs3: fun_int_Code_integer,M: fun_int_o,M1: fun_int_o,M22: fun_int_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_o_o(type_d6188575255521822967unit_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_int_o(type_d8366093980585677751er_int(Rep3,Abs3),top_top_set_int)
         => ( ( comp_i8439899303502915483nteger(comp_o7049022088938261380it_int(Abs2,M),Rep3) = comp_C8146212659816229034nteger(comp_i8439899303502915483nteger(comp_o7049022088938261380it_int(Abs2,M1),Rep),comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs,M22),Rep3)) )
           => ( comp_int_o_int(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4387_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Product_unit_o,Abs2: fun_o_Product_unit,Rep3: fun_Product_unit_o,Abs3: fun_o_Product_unit,M: fun_o_o,M1: fun_int_o,M22: fun_o_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_o_o(type_d6188575255521822967unit_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_o_o(type_d6188575255521822967unit_o(Rep3,Abs3),top_top_set_o)
         => ( ( comp_o7724735231917782709t_unit(comp_o3916225744453444672unit_o(Abs2,M),Rep3) = comp_C2408090164605745152t_unit(comp_i8439899303502915483nteger(comp_o7049022088938261380it_int(Abs2,M1),Rep),comp_o780160699970281785t_unit(comp_i5286552343147518890eger_o(Abs,M22),Rep3)) )
           => ( comp_int_o_o(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4388_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Product_unit_o,Abs2: fun_o_Product_unit,Rep3: fun_Code_natural_nat,Abs3: fun_nat_Code_natural,M: fun_nat_o,M1: fun_int_o,M22: fun_nat_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_o_o(type_d6188575255521822967unit_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( comp_n8360577435903631554atural(comp_o7051512559447311656it_nat(Abs2,M),Rep3) = comp_C6969807811572946549atural(comp_i8439899303502915483nteger(comp_o7049022088938261380it_int(Abs2,M1),Rep),comp_n5699668407433660474atural(comp_i1588355021709917246er_nat(Abs,M22),Rep3)) )
           => ( comp_int_o_nat(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4389_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Code_natural_nat,Abs2: fun_nat_Code_natural,Rep3: fun_Code_integer_int,Abs3: fun_int_Code_integer,M: fun_int_nat,M1: fun_int_nat,M22: fun_int_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep2,Abs2),top_top_set_nat)
       => ( aa_set_int_o(type_d8366093980585677751er_int(Rep3,Abs3),top_top_set_int)
         => ( ( comp_i455893257298015894nteger(comp_n9160511493981425161al_int(Abs2,M),Rep3) = comp_C1891057325546811591nteger(comp_i455893257298015894nteger(comp_n9160511493981425161al_int(Abs2,M1),Rep),comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs,M22),Rep3)) )
           => ( comp_int_nat_int(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4390_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Code_natural_nat,Abs2: fun_nat_Code_natural,Rep3: fun_Product_unit_o,Abs3: fun_o_Product_unit,M: fun_o_nat,M1: fun_int_nat,M22: fun_o_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep2,Abs2),top_top_set_nat)
       => ( aa_set_o_o(type_d6188575255521822967unit_o(Rep3,Abs3),top_top_set_o)
         => ( ( comp_o2643362028263920430t_unit(comp_n2857367870785791611ural_o(Abs2,M),Rep3) = comp_C4153580447849560611t_unit(comp_i455893257298015894nteger(comp_n9160511493981425161al_int(Abs2,M1),Rep),comp_o780160699970281785t_unit(comp_i5286552343147518890eger_o(Abs,M22),Rep3)) )
           => ( comp_int_nat_o(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4391_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,Rep2: fun_Code_natural_nat,Abs2: fun_nat_Code_natural,Rep3: fun_Code_natural_nat,Abs3: fun_nat_Code_natural,M: fun_nat_nat,M1: fun_int_nat,M22: fun_nat_int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep2,Abs2),top_top_set_nat)
       => ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( comp_n140768472405220869atural(comp_n9163001964490475437al_nat(Abs2,M),Rep3) = comp_C714652477303529106atural(comp_i455893257298015894nteger(comp_n9160511493981425161al_int(Abs2,M1),Rep),comp_n5699668407433660474atural(comp_i1588355021709917246er_nat(Abs,M22),Rep3)) )
           => ( comp_int_nat_nat(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4392_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,Rep2: fun_Code_integer_int,Abs2: fun_int_Code_integer,Rep3: fun_Code_integer_int,Abs3: fun_int_Code_integer,M: fun_int_int,M1: fun_o_int,M22: fun_int_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_int_o(type_d8366093980585677751er_int(Rep2,Abs2),top_top_set_int)
       => ( aa_set_int_o(type_d8366093980585677751er_int(Rep3,Abs3),top_top_set_int)
         => ( ( comp_i6014793192326455499nteger(comp_i1585864551200866970er_int(Abs2,M),Rep3) = comp_P6790823133229852966nteger(comp_o780160699970281785t_unit(comp_i5286552343147518890eger_o(Abs2,M1),Rep),comp_i8439899303502915483nteger(comp_o7049022088938261380it_int(Abs,M22),Rep3)) )
           => ( comp_o_int_int(M1,M22) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4393_wf__bounded__measure,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Ub: fun_nat_nat,F: fun_nat_nat] :
      ( ! [A3: nat,B2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),A3),R2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Ub,B2)),aa_nat_nat(Ub,A3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,B2)),aa_nat_nat(Ub,A3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,A3)),aa_nat_nat(F,B2)) ) )
     => wf_nat(R2) ) ).

% wf_bounded_measure
tff(fact_4394_wf__bounded__measure,axiom,
    ! [R2: set_Pr958786334691620121nt_int,Ub: fun_int_nat,F: fun_int_nat] :
      ( ! [A3: int,B2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(B2),A3),R2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Ub,B2)),aa_int_nat(Ub,A3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,B2)),aa_int_nat(Ub,A3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,A3)),aa_int_nat(F,B2)) ) )
     => wf_int(R2) ) ).

% wf_bounded_measure
tff(fact_4395_less__assn__def,axiom,
    ! [A2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A2),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A2),B)
        & ( A2 != B ) ) ) ).

% less_assn_def
tff(fact_4396_top__assn__def,axiom,
    top_top_assn = aa_fun5526454687433275355o_assn(abs_assn,in_range) ).

% top_assn_def
tff(fact_4397_pred__on_Ochain__extend,axiom,
    ! [A: set_Product_unit,P: fun_Pr1520749766685617582unit_o,C5: set_Product_unit,Z: product_unit] :
      ( aa_set8780210009671629607unit_o(pred_c6915491785556283576t_unit(A,P),C5)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(Z),A)
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),C5)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(sup_su8300244955164173186unit_o(P,fequal_Product_unit),X3),Z) )
         => aa_set8780210009671629607unit_o(pred_c6915491785556283576t_unit(A,P),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Z),bot_bo3957492148770167129t_unit)),C5)) ) ) ) ).

% pred_on.chain_extend
tff(fact_4398_pred__on_Ochain__extend,axiom,
    ! [A: set_o,P: fun_o_fun_o_o,C5: set_o,Z: $o] :
      ( aa_set_o_o(pred_chain_o(A,P),C5)
     => ( aa_set_o_o(member_o((Z)),A)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),C5)
             => aa_o_o(aa_o_fun_o_o(sup_su8537505864287452994un_o_o(P,fequal_o),(X3)),(Z)) )
         => aa_set_o_o(pred_chain_o(A,P),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Z)),bot_bot_set_o)),C5)) ) ) ) ).

% pred_on.chain_extend
tff(fact_4399_pred__on_Ochain__extend,axiom,
    ! [A: set_nat,P: fun_nat_fun_nat_o,C5: set_nat,Z: nat] :
      ( aa_set_nat_o(pred_chain_nat(A,P),C5)
     => ( aa_set_nat_o(member_nat(Z),A)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),C5)
             => aa_nat_o(aa_nat_fun_nat_o(sup_su8083968640029086462_nat_o(P,fequal_nat),X3),Z) )
         => aa_set_nat_o(pred_chain_nat(A,P),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Z),bot_bot_set_nat)),C5)) ) ) ) ).

% pred_on.chain_extend
tff(fact_4400_pred__on_Ochain__extend,axiom,
    ! [A: set_int,P: fun_int_fun_int_o,C5: set_int,Z: int] :
      ( aa_set_int_o(pred_chain_int(A,P),C5)
     => ( aa_set_int_o(member_int(Z),A)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),C5)
             => aa_int_o(aa_int_fun_int_o(sup_su7393609867835711046_int_o(P,fequal_int),X3),Z) )
         => aa_set_int_o(pred_chain_int(A,P),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Z),bot_bot_set_int)),C5)) ) ) ) ).

% pred_on.chain_extend
tff(fact_4401_pred__on_Ochain__extend,axiom,
    ! [A: set_fu3555266933746557007_nat_o,P: fun_fu4164567611050918574at_o_o,C5: set_fu3555266933746557007_nat_o,Z: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(pred_c5982856694712694810_nat_o(A,P),C5)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Z),A)
       => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),C5)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(sup_su3092406352011465858at_o_o(P,fequal1890233700291625924_nat_o),X3),Z) )
         => aa_set5091322549519198405at_o_o(pred_c5982856694712694810_nat_o(A,P),aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Z),bot_bo8632255151191677115_nat_o)),C5)) ) ) ) ).

% pred_on.chain_extend
tff(fact_4402_subset__Zorn__nonempty,axiom,
    ! [A11: set_set_nat] :
      ( ( A11 != bot_bot_set_set_nat )
     => ( ! [C7: set_set_nat] :
            ( ( C7 != bot_bot_set_set_nat )
           => ( pred_chain_set_nat(A11,ord_less_set_nat,C7)
             => member_set_nat(comple7399068483239264473et_nat(C7),A11) ) )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A11)
            & ! [Xa3: set_nat] :
                ( member_set_nat(Xa3,A11)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X3),Xa3)
                 => ( Xa3 = X3 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_4403_in__measure,axiom,
    ! [X: nat,Y: nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),measure_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y)) ) ).

% in_measure
tff(fact_4404_in__measure,axiom,
    ! [X: int,Y: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),measure_int(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(F,Y)) ) ).

% in_measure
tff(fact_4405_top_Oordering__top__axioms,axiom,
    aa_set8780210009671629607unit_o(orderi2935705153647960990t_unit(ord_le3507040750410214029t_unit,ord_le8056459307392131481t_unit),top_to1996260823553986621t_unit) ).

% top.ordering_top_axioms
tff(fact_4406_top_Oordering__top__axioms,axiom,
    aa_set_o_o(ordering_top_set_o(ord_less_eq_set_o,ord_less_set_o),top_top_set_o) ).

% top.ordering_top_axioms
tff(fact_4407_top_Oordering__top__axioms,axiom,
    aa_set_int_o(ordering_top_set_int(ord_less_eq_set_int,ord_less_set_int),top_top_set_int) ).

% top.ordering_top_axioms
tff(fact_4408_top_Oordering__top__axioms,axiom,
    aa_assn_o(ordering_top_assn(ord_less_eq_assn,ord_less_assn),top_top_assn) ).

% top.ordering_top_axioms
tff(fact_4409_top_Oordering__top__axioms,axiom,
    orderi2807725203697798511er_nat(ord_le2510731241096832064er_nat,ord_less_filter_nat,top_top_filter_nat) ).

% top.ordering_top_axioms
tff(fact_4410_top_Oordering__top__axioms,axiom,
    aa_set_nat_o(ordering_top_set_nat(ord_less_eq_set_nat,ord_less_set_nat),top_top_set_nat) ).

% top.ordering_top_axioms
tff(fact_4411_Max_Oinsert__remove,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( lattic4901227151466704046nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = $ite(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)) = bot_bo3990330152332043303nteger,X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X),lattic4901227151466704046nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger))))) ) ) ).

% Max.insert_remove
tff(fact_4412_Max_Oinsert__remove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X),lattic1210392890313158908t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ) ) ).

% Max.insert_remove
tff(fact_4413_Max_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
      <=> $ite(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ) ) ).

% Max.insert_remove
tff(fact_4414_Max_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( lattic8263393255366662781ax_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(ord_max_int,X),lattic8263393255366662781ax_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ) ) ).

% Max.insert_remove
tff(fact_4415_Max_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ) ) ).

% Max.insert_remove
tff(fact_4416_Max_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( lattic4901227151466704046nteger(A) = $ite(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)) = bot_bo3990330152332043303nteger,X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X),lattic4901227151466704046nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger))))) ) ) ) ).

% Max.remove
tff(fact_4417_Max_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( lattic1210392890313158908t_unit(A) = $ite(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X),lattic1210392890313158908t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))))) ) ) ) ).

% Max.remove
tff(fact_4418_Max_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,A)
        <=> $ite(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) = bot_bot_set_o,(X),aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))))) ) ) ) ).

% Max.remove
tff(fact_4419_Max_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( lattic8263393255366662781ax_int(A) = $ite(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(ord_max_int,X),lattic8263393255366662781ax_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))))) ) ) ) ).

% Max.remove
tff(fact_4420_Max_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A) = $ite(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))))) ) ) ) ).

% Max.remove
tff(fact_4421_inj__on__Un,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),sup_su848401254843788991nteger(A,B4))
    <=> ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),A)
        & aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),B4)
        & ( inf_in1364745209274528805nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),minus_2355218937544613996nteger(A,B4)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),minus_2355218937544613996nteger(B4,A))) = bot_bo3990330152332043303nteger ) ) ) ).

% inj_on_Un
tff(fact_4422_inj__on__Un,axiom,
    ! [F: fun_nat_set_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_set_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on_nat_set_nat(F),A)
        & aa_set_nat_o(inj_on_nat_set_nat(F),B4)
        & ( inf_inf_set_set_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bot_set_set_nat ) ) ) ).

% inj_on_Un
tff(fact_4423_inj__on__Un,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on2245834603434889120nteger(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on2245834603434889120nteger(F),A)
        & aa_set_nat_o(inj_on2245834603434889120nteger(F),B4)
        & ( inf_in1364745209274528805nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bo3990330152332043303nteger ) ) ) ).

% inj_on_Un
tff(fact_4424_inj__on__Un,axiom,
    ! [F: fun_nat_Product_unit,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on7061601236592826506t_unit(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on7061601236592826506t_unit(F),A)
        & aa_set_nat_o(inj_on7061601236592826506t_unit(F),B4)
        & ( inf_in4660618365625256667t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),image_8730104196221521654t_unit(F,aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bo3957492148770167129t_unit ) ) ) ).

% inj_on_Un
tff(fact_4425_inj__on__Un,axiom,
    ! [F: fun_nat_o,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on_nat_o(F),A)
        & aa_set_nat_o(inj_on_nat_o(F),B4)
        & ( inf_inf_set_o(image_nat_o2(F,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),image_nat_o2(F,aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bot_set_o ) ) ) ).

% inj_on_Un
tff(fact_4426_inj__on__Un,axiom,
    ! [F: fun_nat_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on_nat_nat(F),A)
        & aa_set_nat_o(inj_on_nat_nat(F),B4)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bot_set_nat ) ) ) ).

% inj_on_Un
tff(fact_4427_inj__on__Un,axiom,
    ! [F: fun_nat_int,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on_nat_int(F),A)
        & aa_set_nat_o(inj_on_nat_int(F),B4)
        & ( inf_inf_set_int(image_nat_int2(F,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),image_nat_int2(F,aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bot_set_int ) ) ) ).

% inj_on_Un
tff(fact_4428_inj__on__Un,axiom,
    ! [F: fun_na5781657914532371832_nat_o,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on7336542403725402860_nat_o(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))
    <=> ( aa_set_nat_o(inj_on7336542403725402860_nat_o(F),A)
        & aa_set_nat_o(inj_on7336542403725402860_nat_o(F),B4)
        & ( inf_in3376754634429691197_nat_o(image_7177977788501158744_nat_o(F,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)),image_7177977788501158744_nat_o(F,aa_set_nat_set_nat(minus_minus_set_nat(B4),A))) = bot_bo8632255151191677115_nat_o ) ) ) ).

% inj_on_Un
tff(fact_4429_Max__insert,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( lattic4901227151466704046nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X),lattic4901227151466704046nteger(A)) ) ) ) ).

% Max_insert
tff(fact_4430_Max__insert,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X),lattic1210392890313158908t_unit(A)) ) ) ) ).

% Max_insert
tff(fact_4431_Max__insert,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
        <=> aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A)) ) ) ) ).

% Max_insert
tff(fact_4432_Max__insert,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( lattic8263393255366662781ax_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_int_int(aa_int_fun_int_int(ord_max_int,X),lattic8263393255366662781ax_int(A)) ) ) ) ).

% Max_insert
tff(fact_4433_Max__insert,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),aa_set_nat_nat(lattic8265883725875713057ax_nat,A)) ) ) ) ).

% Max_insert
tff(fact_4434_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(G),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
       => ( image_7082141833288935621nteger(aa_fun2509385170507305080nteger(produc1802105962441033582nteger(F),G),top_to2363841527343591755nteger) = top_to2363841527343591755nteger ) ) ) ).

% map_prod_surj
tff(fact_4435_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Pr2191293291710260995t_unit] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_405062704495631173t_unit(G,top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit )
       => ( image_3456600171786721233t_unit(produc3346700480963112818t_unit(F,G),top_to3244471725241573579t_unit) = top_to3244471725241573579t_unit ) ) ) ).

% map_prod_surj
tff(fact_4436_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Product_unit_o] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_Product_unit_o2(G,top_to1996260823553986621t_unit) = top_top_set_o )
       => ( image_7060974388358999388eger_o(produc8116347370398923901unit_o(F,G),top_to3244471725241573579t_unit) = top_to2658285827425157974eger_o ) ) ) ).

% map_prod_surj
tff(fact_4437_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Product_unit_nat] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_875570014554754200it_nat(G,top_to1996260823553986621t_unit) = top_top_set_nat )
       => ( image_8236373689398998434er_nat(produc4022400886476617259it_nat(F,G),top_to3244471725241573579t_unit) = top_to8760411888770933182er_nat ) ) ) ).

% map_prod_surj
tff(fact_4438_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Product_unit_int] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_873079544045703924it_int(G,top_to1996260823553986621t_unit) = top_top_set_int )
       => ( image_4058522669889801726er_int(produc4019910415967566983it_int(F,G),top_to3244471725241573579t_unit) = top_to6270328743839206426er_int ) ) ) ).

% map_prod_surj
tff(fact_4439_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_o_Product_unit] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_o_Product_unit2(G,top_top_set_o) = top_to1996260823553986621t_unit )
       => ( image_2425804817718344284t_unit(produc3530376873132787239t_unit(F,G),top_to2658285827425157974eger_o) = top_to3244471725241573579t_unit ) ) ) ).

% map_prod_surj
tff(fact_4440_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_o_o] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_o_o2(G,top_top_set_o) = top_top_set_o )
       => ( image_227735840630799463eger_o(produc1084426483498010290er_o_o(F,G),top_to2658285827425157974eger_o) = top_to2658285827425157974eger_o ) ) ) ).

% map_prod_surj
tff(fact_4441_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_o_nat] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_o_nat2(G,top_top_set_o) = top_top_set_nat )
       => ( image_1252617143230709591er_nat(produc7550393335539917622_o_nat(F,G),top_to2658285827425157974eger_o) = top_to8760411888770933182er_nat ) ) ) ).

% map_prod_surj
tff(fact_4442_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_o_int] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( image_o_int2(G,top_top_set_o) = top_top_set_int )
       => ( image_6298138160576288691er_int(produc7547902865030867346_o_int(F,G),top_to2658285827425157974eger_o) = top_to6270328743839206426er_int ) ) ) ).

% map_prod_surj
tff(fact_4443_map__prod__surj,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_nat_Code_integer] :
      ( ( aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),top_to4645266643341252675nteger) = top_to4645266643341252675nteger )
     => ( ( aa_set7438092518518805847nteger(image_1215581382706833972nteger(G),top_top_set_nat) = top_to4645266643341252675nteger )
       => ( image_4921214744877245202nteger(produc1622059938646302433nteger(F,G),top_to8760411888770933182er_nat) = top_to2363841527343591755nteger ) ) ) ).

% map_prod_surj
tff(fact_4444_inj__on__insert,axiom,
    ! [F: fun_Co8499238673658658103nteger,A2: code_integer,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),A))
    <=> ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),A)
        & ~ aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(F,A2)),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),bot_bo3990330152332043303nteger)))) ) ) ).

% inj_on_insert
tff(fact_4445_inj__on__insert,axiom,
    ! [F: fun_Product_unit_nat,A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(inj_on8430439091780834860it_nat(F),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> ( aa_set8780210009671629607unit_o(inj_on8430439091780834860it_nat(F),A)
        & ~ aa_set_nat_o(member_nat(aa_Product_unit_nat(F,A2)),image_875570014554754200it_nat(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_4446_inj__on__insert,axiom,
    ! [F: fun_Product_unit_int,A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(inj_on8427948621271784584it_int(F),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> ( aa_set8780210009671629607unit_o(inj_on8427948621271784584it_int(F),A)
        & ~ aa_set_int_o(member_int(aa_Product_unit_int(F,A2)),image_873079544045703924it_int(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_4447_inj__on__insert,axiom,
    ! [F: fun_Product_unit_o,A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(inj_on8854722734121536060unit_o(F),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> ( aa_set8780210009671629607unit_o(inj_on8854722734121536060unit_o(F),A)
        & ~ aa_set_o_o(member_o(aa_Product_unit_o(F,A2)),image_Product_unit_o2(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_4448_inj__on__insert,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(inj_on8151373323710067377t_unit(F),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> ( aa_set8780210009671629607unit_o(inj_on8151373323710067377t_unit(F),A)
        & ~ aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(F,A2)),image_405062704495631173t_unit(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_4449_inj__on__insert,axiom,
    ! [F: fun_o_nat,A2: $o,A: set_o] :
      ( aa_set_o_o(inj_on_o_nat(F),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> ( aa_set_o_o(inj_on_o_nat(F),A)
        & ~ aa_set_nat_o(member_nat(aa_o_nat(F,(A2))),image_o_nat2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_4450_inj__on__insert,axiom,
    ! [F: fun_o_int,A2: $o,A: set_o] :
      ( aa_set_o_o(inj_on_o_int(F),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> ( aa_set_o_o(inj_on_o_int(F),A)
        & ~ aa_set_int_o(member_int(aa_o_int(F,(A2))),image_o_int2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_4451_inj__on__insert,axiom,
    ! [F: fun_o_o,A2: $o,A: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> ( aa_set_o_o(inj_on_o_o(F),A)
        & ~ aa_set_o_o(member_o(aa_o_o(F,(A2))),image_o_o2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_4452_inj__on__insert,axiom,
    ! [F: fun_o_Product_unit,A2: $o,A: set_o] :
      ( aa_set_o_o(inj_on4268752236855399398t_unit(F),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> ( aa_set_o_o(inj_on4268752236855399398t_unit(F),A)
        & ~ aa_set8780210009671629607unit_o(member_Product_unit(aa_o_Product_unit(F,(A2))),image_o_Product_unit2(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_4453_inj__on__insert,axiom,
    ! [F: fun_int_nat,A2: int,A: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))
    <=> ( aa_set_int_o(inj_on_int_nat(F),A)
        & ~ aa_set_nat_o(member_nat(aa_int_nat(F,A2)),image_int_nat2(F,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int)))) ) ) ).

% inj_on_insert
tff(fact_4454_max__bot,axiom,
    ! [X: set_Product_unit] : ord_ma508538259134630082t_unit(bot_bo3957492148770167129t_unit,X) = X ).

% max_bot
tff(fact_4455_max__bot,axiom,
    ! [X: set_o] : ord_max_set_o(bot_bot_set_o,X) = X ).

% max_bot
tff(fact_4456_max__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(ord_max_set_nat(bot_bot_set_nat),X) = X ).

% max_bot
tff(fact_4457_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_4458_max__bot,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : ord_ma9117053559795956132_nat_o(bot_bo8632255151191677115_nat_o,X) = X ).

% max_bot
tff(fact_4459_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_4460_max__bot2,axiom,
    ! [X: set_Product_unit] : ord_ma508538259134630082t_unit(X,bot_bo3957492148770167129t_unit) = X ).

% max_bot2
tff(fact_4461_max__bot2,axiom,
    ! [X: set_o] : ord_max_set_o(X,bot_bot_set_o) = X ).

% max_bot2
tff(fact_4462_max__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(ord_max_set_nat(X),bot_bot_set_nat) = X ).

% max_bot2
tff(fact_4463_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_4464_max__bot2,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : ord_ma9117053559795956132_nat_o(X,bot_bo8632255151191677115_nat_o) = X ).

% max_bot2
tff(fact_4465_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_4466_max__top,axiom,
    ! [X: assn] : aa_assn_assn(ord_max_assn(top_top_assn),X) = top_top_assn ).

% max_top
tff(fact_4467_max__top,axiom,
    ! [X: set_Product_unit] : ord_ma508538259134630082t_unit(top_to1996260823553986621t_unit,X) = top_to1996260823553986621t_unit ).

% max_top
tff(fact_4468_max__top,axiom,
    ! [X: set_o] : ord_max_set_o(top_top_set_o,X) = top_top_set_o ).

% max_top
tff(fact_4469_max__top,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(ord_max_set_nat(top_top_set_nat),X) = top_top_set_nat ).

% max_top
tff(fact_4470_max__top,axiom,
    ! [X: set_int] : ord_max_set_int(top_top_set_int,X) = top_top_set_int ).

% max_top
tff(fact_4471_max__top2,axiom,
    ! [X: assn] : aa_assn_assn(ord_max_assn(X),top_top_assn) = top_top_assn ).

% max_top2
tff(fact_4472_max__top2,axiom,
    ! [X: set_Product_unit] : ord_ma508538259134630082t_unit(X,top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).

% max_top2
tff(fact_4473_max__top2,axiom,
    ! [X: set_o] : ord_max_set_o(X,top_top_set_o) = top_top_set_o ).

% max_top2
tff(fact_4474_max__top2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(ord_max_set_nat(X),top_top_set_nat) = top_top_set_nat ).

% max_top2
tff(fact_4475_max__top2,axiom,
    ! [X: set_int] : ord_max_set_int(X,top_top_set_int) = top_top_set_int ).

% max_top2
tff(fact_4476_map__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A2: nat,B: nat] : aa_Pro262675715343573244at_nat(produc6977886695330630970at_nat(F,G),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,A2)),aa_nat_nat(G,B)) ).

% map_prod_simp
tff(fact_4477_map__prod__simp,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,A2: nat,B: nat] : aa_Pro6307550444590206900nt_int(produc7347049785058425330at_int(F,G),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,A2)),aa_nat_int(G,B)) ).

% map_prod_simp
tff(fact_4478_map__prod__simp,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A2: int,B: int] : aa_Pro925629398452685892at_nat(produc5667422475138734578nt_nat(F,G),aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,A2)),aa_int_nat(G,B)) ).

% map_prod_simp
tff(fact_4479_map__prod__simp,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: int,B: int] : aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,A2)),aa_int_int(G,B)) ).

% map_prod_simp
tff(fact_4480_map__prod__simp,axiom,
    ! [F: fun_he1033112309995878576it_nat,G: fun_set_nat_nat,A2: heap_e7401611519738050253t_unit,B: set_nat] : aa_Pro2139006781426367169at_nat(produc8285742784527409791at_nat(F,G),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_hea5848002036693495265it_nat(F,A2)),aa_set_nat_nat(G,B)) ).

% map_prod_simp
tff(fact_4481_map__prod__simp,axiom,
    ! [F: fun_he6078633327341457676it_int,G: fun_set_nat_int,A2: heap_e7401611519738050253t_unit,B: set_nat] : aa_Pro8183881510673000825nt_int(produc5973434381925408567at_int(F,G),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_hea5845511566184444989it_int(F,A2)),aa_set_nat_int(G,B)) ).

% map_prod_simp
tff(fact_4482_map__prod__simp,axiom,
    ! [F: fun_na8669532458821385046t_unit,G: fun_nat_set_nat,A2: nat,B: nat] : aa_Pro3449680268614826951et_nat(produc2064153442785501et_nat(F,G),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_nat4191430056936421439t_unit(F,A2)),aa_nat_set_nat(G,B)) ).

% map_prod_simp
tff(fact_4483_map__prod__simp,axiom,
    ! [F: fun_in7572449478177386490t_unit,G: fun_int_set_nat,A2: int,B: int] : aa_Pro5732104609077762687et_nat(produc3152090231653401565et_nat(F,G),aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_int3094347076292422883t_unit(F,A2)),aa_int_set_nat(G,B)) ).

% map_prod_simp
tff(fact_4484_map__prod__simp,axiom,
    ! [F: fun_he2331061685413951235t_unit,G: fun_set_nat_set_nat,A2: heap_e7401611519738050253t_unit,B: set_nat] : aa_Pro3679490825741336130et_nat(produc8234977478398262212et_nat(F,G),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(F,A2)),aa_set_nat_set_nat(G,B)) ).

% map_prod_simp
tff(fact_4485_map__prod__simp,axiom,
    ! [F: fun_na5781657914532371832_nat_o,G: fun_na1841593615831545295et_nat,A2: nat,B: nat] : aa_Pro3269747882002560613et_nat(produc9136172729928082811et_nat(F,G),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_nat2198927446035044705_nat_o(F,A2)),aa_nat6208498561624090640et_nat(G,B)) ).

% map_prod_simp
tff(fact_4486_snd__comp__map__prod,axiom,
    ! [F: fun_he2331061685413951235t_unit,G: fun_set_nat_set_nat] : comp_P2930521774885306735et_nat(produc8586169260539613262et_nat,produc8234977478398262212et_nat(F,G)) = comp_s5696514601250091763et_nat(G,produc8586169260539613262et_nat) ).

% snd_comp_map_prod
tff(fact_4487_snd__comp__map__prod,axiom,
    ! [F: fun_in7572449478177386490t_unit,G: fun_int_set_nat] : comp_P8606889026643445772nt_int(produc8586169260539613262et_nat,produc3152090231653401565et_nat(F,G)) = comp_i6200299785623974754nt_int(G,product_snd_int_int) ).

% snd_comp_map_prod
tff(fact_4488_snd__comp__map__prod,axiom,
    ! [F: fun_he6078633327341457676it_int,G: fun_set_nat_int] : comp_P8308861049717174982et_nat(product_snd_int_int,produc5973434381925408567at_int(F,G)) = comp_s1601710177542341017et_nat(G,produc8586169260539613262et_nat) ).

% snd_comp_map_prod
tff(fact_4489_snd__comp__map__prod,axiom,
    ! [F: fun_int_int,G: fun_int_int] : comp_P5356232565778667509nt_int(product_snd_int_int,produc6036585564866528938nt_int(F,G)) = comp_i4330622015127193916nt_int(G,product_snd_int_int) ).

% snd_comp_map_prod
tff(fact_4490_snd__map__prod,axiom,
    ! [F: fun_he2331061685413951235t_unit,G: fun_set_nat_set_nat,X: produc3658429121746597890et_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_Pro3679490825741336130et_nat(produc8234977478398262212et_nat(F,G),X)) = aa_set_nat_set_nat(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X)) ).

% snd_map_prod
tff(fact_4491_snd__map__prod,axiom,
    ! [F: fun_in7572449478177386490t_unit,G: fun_int_set_nat,X: product_prod_int_int] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_Pro5732104609077762687et_nat(produc3152090231653401565et_nat(F,G),X)) = aa_int_set_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ).

% snd_map_prod
tff(fact_4492_snd__map__prod,axiom,
    ! [F: fun_he6078633327341457676it_int,G: fun_set_nat_int,X: produc3658429121746597890et_nat] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro8183881510673000825nt_int(produc5973434381925408567at_int(F,G),X)) = aa_set_nat_int(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X)) ).

% snd_map_prod
tff(fact_4493_snd__map__prod,axiom,
    ! [F: fun_int_int,G: fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),X)) = aa_int_int(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ).

% snd_map_prod
tff(fact_4494_map__prod__imageI,axiom,
    ! [A2: nat,B: nat,R: set_Pr1261947904930325089at_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,A2)),aa_nat_nat(G,B)),image_5168914502847457605at_nat(produc6977886695330630970at_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4495_map__prod__imageI,axiom,
    ! [A2: nat,B: nat,R: set_Pr1261947904930325089at_nat,F: fun_nat_int,G: fun_nat_int] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,A2)),aa_nat_int(G,B)),image_1990417195239315453nt_int(produc7347049785058425330at_int(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4496_map__prod__imageI,axiom,
    ! [A2: int,B: int,R: set_Pr958786334691620121nt_int,F: fun_int_nat,G: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,A2)),aa_int_nat(G,B)),image_5831868185956570253at_nat(produc5667422475138734578nt_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4497_map__prod__imageI,axiom,
    ! [A2: int,B: int,R: set_Pr958786334691620121nt_int,F: fun_int_int,G: fun_int_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,A2)),aa_int_int(G,B)),image_2653370878348428101nt_int(produc6036585564866528938nt_int(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4498_map__prod__imageI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R: set_Pr3948176798113811640et_nat,F: fun_he1033112309995878576it_nat,G: fun_set_nat_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_hea5848002036693495265it_nat(F,A2)),aa_set_nat_nat(G,B)),image_7757775281999735736at_nat(produc8285742784527409791at_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4499_map__prod__imageI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R: set_Pr3948176798113811640et_nat,F: fun_he6078633327341457676it_int,G: fun_set_nat_int] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R)
     => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_hea5845511566184444989it_int(F,A2)),aa_set_nat_int(G,B)),image_4579277974391593584nt_int(produc5973434381925408567at_int(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4500_map__prod__imageI,axiom,
    ! [A2: nat,B: nat,R: set_Pr1261947904930325089at_nat,F: fun_na8669532458821385046t_unit,G: fun_nat_set_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R)
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_nat4191430056936421439t_unit(F,A2)),aa_nat_set_nat(G,B)),image_9068448769188195518et_nat(produc2064153442785501et_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4501_map__prod__imageI,axiom,
    ! [A2: int,B: int,R: set_Pr958786334691620121nt_int,F: fun_in7572449478177386490t_unit,G: fun_int_set_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),R)
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_int3094347076292422883t_unit(F,A2)),aa_int_set_nat(G,B)),image_2127501072796355446et_nat(produc3152090231653401565et_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4502_map__prod__imageI,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R: set_Pr3948176798113811640et_nat,F: fun_he2331061685413951235t_unit,G: fun_set_nat_set_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),R)
     => member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(F,A2)),aa_set_nat_set_nat(G,B)),image_7426123528223433739et_nat(produc8234977478398262212et_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4503_map__prod__imageI,axiom,
    ! [A2: nat,B: nat,R: set_Pr1261947904930325089at_nat,F: fun_na5781657914532371832_nat_o,G: fun_na1841593615831545295et_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),R)
     => member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_nat2198927446035044705_nat_o(F,A2)),aa_nat6208498561624090640et_nat(G,B)),image_3859208966192563164et_nat(produc9136172729928082811et_nat(F,G),R)) ) ).

% map_prod_imageI
tff(fact_4504_exists__leI,axiom,
    ! [N2: nat,P: fun_nat_o] :
      ( ( ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),N2)
           => ~ aa_nat_o(P,N5) )
       => aa_nat_o(P,N2) )
     => ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
          & aa_nat_o(P,N6) ) ) ).

% exists_leI
tff(fact_4505_max__def,axiom,
    ! [A2: filter_nat,B: filter_nat] :
      ord_max_filter_nat(A2,B) = $ite(aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),B),B,A2) ).

% max_def
tff(fact_4506_max__def,axiom,
    ! [A2: set_nat,B: set_nat] :
      aa_set_nat_set_nat(ord_max_set_nat(A2),B) = $ite(aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B),B,A2) ).

% max_def
tff(fact_4507_max__def,axiom,
    ! [A2: rat,B: rat] :
      aa_rat_rat(ord_max_rat(A2),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B),B,A2) ).

% max_def
tff(fact_4508_max__def,axiom,
    ! [A2: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B),B,A2) ).

% max_def
tff(fact_4509_max__def,axiom,
    ! [A2: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B),B,A2) ).

% max_def
tff(fact_4510_max__absorb1,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y),X)
     => ( ord_max_filter_nat(X,Y) = X ) ) ).

% max_absorb1
tff(fact_4511_max__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(ord_max_set_nat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_4512_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X)
     => ( aa_rat_rat(ord_max_rat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_4513_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_4514_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_4515_max__absorb2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X),Y)
     => ( ord_max_filter_nat(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_4516_max__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(ord_max_set_nat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4517_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y)
     => ( aa_rat_rat(ord_max_rat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4518_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_4519_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_4520_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_4521_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(minus_minus_int(X),Z)),aa_int_int(minus_minus_int(Y),Z)) ).

% max_diff_distrib_left
tff(fact_4522_map__prod_Ocomp,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_Co1561183069634336733nteger,Ha: fun_Co8499238673658658103nteger,I2: fun_Co8499238673658658103nteger] : comp_P2623459279333308012nteger(produc3249263839394015674nteger(F,G),aa_fun2509385170507305080nteger(produc1802105962441033582nteger(Ha),I2)) = produc3249263839394015674nteger(comp_C7848664941760151970nteger(F,Ha),comp_C7848664941760151970nteger(G,I2)) ).

% map_prod.comp
tff(fact_4523_map__prod_Ocomp,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_fu4800041728187187127nteger,Ha: fun_Co8499238673658658103nteger,I2: fun_Co1981900195452921617nteger] : comp_P7479062676170010950nteger(produc2109198901873568148nteger(F,G),produc4699376468482697800nteger(Ha,I2)) = produc3249263839394015674nteger(comp_C7848664941760151970nteger(F,Ha),comp_f6739083541385432444nteger(G,I2)) ).

% map_prod.comp
tff(fact_4524_map__prod_Ocomp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1561183069634336733nteger,Ha: fun_Co1981900195452921617nteger,I2: fun_Co8499238673658658103nteger] : comp_P4540793059153670726nteger(produc1996466970500618900nteger(F,G),produc8108006236828656200nteger(Ha,I2)) = produc3249263839394015674nteger(comp_f6739083541385432444nteger(F,Ha),comp_C7848664941760151970nteger(G,I2)) ).

% map_prod.comp
tff(fact_4525_map__prod_Ocomp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu4800041728187187127nteger,Ha: fun_Co1981900195452921617nteger,I2: fun_Co1981900195452921617nteger] : comp_P7572596982727354656nteger(produc7836332429974410862nteger(F,G),produc8119697944725045538nteger(Ha,I2)) = produc3249263839394015674nteger(comp_f6739083541385432444nteger(F,Ha),comp_f6739083541385432444nteger(G,I2)) ).

% map_prod.comp
tff(fact_4526_map__prod_Ocompositionality,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_Co1561183069634336733nteger,Ha: fun_Co8499238673658658103nteger,I2: fun_Co8499238673658658103nteger,Prod: produc8923325533196201883nteger] : aa_Pro552286414676239304nteger(produc3249263839394015674nteger(F,G),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc1802105962441033582nteger(Ha),I2),Prod)) = aa_Pro552286414676239304nteger(produc3249263839394015674nteger(comp_C7848664941760151970nteger(F,Ha),comp_C7848664941760151970nteger(G,I2)),Prod) ).

% map_prod.compositionality
tff(fact_4527_map__prod_Ocompositionality,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_fu4800041728187187127nteger,Ha: fun_Co8499238673658658103nteger,I2: fun_Co1981900195452921617nteger,Prod: produc8923325533196201883nteger] : aa_Pro4358925961699788142nteger(produc2109198901873568148nteger(F,G),aa_Pro2514201178793240662nteger(produc4699376468482697800nteger(Ha,I2),Prod)) = aa_Pro552286414676239304nteger(produc3249263839394015674nteger(comp_C7848664941760151970nteger(F,Ha),comp_f6739083541385432444nteger(G,I2)),Prod) ).

% map_prod.compositionality
tff(fact_4528_map__prod_Ocompositionality,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1561183069634336733nteger,Ha: fun_Co1981900195452921617nteger,I2: fun_Co8499238673658658103nteger,Prod: produc8923325533196201883nteger] : aa_Pro1931126632642659438nteger(produc1996466970500618900nteger(F,G),aa_Pro6222166416904055126nteger(produc8108006236828656200nteger(Ha,I2),Prod)) = aa_Pro552286414676239304nteger(produc3249263839394015674nteger(comp_f6739083541385432444nteger(F,Ha),comp_C7848664941760151970nteger(G,I2)),Prod) ).

% map_prod.compositionality
tff(fact_4529_map__prod_Ocompositionality,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu4800041728187187127nteger,Ha: fun_Co1981900195452921617nteger,I2: fun_Co1981900195452921617nteger,Prod: produc8923325533196201883nteger] : aa_Pro3873633512513284628nteger(produc7836332429974410862nteger(F,G),aa_Pro4416545769347374384nteger(produc8119697944725045538nteger(Ha,I2),Prod)) = aa_Pro552286414676239304nteger(produc3249263839394015674nteger(comp_f6739083541385432444nteger(F,Ha),comp_f6739083541385432444nteger(G,I2)),Prod) ).

% map_prod.compositionality
tff(fact_4530_map__prod__compose,axiom,
    ! [F1: fun_Co1561183069634336733nteger,F22: fun_Co8499238673658658103nteger,G1: fun_Co1561183069634336733nteger,G2: fun_Co8499238673658658103nteger] : produc3249263839394015674nteger(comp_C7848664941760151970nteger(F1,F22),comp_C7848664941760151970nteger(G1,G2)) = comp_P2623459279333308012nteger(produc3249263839394015674nteger(F1,G1),aa_fun2509385170507305080nteger(produc1802105962441033582nteger(F22),G2)) ).

% map_prod_compose
tff(fact_4531_map__prod__compose,axiom,
    ! [F1: fun_Co1561183069634336733nteger,F22: fun_Co8499238673658658103nteger,G1: fun_fu4800041728187187127nteger,G2: fun_Co1981900195452921617nteger] : produc3249263839394015674nteger(comp_C7848664941760151970nteger(F1,F22),comp_f6739083541385432444nteger(G1,G2)) = comp_P7479062676170010950nteger(produc2109198901873568148nteger(F1,G1),produc4699376468482697800nteger(F22,G2)) ).

% map_prod_compose
tff(fact_4532_map__prod__compose,axiom,
    ! [F1: fun_fu4800041728187187127nteger,F22: fun_Co1981900195452921617nteger,G1: fun_Co1561183069634336733nteger,G2: fun_Co8499238673658658103nteger] : produc3249263839394015674nteger(comp_f6739083541385432444nteger(F1,F22),comp_C7848664941760151970nteger(G1,G2)) = comp_P4540793059153670726nteger(produc1996466970500618900nteger(F1,G1),produc8108006236828656200nteger(F22,G2)) ).

% map_prod_compose
tff(fact_4533_map__prod__compose,axiom,
    ! [F1: fun_fu4800041728187187127nteger,F22: fun_Co1981900195452921617nteger,G1: fun_fu4800041728187187127nteger,G2: fun_Co1981900195452921617nteger] : produc3249263839394015674nteger(comp_f6739083541385432444nteger(F1,F22),comp_f6739083541385432444nteger(G1,G2)) = comp_P7572596982727354656nteger(produc7836332429974410862nteger(F1,G1),produc8119697944725045538nteger(F22,G2)) ).

% map_prod_compose
tff(fact_4534_inj__img__insertE,axiom,
    ! [F: fun_Co8499238673658658103nteger,A: set_Code_integer,X: code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
       => ( ( aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),B4) = aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A) )
         => ~ ! [X8: code_integer,A6: set_Code_integer] :
                ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X8),A6)
               => ( ( A = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X8),A6) )
                 => ( ( X = aa_Cod1783974652533396728nteger(F,X8) )
                   => ( B4 != aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4535_inj__img__insertE,axiom,
    ! [F: fun_nat_Code_integer,A: set_nat,X: code_integer,B4: set_Code_integer] :
      ( aa_set_nat_o(inj_on2245834603434889120nteger(F),A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
       => ( ( aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),B4) = aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A) )
         => ~ ! [X8: nat,A6: set_nat] :
                ( ~ aa_set_nat_o(member_nat(X8),A6)
               => ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X8),A6) )
                 => ( ( X = aa_nat_Code_integer(F,X8) )
                   => ( B4 != aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4536_inj__img__insertE,axiom,
    ! [F: fun_nat_nat,A: set_nat,X: nat,B4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A)
     => ( ~ aa_set_nat_o(member_nat(X),B4)
       => ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4) = aa_set_nat_set_nat(image_nat_nat2(F),A) )
         => ~ ! [X8: nat,A6: set_nat] :
                ( ~ aa_set_nat_o(member_nat(X8),A6)
               => ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X8),A6) )
                 => ( ( X = aa_nat_nat(F,X8) )
                   => ( B4 != aa_set_nat_set_nat(image_nat_nat2(F),A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4537_inj__img__insertE,axiom,
    ! [F: fun_int_nat,A: set_int,X: nat,B4: set_nat] :
      ( aa_set_int_o(inj_on_int_nat(F),A)
     => ( ~ aa_set_nat_o(member_nat(X),B4)
       => ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4) = image_int_nat2(F,A) )
         => ~ ! [X8: int,A6: set_int] :
                ( ~ aa_set_int_o(member_int(X8),A6)
               => ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X8),A6) )
                 => ( ( X = aa_int_nat(F,X8) )
                   => ( B4 != image_int_nat2(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4538_inj__img__insertE,axiom,
    ! [F: fun_o_nat,A: set_o,X: nat,B4: set_nat] :
      ( aa_set_o_o(inj_on_o_nat(F),A)
     => ( ~ aa_set_nat_o(member_nat(X),B4)
       => ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4) = image_o_nat2(F,A) )
         => ~ ! [X8: $o,A6: set_o] :
                ( ~ aa_set_o_o(member_o((X8)),A6)
               => ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X8)),A6) )
                 => ( ( X = aa_o_nat(F,(X8)) )
                   => ( B4 != image_o_nat2(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4539_inj__img__insertE,axiom,
    ! [F: fun_Product_unit_nat,A: set_Product_unit,X: nat,B4: set_nat] :
      ( aa_set8780210009671629607unit_o(inj_on8430439091780834860it_nat(F),A)
     => ( ~ aa_set_nat_o(member_nat(X),B4)
       => ( ( aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),B4) = image_875570014554754200it_nat(F,A) )
         => ~ ! [X8: product_unit,A6: set_Product_unit] :
                ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X8),A6)
               => ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X8),A6) )
                 => ( ( X = aa_Product_unit_nat(F,X8) )
                   => ( B4 != image_875570014554754200it_nat(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4540_inj__img__insertE,axiom,
    ! [F: fun_nat_int,A: set_nat,X: int,B4: set_int] :
      ( aa_set_nat_o(inj_on_nat_int(F),A)
     => ( ~ aa_set_int_o(member_int(X),B4)
       => ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4) = image_nat_int2(F,A) )
         => ~ ! [X8: nat,A6: set_nat] :
                ( ~ aa_set_nat_o(member_nat(X8),A6)
               => ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X8),A6) )
                 => ( ( X = aa_nat_int(F,X8) )
                   => ( B4 != image_nat_int2(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4541_inj__img__insertE,axiom,
    ! [F: fun_int_int,A: set_int,X: int,B4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A)
     => ( ~ aa_set_int_o(member_int(X),B4)
       => ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4) = image_int_int2(F,A) )
         => ~ ! [X8: int,A6: set_int] :
                ( ~ aa_set_int_o(member_int(X8),A6)
               => ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X8),A6) )
                 => ( ( X = aa_int_int(F,X8) )
                   => ( B4 != image_int_int2(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4542_inj__img__insertE,axiom,
    ! [F: fun_o_int,A: set_o,X: int,B4: set_int] :
      ( aa_set_o_o(inj_on_o_int(F),A)
     => ( ~ aa_set_int_o(member_int(X),B4)
       => ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4) = image_o_int2(F,A) )
         => ~ ! [X8: $o,A6: set_o] :
                ( ~ aa_set_o_o(member_o((X8)),A6)
               => ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X8)),A6) )
                 => ( ( X = aa_o_int(F,(X8)) )
                   => ( B4 != image_o_int2(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4543_inj__img__insertE,axiom,
    ! [F: fun_Product_unit_int,A: set_Product_unit,X: int,B4: set_int] :
      ( aa_set8780210009671629607unit_o(inj_on8427948621271784584it_int(F),A)
     => ( ~ aa_set_int_o(member_int(X),B4)
       => ( ( aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),B4) = image_873079544045703924it_int(F,A) )
         => ~ ! [X8: product_unit,A6: set_Product_unit] :
                ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X8),A6)
               => ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X8),A6) )
                 => ( ( X = aa_Product_unit_int(F,X8) )
                   => ( B4 != image_873079544045703924it_int(F,A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4544_type__copy__wit,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,X: nat,S: fun_int_set_nat,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_nat_o(member_nat(X),aa_Cod6142706534279259169et_nat(comp_i5102403792180996084nteger(S,Rep),aa_int_Code_integer(Abs,Y)))
       => aa_set_nat_o(member_nat(X),aa_int_set_nat(S,Y)) ) ) ).

% type_copy_wit
tff(fact_4545_type__copy__wit,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,X: int,S: fun_int_set_int,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_int_o(member_int(X),aa_Cod1964855514770062461et_int(comp_i1887841406321795152nteger(S,Rep),aa_int_Code_integer(Abs,Y)))
       => aa_set_int_o(member_int(X),aa_int_set_int(S,Y)) ) ) ).

% type_copy_wit
tff(fact_4546_type__copy__wit,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,X: $o,S: fun_int_set_o,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set_o_o(member_o((X)),aa_Cod777817632936341789_set_o(comp_i5479505385629679622nteger(S,Rep),aa_int_Code_integer(Abs,Y)))
       => aa_set_o_o(member_o((X)),aa_int_set_o(S,Y)) ) ) ).

% type_copy_wit
tff(fact_4547_type__copy__wit,axiom,
    ! [Rep: fun_Code_integer_int,Abs: fun_int_Code_integer,X: product_unit,S: fun_in4731230917697954842t_unit,Y: int] :
      ( aa_set_int_o(type_d8366093980585677751er_int(Rep,Abs),top_top_set_int)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),aa_Cod82315800264730130t_unit(comp_i3319814986458681467nteger(S,Rep),aa_int_Code_integer(Abs,Y)))
       => aa_set8780210009671629607unit_o(member_Product_unit(X),aa_int6548451963861500163t_unit(S,Y)) ) ) ).

% type_copy_wit
tff(fact_4548_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,X: nat,S: fun_o_set_nat,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_nat_o(member_nat(X),aa_Pro6064029160526250199et_nat(comp_o9190160103024335696t_unit(S,Rep),aa_o_Product_unit(Abs,(Y))))
       => aa_set_nat_o(member_nat(X),aa_o_set_nat(S,(Y))) ) ) ).

% type_copy_wit
tff(fact_4549_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,X: int,S: fun_o_set_int,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_int_o(member_int(X),aa_Pro1886178141017053491et_int(comp_o1521574261646630388t_unit(S,Rep),aa_o_Product_unit(Abs,(Y))))
       => aa_set_int_o(member_int(X),aa_o_set_int(S,(Y))) ) ) ).

% type_copy_wit
tff(fact_4550_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,X: $o,S: fun_o_set_o,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(member_o((X)),aa_Pro536174410435844839_set_o(comp_o6895741221744486154t_unit(S,Rep),aa_o_Product_unit(Abs,(Y))))
       => aa_set_o_o(member_o((X)),aa_o_set_o(S,(Y))) ) ) ).

% type_copy_wit
tff(fact_4551_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,X: product_unit,S: fun_o_1833192751255691672t_unit,Y: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),aa_Pro234974305641502940t_unit(comp_o2086126948152544725t_unit(S,Rep),aa_o_Product_unit(Abs,(Y))))
       => aa_set8780210009671629607unit_o(member_Product_unit(X),aa_o_s3492205133957624465t_unit(S,(Y))) ) ) ).

% type_copy_wit
tff(fact_4552_type__copy__wit,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,X: nat,S: fun_nat_set_nat,Y: nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),top_top_set_nat)
     => ( aa_set_nat_o(member_nat(X),comp_n4787279007288201059atural(S,Rep,aa_nat_Code_natural(Abs,Y)))
       => aa_set_nat_o(member_nat(X),aa_nat_set_nat(S,Y)) ) ) ).

% type_copy_wit
tff(fact_4553_type__copy__wit,axiom,
    ! [Rep: fun_Code_natural_nat,Abs: fun_nat_Code_natural,X: int,S: fun_nat_set_int,Y: nat] :
      ( aa_set_nat_o(type_d4410041424927559462al_nat(Rep,Abs),top_top_set_nat)
     => ( aa_set_int_o(member_int(X),comp_n1572716621429000127atural(S,Rep,aa_nat_Code_natural(Abs,Y)))
       => aa_set_int_o(member_int(X),aa_nat_set_int(S,Y)) ) ) ).

% type_copy_wit
tff(fact_4554_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_nat_nat,G: fun_nat_nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,image_5168914502847457605at_nat(produc6977886695330630970at_nat(F,G),R))
     => ~ ! [X3: nat,Y2: nat] :
            ( ( C = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,X3)),aa_nat_nat(G,Y2)) )
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4555_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_int_nat,G: fun_int_nat,R: set_Pr958786334691620121nt_int] :
      ( member8440522571783428010at_nat(C,image_5831868185956570253at_nat(produc5667422475138734578nt_nat(F,G),R))
     => ~ ! [X3: int,Y2: int] :
            ( ( C = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,X3)),aa_int_nat(G,Y2)) )
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4556_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: fun_nat_int,G: fun_nat_int,R: set_Pr1261947904930325089at_nat] :
      ( member5262025264175285858nt_int(C,image_1990417195239315453nt_int(produc7347049785058425330at_int(F,G),R))
     => ~ ! [X3: nat,Y2: nat] :
            ( ( C = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,X3)),aa_nat_int(G,Y2)) )
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4557_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: fun_int_int,G: fun_int_int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(C,image_2653370878348428101nt_int(produc6036585564866528938nt_int(F,G),R))
     => ~ ! [X3: int,Y2: int] :
            ( ( C = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,X3)),aa_int_int(G,Y2)) )
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4558_prod__fun__imageE,axiom,
    ! [C: produc3658429121746597890et_nat,F: fun_na8669532458821385046t_unit,G: fun_nat_set_nat,R: set_Pr1261947904930325089at_nat] :
      ( member6260224972018164377et_nat(C,image_9068448769188195518et_nat(produc2064153442785501et_nat(F,G),R))
     => ~ ! [X3: nat,Y2: nat] :
            ( ( C = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_nat4191430056936421439t_unit(F,X3)),aa_nat_set_nat(G,Y2)) )
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4559_prod__fun__imageE,axiom,
    ! [C: produc3658429121746597890et_nat,F: fun_in7572449478177386490t_unit,G: fun_int_set_nat,R: set_Pr958786334691620121nt_int] :
      ( member6260224972018164377et_nat(C,image_2127501072796355446et_nat(produc3152090231653401565et_nat(F,G),R))
     => ~ ! [X3: int,Y2: int] :
            ( ( C = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_int3094347076292422883t_unit(F,X3)),aa_int_set_nat(G,Y2)) )
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4560_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_he1033112309995878576it_nat,G: fun_set_nat_nat,R: set_Pr3948176798113811640et_nat] :
      ( member8440522571783428010at_nat(C,image_7757775281999735736at_nat(produc8285742784527409791at_nat(F,G),R))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
            ( ( C = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_hea5848002036693495265it_nat(F,X3)),aa_set_nat_nat(G,Y2)) )
           => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4561_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: fun_he6078633327341457676it_int,G: fun_set_nat_int,R: set_Pr3948176798113811640et_nat] :
      ( member5262025264175285858nt_int(C,image_4579277974391593584nt_int(produc5973434381925408567at_int(F,G),R))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
            ( ( C = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_hea5845511566184444989it_int(F,X3)),aa_set_nat_int(G,Y2)) )
           => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4562_prod__fun__imageE,axiom,
    ! [C: produc3658429121746597890et_nat,F: fun_he2331061685413951235t_unit,G: fun_set_nat_set_nat,R: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(C,image_7426123528223433739et_nat(produc8234977478398262212et_nat(F,G),R))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
            ( ( C = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(F,X3)),aa_set_nat_set_nat(G,Y2)) )
           => ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4563_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_fu3504661237974713170_o_nat,G: fun_Pr6529957239379720399at_nat,R: set_Pr1312406793078187350et_nat] :
      ( member8440522571783428010at_nat(C,image_3566234039933809690at_nat(produc8263063249334093469at_nat(F,G),R))
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( C = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_fun767828674014743683_o_nat(F,X3)),aa_Pro8901340430247369232at_nat(G,Y2)) )
           => ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2),R) ) ) ).

% prod_fun_imageE
tff(fact_4564_inj__on__iff__surj,axiom,
    ! [A: set_Code_integer,A7: set_Code_integer] :
      ( ( A != bot_bo3990330152332043303nteger )
     => ( ? [F4: fun_Co8499238673658658103nteger] :
            ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F4),A)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(F4),A)),A7) )
      <=> ? [G3: fun_Co8499238673658658103nteger] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(G3),A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4565_inj__on__iff__surj,axiom,
    ! [A: set_nat,A7: set_set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ? [F4: fun_nat_set_nat] :
            ( aa_set_nat_o(inj_on_nat_set_nat(F4),A)
            & ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F4),A),A7) )
      <=> ? [G3: fun_set_nat_nat] : image_set_nat_nat(G3,A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4566_inj__on__iff__surj,axiom,
    ! [A: set_nat,A7: set_Code_integer] :
      ( ( A != bot_bot_set_nat )
     => ( ? [F4: fun_nat_Code_integer] :
            ( aa_set_nat_o(inj_on2245834603434889120nteger(F4),A)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(aa_set7438092518518805847nteger(image_1215581382706833972nteger(F4),A)),A7) )
      <=> ? [G3: fun_Code_integer_nat] : image_951025933927791156er_nat(G3,A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4567_inj__on__iff__surj,axiom,
    ! [A: set_set_nat,A7: set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( ? [F4: fun_set_nat_nat] :
            ( inj_on_set_nat_nat(F4,A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_set_nat_nat(F4,A)),A7) )
      <=> ? [G3: fun_nat_set_nat] : aa_set7246812951530217472et_nat(image_nat_set_nat(G3),A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4568_inj__on__iff__surj,axiom,
    ! [A: set_Code_integer,A7: set_nat] :
      ( ( A != bot_bo3990330152332043303nteger )
     => ( ? [F4: fun_Code_integer_nat] :
            ( aa_set5643387332196575111eger_o(inj_on1981279154655846304er_nat(F4),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_951025933927791156er_nat(F4,A)),A7) )
      <=> ? [G3: fun_nat_Code_integer] : aa_set7438092518518805847nteger(image_1215581382706833972nteger(G3),A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4569_inj__on__iff__surj,axiom,
    ! [A: set_Product_unit,A7: set_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ? [F4: fun_Product_unit_nat] :
            ( aa_set8780210009671629607unit_o(inj_on8430439091780834860it_nat(F4),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_875570014554754200it_nat(F4,A)),A7) )
      <=> ? [G3: fun_nat_Product_unit] : image_8730104196221521654t_unit(G3,A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4570_inj__on__iff__surj,axiom,
    ! [A: set_o,A7: set_nat] :
      ( ( A != bot_bot_set_o )
     => ( ? [F4: fun_o_nat] :
            ( aa_set_o_o(inj_on_o_nat(F4),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_o_nat2(F4,A)),A7) )
      <=> ? [G3: fun_nat_o] : image_nat_o2(G3,A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4571_inj__on__iff__surj,axiom,
    ! [A: set_nat,A7: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ? [F4: fun_nat_nat] :
            ( aa_set_nat_o(inj_on_nat_nat(F4),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat2(F4),A)),A7) )
      <=> ? [G3: fun_nat_nat] : aa_set_nat_set_nat(image_nat_nat2(G3),A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4572_inj__on__iff__surj,axiom,
    ! [A: set_int,A7: set_nat] :
      ( ( A != bot_bot_set_int )
     => ( ? [F4: fun_int_nat] :
            ( aa_set_int_o(inj_on_int_nat(F4),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_int_nat2(F4,A)),A7) )
      <=> ? [G3: fun_nat_int] : image_nat_int2(G3,A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4573_inj__on__iff__surj,axiom,
    ! [A: set_fu3555266933746557007_nat_o,A7: set_nat] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( ? [F4: fun_fu3504661237974713170_o_nat] :
            ( aa_set5091322549519198405at_o_o(inj_on5905443631705101838_o_nat(F4),A)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_5746879016480857722_o_nat(F4,A)),A7) )
      <=> ? [G3: fun_na5781657914532371832_nat_o] : image_7177977788501158744_nat_o(G3,A7) = A ) ) ).

% inj_on_iff_surj
tff(fact_4574_Sup__insert__finite,axiom,
    ! [S: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( complete_Sup_Sup_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),S)) = $ite(S = bot_bot_set_int,X,aa_int_int(aa_int_fun_int_int(ord_max_int,X),complete_Sup_Sup_int(S))) ) ) ).

% Sup_insert_finite
tff(fact_4575_Sup__insert__finite,axiom,
    ! [S: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),S)) = $ite(S = bot_bot_set_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),aa_set_nat_nat(complete_Sup_Sup_nat,S))) ) ) ).

% Sup_insert_finite
tff(fact_4576_Sup__insert__finite,axiom,
    ! [S: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( comple4687483117567863418t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),S)) = $ite(S = bot_bo3957492148770167129t_unit,X,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X),comple4687483117567863418t_unit(S))) ) ) ).

% Sup_insert_finite
tff(fact_4577_hom__Max__commute,axiom,
    ! [Ha: fun_Co8499238673658658103nteger,N: set_Code_integer] :
      ( ! [X3: code_integer,Y2: code_integer] : aa_Cod1783974652533396728nteger(Ha,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X3),Y2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_Cod1783974652533396728nteger(Ha,X3)),aa_Cod1783974652533396728nteger(Ha,Y2))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,N)
       => ( ( N != bot_bo3990330152332043303nteger )
         => ( aa_Cod1783974652533396728nteger(Ha,lattic4901227151466704046nteger(N)) = lattic4901227151466704046nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(Ha),N)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4578_hom__Max__commute,axiom,
    ! [Ha: fun_Pr2191293291710260995t_unit,N: set_Product_unit] :
      ( ! [X3: product_unit,Y2: product_unit] : aa_Pro8294289767544744956t_unit(Ha,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X3),Y2)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,aa_Pro8294289767544744956t_unit(Ha,X3)),aa_Pro8294289767544744956t_unit(Ha,Y2))
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,N)
       => ( ( N != bot_bo3957492148770167129t_unit )
         => ( aa_Pro8294289767544744956t_unit(Ha,lattic1210392890313158908t_unit(N)) = lattic1210392890313158908t_unit(image_405062704495631173t_unit(Ha,N)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4579_hom__Max__commute,axiom,
    ! [Ha: fun_o_o,N: set_o] :
      ( ! [X3: $o,Y2: $o] :
          ( aa_o_o(Ha,aa_o_o(aa_o_fun_o_o(ord_max_o,(X3)),(Y2)))
        <=> aa_o_o(aa_o_fun_o_o(ord_max_o,aa_o_o(Ha,(X3))),aa_o_o(Ha,(Y2))) )
     => ( aa_set_o_o(finite_finite_o,N)
       => ( ( N != bot_bot_set_o )
         => ( aa_o_o(Ha,aa_set_o_o(lattic1921953407002678535_Max_o,N))
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,image_o_o2(Ha,N)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4580_hom__Max__commute,axiom,
    ! [Ha: fun_int_int,N: set_int] :
      ( ! [X3: int,Y2: int] : aa_int_int(Ha,aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(Ha,X3)),aa_int_int(Ha,Y2))
     => ( aa_set_int_o(finite_finite_int,N)
       => ( ( N != bot_bot_set_int )
         => ( aa_int_int(Ha,lattic8263393255366662781ax_int(N)) = lattic8263393255366662781ax_int(image_int_int2(Ha,N)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4581_hom__Max__commute,axiom,
    ! [Ha: fun_nat_nat,N: set_nat] :
      ( ! [X3: nat,Y2: nat] : aa_nat_nat(Ha,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(Ha,X3)),aa_nat_nat(Ha,Y2))
     => ( aa_set_nat_o(finite_finite_nat,N)
       => ( ( N != bot_bot_set_nat )
         => ( aa_nat_nat(Ha,aa_set_nat_nat(lattic8265883725875713057ax_nat,N)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat2(Ha),N)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4582_Max_Osubset,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( B4 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A)
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,lattic4901227151466704046nteger(B4)),lattic4901227151466704046nteger(A)) = lattic4901227151466704046nteger(A) ) ) ) ) ).

% Max.subset
tff(fact_4583_Max_Osubset,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( B4 != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),A)
         => ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,lattic1210392890313158908t_unit(B4)),lattic1210392890313158908t_unit(A)) = lattic1210392890313158908t_unit(A) ) ) ) ) ).

% Max.subset
tff(fact_4584_Max_Osubset,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( B4 != bot_bot_set_o )
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),A)
         => ( aa_o_o(aa_o_fun_o_o(ord_max_o,aa_set_o_o(lattic1921953407002678535_Max_o,B4)),aa_set_o_o(lattic1921953407002678535_Max_o,A))
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,A) ) ) ) ) ).

% Max.subset
tff(fact_4585_Max_Osubset,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( B4 != bot_bot_set_int )
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),A)
         => ( aa_int_int(aa_int_fun_int_int(ord_max_int,lattic8263393255366662781ax_int(B4)),lattic8263393255366662781ax_int(A)) = lattic8263393255366662781ax_int(A) ) ) ) ) ).

% Max.subset
tff(fact_4586_Max_Osubset,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( B4 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,B4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,A)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,A) ) ) ) ) ).

% Max.subset
tff(fact_4587_Max_Oinsert__not__elem,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( ( A != bot_bo3990330152332043303nteger )
         => ( lattic4901227151466704046nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X),lattic4901227151466704046nteger(A)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4588_Max_Oinsert__not__elem,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( ( A != bot_bo3957492148770167129t_unit )
         => ( lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X),lattic1210392890313158908t_unit(A)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4589_Max_Oinsert__not__elem,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( ( A != bot_bot_set_o )
         => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
          <=> aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),aa_set_o_o(lattic1921953407002678535_Max_o,A)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4590_Max_Oinsert__not__elem,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( ( A != bot_bot_set_int )
         => ( lattic8263393255366662781ax_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_int_int(aa_int_fun_int_int(ord_max_int,X),lattic8263393255366662781ax_int(A)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4591_Max_Oinsert__not__elem,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( ( A != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),aa_set_nat_nat(lattic8265883725875713057ax_nat,A)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4592_Max_Oclosed,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,Y2: code_integer] : aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X3),Y2)),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X3),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y2),bot_bo3990330152332043303nteger)))
         => aa_set5643387332196575111eger_o(member_Code_integer(lattic4901227151466704046nteger(A)),A) ) ) ) ).

% Max.closed
tff(fact_4593_Max_Oclosed,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit,Y2: product_unit] : aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,X3),Y2)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y2),bot_bo3957492148770167129t_unit)))
         => aa_set8780210009671629607unit_o(member_Product_unit(lattic1210392890313158908t_unit(A)),A) ) ) ) ).

% Max.closed
tff(fact_4594_Max_Oclosed,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y2: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(ord_max_o,(X3)),(Y2))),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y2)),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1921953407002678535_Max_o,A)),A) ) ) ) ).

% Max.closed
tff(fact_4595_Max_Oclosed,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X3: int,Y2: int] : aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Y2)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y2),bot_bot_set_int)))
         => aa_set_int_o(member_int(lattic8263393255366662781ax_int(A)),A) ) ) ) ).

% Max.closed
tff(fact_4596_Max_Oclosed,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [X3: nat,Y2: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Y2)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y2),bot_bot_set_nat)))
         => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),A) ) ) ) ).

% Max.closed
tff(fact_4597_Max_Ounion,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => ( ( B4 != bot_bo3990330152332043303nteger )
           => ( lattic4901227151466704046nteger(sup_su848401254843788991nteger(A,B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,lattic4901227151466704046nteger(A)),lattic4901227151466704046nteger(B4)) ) ) ) ) ) ).

% Max.union
tff(fact_4598_Max_Ounion,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => ( ( B4 != bot_bo3957492148770167129t_unit )
           => ( lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(ord_max_Product_unit,lattic1210392890313158908t_unit(A)),lattic1210392890313158908t_unit(B4)) ) ) ) ) ) ).

% Max.union
tff(fact_4599_Max_Ounion,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => ( ( B4 != bot_bot_set_o )
           => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
            <=> aa_o_o(aa_o_fun_o_o(ord_max_o,aa_set_o_o(lattic1921953407002678535_Max_o,A)),aa_set_o_o(lattic1921953407002678535_Max_o,B4)) ) ) ) ) ) ).

% Max.union
tff(fact_4600_Max_Ounion,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => ( ( B4 != bot_bot_set_int )
           => ( lattic8263393255366662781ax_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_int_int(aa_int_fun_int_int(ord_max_int,lattic8263393255366662781ax_int(A)),lattic8263393255366662781ax_int(B4)) ) ) ) ) ) ).

% Max.union
tff(fact_4601_Max_Ounion,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( B4 != bot_bot_set_nat )
           => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A)),aa_set_nat_nat(lattic8265883725875713057ax_nat,B4)) ) ) ) ) ) ).

% Max.union
tff(fact_4602_Max_Oeq__fold,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))
      <=> aa_set_o_o(finite_fold_o_o(ord_max_o,(X)),A) ) ) ).

% Max.eq_fold
tff(fact_4603_Max_Oeq__fold,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( lattic1210392890313158908t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = finite8834431295148555784t_unit(ord_max_Product_unit,X,A) ) ) ).

% Max.eq_fold
tff(fact_4604_Max_Oeq__fold,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( lattic4901227151466704046nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = finite6454954857057555716nteger(ord_max_Code_integer,X,A) ) ) ).

% Max.eq_fold
tff(fact_4605_Max_Oeq__fold,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( lattic8263393255366662781ax_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = finite_fold_int_int(ord_max_int,X,A) ) ) ).

% Max.eq_fold
tff(fact_4606_Max_Oeq__fold,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_set_nat_nat(finite_fold_nat_nat(ord_max_nat,X),A) ) ) ).

% Max.eq_fold
tff(fact_4607_finite__range__prod,axiom,
    ! [F: fun_Co2652536811680367946nteger] :
      ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(comp_P5725896914063526938nteger(produc8897218493789310642nteger,F),top_to4645266643341252675nteger))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(comp_P6122662217367900327nteger(produc1781891006948753140nteger,F)),top_to4645266643341252675nteger))
       => finite1154042800025562295nteger(image_3408592110290467754nteger(F,top_to4645266643341252675nteger)) ) ) ).

% finite_range_prod
tff(fact_4608_finite__range__prod,axiom,
    ! [F: fun_Co9221593637010378er_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(comp_P7101464685127059239nteger(produc8632663045010267826er_nat,F)),top_to4645266643341252675nteger))
     => ( aa_set_nat_o(finite_finite_nat,image_951025933927791156er_nat(comp_P5964733648194011802nteger(produc1517335558169710324er_nat,F),top_to4645266643341252675nteger))
       => finite4996374594527322935er_nat(image_7250923904792228394er_nat(F,top_to4645266643341252675nteger)) ) ) ).

% finite_range_prod
tff(fact_4609_finite__range__prod,axiom,
    ! [F: fun_Co6597186368970283863nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(comp_P7111796090210262324nteger(produc8508995932063986495nteger,F)),top_to4645266643341252675nteger))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_set8622560600277679972nteger(image_4470545334726330049nteger(comp_P7111796090210262324nteger(produc6174133586879617921nteger,F)),top_to4645266643341252675nteger))
       => finite2154694888287295812nteger(image_8049062792287394871nteger(F,top_to4645266643341252675nteger)) ) ) ).

% finite_range_prod
tff(fact_4610_finite__range__prod,axiom,
    ! [F: fun_Pr7036769738611282295at_nat] :
      ( aa_set_nat_o(finite_finite_nat,image_875570014554754200it_nat(comp_P2980510830910040157t_unit(product_fst_nat_nat,F),top_to1996260823553986621t_unit))
     => ( aa_set_nat_o(finite_finite_nat,image_875570014554754200it_nat(comp_P2980510830910040157t_unit(product_snd_nat_nat,F),top_to1996260823553986621t_unit))
       => finite6177210948735845034at_nat(image_7281959405765017145at_nat(F,top_to1996260823553986621t_unit)) ) ) ).

% finite_range_prod
tff(fact_4611_finite__range__prod,axiom,
    ! [F: fun_Pr7106788231463976196nteger] :
      ( aa_set_nat_o(finite_finite_nat,image_875570014554754200it_nat(comp_P7621350044865347728t_unit(produc8897218493789310642nteger,F),top_to1996260823553986621t_unit))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_3815118110082124069nteger(comp_P7320554561093743171t_unit(produc1781891006948753140nteger,F),top_to1996260823553986621t_unit))
       => finite1154042800025562295nteger(image_4946983642536848454nteger(F,top_to1996260823553986621t_unit)) ) ) ).

% finite_range_prod
tff(fact_4612_finite__range__prod,axiom,
    ! [F: fun_Pr4463473013420618628er_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_3815118110082124069nteger(comp_P7581484221039074243t_unit(produc8632663045010267826er_nat,F),top_to1996260823553986621t_unit))
     => ( aa_set_nat_o(finite_finite_nat,image_875570014554754200it_nat(comp_P6821003776282389008t_unit(produc1517335558169710324er_nat,F),top_to1996260823553986621t_unit))
       => finite4996374594527322935er_nat(image_8789315437038609094er_nat(F,top_to1996260823553986621t_unit)) ) ) ).

% finite_range_prod
tff(fact_4613_finite__range__prod,axiom,
    ! [F: fun_Pr864343514034809617nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_3815118110082124069nteger(comp_P659088277599099894t_unit(produc8508995932063986495nteger,F),top_to1996260823553986621t_unit))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_3815118110082124069nteger(comp_P659088277599099894t_unit(produc6174133586879617921nteger,F),top_to1996260823553986621t_unit))
       => finite2154694888287295812nteger(image_8089323762101378259nteger(F,top_to1996260823553986621t_unit)) ) ) ).

% finite_range_prod
tff(fact_4614_finite__range__prod,axiom,
    ! [F: fun_Pr6733608168372577327nt_int] :
      ( aa_set_int_o(finite_finite_int,image_873079544045703924it_int(comp_P6796073005956588617t_unit(product_fst_int_int,F),top_to1996260823553986621t_unit))
     => ( aa_set_int_o(finite_finite_int,image_873079544045703924it_int(comp_P6796073005956588617t_unit(product_snd_int_int,F),top_to1996260823553986621t_unit))
       => finite2998713641127702882nt_int(image_4103462098156874993nt_int(F,top_to1996260823553986621t_unit)) ) ) ).

% finite_range_prod
tff(fact_4615_finite__range__prod,axiom,
    ! [F: fun_o_67843798866261420at_nat] :
      ( aa_set_nat_o(finite_finite_nat,image_o_nat2(comp_P4394204175247225320_nat_o(product_fst_nat_nat,F),top_top_set_o))
     => ( aa_set_nat_o(finite_finite_nat,image_o_nat2(comp_P4394204175247225320_nat_o(product_snd_nat_nat,F),top_top_set_o))
       => finite6177210948735845034at_nat(image_3855930084881510382at_nat(F,top_top_set_o)) ) ) ).

% finite_range_prod
tff(fact_4616_finite__range__prod,axiom,
    ! [F: fun_o_8405014634842623289nteger] :
      ( aa_set_nat_o(finite_finite_nat,image_o_nat2(comp_P7179991235626930843_nat_o(produc8897218493789310642nteger,F),top_top_set_o))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_o_Code_integer(comp_P7177471617286032590eger_o(produc1781891006948753140nteger,F),top_top_set_o))
       => finite1154042800025562295nteger(image_8407096251609565691nteger(F,top_top_set_o)) ) ) ).

% finite_range_prod
tff(fact_4617_inj__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] :
      ( inj_on20428373856169009nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),top_to2363841527343591755nteger)
    <=> aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(F),top_to4645266643341252675nteger) ) ).

% inj_apsnd
tff(fact_4618_total__inv__image,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,R2: set_Pr5094982260447487303t_unit] :
      ( aa_set8780210009671629607unit_o(inj_on8151373323710067377t_unit(F),top_to1996260823553986621t_unit)
     => ( total_5665700600833547665t_unit(top_to1996260823553986621t_unit,R2)
       => total_5665700600833547665t_unit(top_to1996260823553986621t_unit,inv_im8770536620514187154t_unit(R2,F)) ) ) ).

% total_inv_image
tff(fact_4619_total__inv__image,axiom,
    ! [F: fun_Product_unit_o,R2: set_Product_prod_o_o] :
      ( aa_set8780210009671629607unit_o(inj_on8854722734121536060unit_o(F),top_to1996260823553986621t_unit)
     => ( total_on_o(top_top_set_o,R2)
       => total_5665700600833547665t_unit(top_to1996260823553986621t_unit,inv_im4602722609573551687t_unit(R2,F)) ) ) ).

% total_inv_image
tff(fact_4620_total__inv__image,axiom,
    ! [F: fun_Product_unit_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set8780210009671629607unit_o(inj_on8430439091780834860it_nat(F),top_to1996260823553986621t_unit)
     => ( total_on_nat(top_top_set_nat,R2)
       => total_5665700600833547665t_unit(top_to1996260823553986621t_unit,inv_im7624054837341743209t_unit(R2,F)) ) ) ).

% total_inv_image
tff(fact_4621_total__inv__image,axiom,
    ! [F: fun_Product_unit_int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set8780210009671629607unit_o(inj_on8427948621271784584it_int(F),top_to1996260823553986621t_unit)
     => ( total_on_int(top_top_set_int,R2)
       => total_5665700600833547665t_unit(top_to1996260823553986621t_unit,inv_im4740476784543923469t_unit(R2,F)) ) ) ).

% total_inv_image
tff(fact_4622_total__inv__image,axiom,
    ! [F: fun_o_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( aa_set_o_o(inj_on4268752236855399398t_unit(F),top_top_set_o)
     => ( total_5665700600833547665t_unit(top_to1996260823553986621t_unit,R2)
       => total_on_o(top_top_set_o,inv_im9188693106839688349unit_o(R2,F)) ) ) ).

% total_inv_image
tff(fact_4623_total__inv__image,axiom,
    ! [F: fun_o_o,R2: set_Product_prod_o_o] :
      ( aa_set_o_o(inj_on_o_o(F),top_top_set_o)
     => ( total_on_o(top_top_set_o,R2)
       => total_on_o(top_top_set_o,inv_image_o_o(R2,F)) ) ) ).

% total_inv_image
tff(fact_4624_total__inv__image,axiom,
    ! [F: fun_o_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set_o_o(inj_on_o_nat(F),top_top_set_o)
     => ( total_on_nat(top_top_set_nat,R2)
       => total_on_o(top_top_set_o,inv_image_nat_o(R2,F)) ) ) ).

% total_inv_image
tff(fact_4625_total__inv__image,axiom,
    ! [F: fun_o_int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set_o_o(inj_on_o_int(F),top_top_set_o)
     => ( total_on_int(top_top_set_int,R2)
       => total_on_o(top_top_set_o,inv_image_int_o(R2,F)) ) ) ).

% total_inv_image
tff(fact_4626_total__inv__image,axiom,
    ! [F: fun_nat_Product_unit,R2: set_Pr5094982260447487303t_unit] :
      ( aa_set_nat_o(inj_on7061601236592826506t_unit(F),top_top_set_nat)
     => ( total_5665700600833547665t_unit(top_to1996260823553986621t_unit,R2)
       => total_on_nat(top_top_set_nat,inv_im8992892692529751563it_nat(R2,F)) ) ) ).

% total_inv_image
tff(fact_4627_total__inv__image,axiom,
    ! [F: fun_nat_o,R2: set_Product_prod_o_o] :
      ( aa_set_nat_o(inj_on_nat_o(F),top_top_set_nat)
     => ( total_on_o(top_top_set_o,R2)
       => total_on_nat(top_top_set_nat,inv_image_o_nat(R2,F)) ) ) ).

% total_inv_image
tff(fact_4628_card__Diff1__less,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))),finite4902975817058060853nteger(A)) ) ) ).

% card_Diff1_less
tff(fact_4629_card__Diff1__less,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A)) ) ) ).

% card_Diff1_less
tff(fact_4630_card__Diff1__less,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),finite_card_o(A)) ) ) ).

% card_Diff1_less
tff(fact_4631_card__Diff1__less,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))),finite_card_int(A)) ) ) ).

% card_Diff1_less
tff(fact_4632_card__Diff1__less,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A)) ) ) ).

% card_Diff1_less
tff(fact_4633_card__Diff1__less,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A)) ) ) ).

% card_Diff1_less
tff(fact_4634_card__Diff2__less,axiom,
    ! [A: set_Code_integer,X: code_integer,Y: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,Y),bot_bo3990330152332043303nteger)))),finite4902975817058060853nteger(A)) ) ) ) ).

% card_Diff2_less
tff(fact_4635_card__Diff2__less,axiom,
    ! [A: set_Product_unit,X: product_unit,Y: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_set8780210009671629607unit_o(member_Product_unit(Y),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,Y),bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A)) ) ) ) ).

% card_Diff2_less
tff(fact_4636_card__Diff2__less,axiom,
    ! [A: set_o,X: $o,Y: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_o(member_o((Y)),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(Y)),bot_bot_set_o)))),finite_card_o(A)) ) ) ) ).

% card_Diff2_less
tff(fact_4637_card__Diff2__less,axiom,
    ! [A: set_int,X: int,Y: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( aa_set_int_o(member_int(Y),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,Y),bot_bot_set_int)))),finite_card_int(A)) ) ) ) ).

% card_Diff2_less
tff(fact_4638_card__Diff2__less,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(Y),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A)) ) ) ) ).

% card_Diff2_less
tff(fact_4639_card__Diff2__less,axiom,
    ! [A: set_nat,X: nat,Y: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_o(member_nat(Y),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,Y),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A)) ) ) ) ).

% card_Diff2_less
tff(fact_4640_card__Diff1__less__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))),finite4902975817058060853nteger(A))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
        & aa_set5643387332196575111eger_o(member_Code_integer(X),A) ) ) ).

% card_Diff1_less_iff
tff(fact_4641_card__Diff1__less__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A))
    <=> ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
        & aa_set8780210009671629607unit_o(member_Product_unit(X),A) ) ) ).

% card_Diff1_less_iff
tff(fact_4642_card__Diff1__less__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),finite_card_o(A))
    <=> ( aa_set_o_o(finite_finite_o,A)
        & aa_set_o_o(member_o((X)),A) ) ) ).

% card_Diff1_less_iff
tff(fact_4643_card__Diff1__less__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))),finite_card_int(A))
    <=> ( aa_set_int_o(finite_finite_int,A)
        & aa_set_int_o(member_int(X),A) ) ) ).

% card_Diff1_less_iff
tff(fact_4644_card__Diff1__less__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A))
    <=> ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
        & aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A) ) ) ).

% card_Diff1_less_iff
tff(fact_4645_card__Diff1__less__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A))
    <=> ( aa_set_nat_o(finite_finite_nat,A)
        & aa_set_nat_o(member_nat(X),A) ) ) ).

% card_Diff1_less_iff
tff(fact_4646_Inter__in__chain,axiom,
    ! [B8: set_set_nat,A11: set_set_nat] :
      ( finite1152437895449049373et_nat(B8)
     => ( ( B8 != bot_bot_set_set_nat )
       => ( pred_chain_set_nat(A11,ord_less_set_nat,B8)
         => member_set_nat(comple7806235888213564991et_nat(B8),B8) ) ) ) ).

% Inter_in_chain
tff(fact_4647_card__eq__UNIV2,axiom,
    ! [S: set_Product_unit] :
      ( ( finite410649719033368117t_unit(top_to1996260823553986621t_unit) = finite410649719033368117t_unit(S) )
    <=> ( S = top_to1996260823553986621t_unit ) ) ).

% card_eq_UNIV2
tff(fact_4648_card__eq__UNIV2,axiom,
    ! [S: set_o] :
      ( ( finite_card_o(top_top_set_o) = finite_card_o(S) )
    <=> ( S = top_top_set_o ) ) ).

% card_eq_UNIV2
tff(fact_4649_card__eq__UNIV,axiom,
    ! [S: set_Product_unit] :
      ( ( finite410649719033368117t_unit(S) = finite410649719033368117t_unit(top_to1996260823553986621t_unit) )
    <=> ( S = top_to1996260823553986621t_unit ) ) ).

% card_eq_UNIV
tff(fact_4650_card__eq__UNIV,axiom,
    ! [S: set_o] :
      ( ( finite_card_o(S) = finite_card_o(top_top_set_o) )
    <=> ( S = top_top_set_o ) ) ).

% card_eq_UNIV
tff(fact_4651_fst__map__prod,axiom,
    ! [F: fun_int_int,G: fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),X)) = aa_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)) ).

% fst_map_prod
tff(fact_4652_apfst__conv,axiom,
    ! [F: fun_he2331061685413951235t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] : aa_Pro3679490825741336130et_nat(produc4172003873565521026et_nat(F),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(F,X)),Y) ).

% apfst_conv
tff(fact_4653_apfst__conv,axiom,
    ! [F: fun_nat_nat,X: nat,Y: nat] : produc986720760941809198at_nat(F,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,X)),Y) ).

% apfst_conv
tff(fact_4654_apfst__conv,axiom,
    ! [F: fun_int_int,X: int,Y: int] : aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,X)),Y) ).

% apfst_conv
tff(fact_4655_apfst__conv,axiom,
    ! [F: fun_fu8419763957888848259_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : aa_Pro5931322043027775230et_nat(produc1926458052860118910et_nat(F),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_fun2638059832795814268_nat_o(F,X)),Y) ).

% apfst_conv
tff(fact_4656_apfst__conv,axiom,
    ! [F: fun_fu8419763957888848259_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : produc7064264152226348700et_nat(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_fun2638059832795814268_nat_o(F,X)),Y) ).

% apfst_conv
tff(fact_4657_apsnd__conv,axiom,
    ! [F: fun_set_nat_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] : aa_Pro3679490825741336130et_nat(produc3286488329990770625t_unit(F),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),aa_set_nat_set_nat(F,Y)) ).

% apsnd_conv
tff(fact_4658_apsnd__conv,axiom,
    ! [F: fun_nat_nat,X: nat,Y: nat] : produc3094765310956027504at_nat(F,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(F,Y)) ).

% apsnd_conv
tff(fact_4659_apsnd__conv,axiom,
    ! [F: fun_int_int,X: int,Y: int] : aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = aa_int7462565939943104719nt_int(product_Pair_int_int(X),aa_int_int(F,Y)) ).

% apsnd_conv
tff(fact_4660_apsnd__conv,axiom,
    ! [F: fun_Pr7091840695141096449et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : aa_Pro5931322043027775230et_nat(produc1991759244683392747_nat_o(F),aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),aa_Pro3679490825741336130et_nat(F,Y)) ).

% apsnd_conv
tff(fact_4661_apsnd__conv,axiom,
    ! [F: fun_Pr4885567672785483807et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] : produc4622684310891945677_nat_o(F,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro434340453207536480et_nat(F,Y)) ).

% apsnd_conv
tff(fact_4662_apsnd__conv,axiom,
    ! [F: fun_Pr2813956079162534559et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : produc174852792182663501_nat_o(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),aa_Pro5541488491759012064et_nat(F,Y)) ).

% apsnd_conv
tff(fact_4663_apsnd__conv,axiom,
    ! [F: fun_Pr836988951548913597et_nat,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] : produc2056317742435807279_nat_o(F,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),aa_Pro5931322043027775230et_nat(F,Y)) ).

% apsnd_conv
tff(fact_4664_apsnd__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: code_integer,Y: code_integer] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X),Y)) = aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(X),aa_Cod1783974652533396728nteger(F,Y)) ).

% apsnd_conv
tff(fact_4665_fst__apfst,axiom,
    ! [F: fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),X)) = aa_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X)) ).

% fst_apfst
tff(fact_4666_snd__apfst,axiom,
    ! [F: fun_he2331061685413951235t_unit,X: produc3658429121746597890et_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_Pro3679490825741336130et_nat(produc4172003873565521026et_nat(F),X)) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X) ).

% snd_apfst
tff(fact_4667_snd__apfst,axiom,
    ! [F: fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),X)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,X) ).

% snd_apfst
tff(fact_4668_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),inv_image_nat_nat(R2,F))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,X)),aa_nat_nat(F,Y)),R2) ) ).

% in_inv_image
tff(fact_4669_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr958786334691620121nt_int,F: fun_nat_int] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),inv_image_int_nat(R2,F))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,X)),aa_nat_int(F,Y)),R2) ) ).

% in_inv_image
tff(fact_4670_in__inv__image,axiom,
    ! [X: int,Y: int,R2: set_Pr1261947904930325089at_nat,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),inv_image_nat_int(R2,F))
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,X)),aa_int_nat(F,Y)),R2) ) ).

% in_inv_image
tff(fact_4671_in__inv__image,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,F: fun_int_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),inv_image_int_int(R2,F))
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,X)),aa_int_int(F,Y)),R2) ) ).

% in_inv_image
tff(fact_4672_fst__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),X)) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,X) ).

% fst_apsnd
tff(fact_4673_fst__apsnd,axiom,
    ! [F: fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),X)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,X) ).

% fst_apsnd
tff(fact_4674_apsnd__eq__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger,G: fun_Co8499238673658658103nteger] :
      ( ( aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),X) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),X) )
    <=> ( aa_Cod1783974652533396728nteger(F,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,X)) = aa_Cod1783974652533396728nteger(G,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,X)) ) ) ).

% apsnd_eq_conv
tff(fact_4675_snd__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),X)) = aa_Cod1783974652533396728nteger(F,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,X)) ).

% snd_apsnd
tff(fact_4676_snd__apsnd,axiom,
    ! [F: fun_set_nat_set_nat,X: produc3658429121746597890et_nat] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_Pro3679490825741336130et_nat(produc3286488329990770625t_unit(F),X)) = aa_set_nat_set_nat(F,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X)) ).

% snd_apsnd
tff(fact_4677_snd__apsnd,axiom,
    ! [F: fun_int_int,X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),X)) = aa_int_int(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ).

% snd_apsnd
tff(fact_4678_Inf__eq__bot__iff,axiom,
    ! [A: set_Product_unit] :
      ( ( comple2584293577114468500t_unit(A) = bot_bot_Product_unit )
    <=> ! [X2: product_unit] :
          ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,bot_bot_Product_unit),X2)
         => ? [Xa2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(Xa2),A)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,Xa2),X2) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_4679_Inf__empty,axiom,
    ( aa_set_o_o(complete_Inf_Inf_o,bot_bot_set_o)
  <=> top_top_o ) ).

% Inf_empty
tff(fact_4680_Inf__empty,axiom,
    comple6093339926577406454_nat_o(bot_bo8632255151191677115_nat_o) = top_to6660859512133667391_nat_o ).

% Inf_empty
tff(fact_4681_Inf__empty,axiom,
    comple7216939928495492468t_unit(bot_bo6284674557190688313t_unit) = top_to1996260823553986621t_unit ).

% Inf_empty
tff(fact_4682_Inf__empty,axiom,
    comple3063163877087187839_set_o(bot_bot_set_set_o) = top_top_set_o ).

% Inf_empty
tff(fact_4683_Inf__empty,axiom,
    comple3628384868704368283et_int(bot_bot_set_set_int) = top_top_set_int ).

% Inf_empty
tff(fact_4684_Inf__empty,axiom,
    comple7806235888213564991et_nat(bot_bot_set_set_nat) = top_top_set_nat ).

% Inf_empty
tff(fact_4685_Inf__empty,axiom,
    comple2584293577114468500t_unit(bot_bo3957492148770167129t_unit) = top_top_Product_unit ).

% Inf_empty
tff(fact_4686_Inf__UNIV,axiom,
    comple7216939928495492468t_unit(top_to1767297665138865437t_unit) = bot_bo3957492148770167129t_unit ).

% Inf_UNIV
tff(fact_4687_Inf__UNIV,axiom,
    comple3063163877087187839_set_o(top_top_set_set_o) = bot_bot_set_o ).

% Inf_UNIV
tff(fact_4688_Inf__UNIV,axiom,
    comple3628384868704368283et_int(top_top_set_set_int) = bot_bot_set_int ).

% Inf_UNIV
tff(fact_4689_Inf__UNIV,axiom,
    comple5146519332901335382_nat_o(top_to7216145672142822655_nat_o) = bot_bo8632255151191677115_nat_o ).

% Inf_UNIV
tff(fact_4690_Inf__UNIV,axiom,
    ( aa_set_o_o(complete_Inf_Inf_o,top_top_set_o)
  <=> bot_bot_o ) ).

% Inf_UNIV
tff(fact_4691_Inf__UNIV,axiom,
    comple7806235888213564991et_nat(top_top_set_set_nat) = bot_bot_set_nat ).

% Inf_UNIV
tff(fact_4692_Inf__UNIV,axiom,
    comple2584293577114468500t_unit(top_to1996260823553986621t_unit) = bot_bot_Product_unit ).

% Inf_UNIV
tff(fact_4693_cInf__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% cInf_singleton
tff(fact_4694_cInf__singleton,axiom,
    ! [X: int] : complete_Inf_Inf_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% cInf_singleton
tff(fact_4695_cInf__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o] : comple6093339926577406454_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) = X ).

% cInf_singleton
tff(fact_4696_cInf__singleton,axiom,
    ! [X: nat] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% cInf_singleton
tff(fact_4697_cInf__singleton,axiom,
    ! [X: set_nat] : comple7806235888213564991et_nat(insert_set_nat(X,bot_bot_set_set_nat)) = X ).

% cInf_singleton
tff(fact_4698_cInf__singleton,axiom,
    ! [X: product_unit] : comple2584293577114468500t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% cInf_singleton
tff(fact_4699_Inf__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(A2)),aa_set_o_o(complete_Inf_Inf_o,A)) ) ).

% Inf_insert
tff(fact_4700_Inf__insert,axiom,
    ! [A2: set_nat,A: set_set_nat] : comple7806235888213564991et_nat(insert_set_nat(A2,A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),comple7806235888213564991et_nat(A)) ).

% Inf_insert
tff(fact_4701_Inf__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] : comple2584293577114468500t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A2),comple2584293577114468500t_unit(A)) ).

% Inf_insert
tff(fact_4702_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_4703_prod_Ocollapse,axiom,
    ! [Prod: produc8799533983703829792et_nat] : aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Prod)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_4704_prod_Ocollapse,axiom,
    ! [Prod: produc3112576180687729086et_nat] : aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,Prod)),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_4705_prod_Ocollapse,axiom,
    ! [Prod: produc3658429121746597890et_nat] : aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_4706_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) = Prod ).

% prod.collapse
tff(fact_4707_card__ge__UNIV,axiom,
    ! [S: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(top_to1996260823553986621t_unit)),finite410649719033368117t_unit(S))
    <=> ( S = top_to1996260823553986621t_unit ) ) ).

% card_ge_UNIV
tff(fact_4708_card__ge__UNIV,axiom,
    ! [S: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(top_top_set_o)),finite_card_o(S))
    <=> ( S = top_top_set_o ) ) ).

% card_ge_UNIV
tff(fact_4709_card__Diff__insert,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),B4)
       => ( finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),B4))) = aa_nat_nat(minus_minus_nat(finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_4710_card__Diff__insert,axiom,
    ! [A2: int,A: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ( ~ aa_set_int_o(member_int(A2),B4)
       => ( finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),B4))) = aa_nat_nat(minus_minus_nat(finite_card_int(minus_minus_set_int(A,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_4711_card__Diff__insert,axiom,
    ! [A2: $o,A: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( ~ aa_set_o_o(member_o((A2)),B4)
       => ( finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),B4))) = aa_nat_nat(minus_minus_nat(finite_card_o(minus_minus_set_o(A,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_4712_card__Diff__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(A2),A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(A2),B4)
       => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),B4))) = aa_nat_nat(minus_minus_nat(finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_4713_card__Diff__insert,axiom,
    ! [A2: nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( ~ aa_set_nat_o(member_nat(A2),B4)
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),B4))) = aa_nat_nat(minus_minus_nat(aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_4714_img__fst,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
      ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B),S)
     => member242389114213480694t_unit(A2,image_6824039242672219652t_unit(produc1824681642469235216et_nat,S)) ) ).

% img_fst
tff(fact_4715_img__fst,axiom,
    ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B),S)
     => aa_set_nat_o(member_nat(A2),image_2486076414777270412at_nat(product_fst_nat_nat,S)) ) ).

% img_fst
tff(fact_4716_img__fst,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S: set_Pr1312406793078187350et_nat] :
      ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B),S)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),image_4444162529858428168_nat_o(produc5325900173150335662et_nat,S)) ) ).

% img_fst
tff(fact_4717_img__fst,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S: set_Pr4040394263813967220et_nat] :
      ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B),S)
     => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),image_8400968272122294250_nat_o(produc4078497350495916492et_nat,S)) ) ).

% img_fst
tff(fact_4718_img__fst,axiom,
    ! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B),S)
     => aa_set_int_o(member_int(A2),image_5042161079198086560nt_int(product_fst_int_int,S)) ) ).

% img_fst
tff(fact_4719_fst__swap,axiom,
    ! [X: produc3658429121746597890et_nat] : produc6012101397789842562t_unit(aa_Pro7370057327950226428t_unit(produc6300697153377358172et_nat,X)) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X) ).

% fst_swap
tff(fact_4720_fst__swap,axiom,
    ! [X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Pro6970504127699319548nt_int(product_swap_int_int,X)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,X) ).

% fst_swap
tff(fact_4721_snd__swap,axiom,
    ! [X: produc7348995623955488188t_unit] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_Pro2092192379717785596et_nat(produc1264744871843189710t_unit,X)) = produc6012101397789842562t_unit(X) ).

% snd_swap
tff(fact_4722_snd__swap,axiom,
    ! [X: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Pro6970504127699319548nt_int(product_swap_int_int,X)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,X) ).

% snd_swap
tff(fact_4723_snd__comp__apfst,axiom,
    ! [F: fun_he2331061685413951235t_unit] : comp_P2930521774885306735et_nat(produc8586169260539613262et_nat,produc4172003873565521026et_nat(F)) = produc8586169260539613262et_nat ).

% snd_comp_apfst
tff(fact_4724_snd__comp__apfst,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_snd_int_int,produc2355237562570658114nt_int(F)) = product_snd_int_int ).

% snd_comp_apfst
tff(fact_4725_fst__comp__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] : comp_P3428181540658402820nteger(produc8508995932063986495nteger,aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F)) = produc8508995932063986495nteger ).

% fst_comp_apsnd
tff(fact_4726_fst__comp__apsnd,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_fst_int_int,produc4463282112584876420nt_int(F)) = product_fst_int_int ).

% fst_comp_apsnd
tff(fact_4727_range__fst,axiom,
    image_5042161079198086560nt_int(product_fst_int_int,top_to4366644338036079209nt_int) = top_top_set_int ).

% range_fst
tff(fact_4728_fst__comp__map__prod,axiom,
    ! [F: fun_int_int,G: fun_int_int] : comp_P5356232565778667509nt_int(product_fst_int_int,produc6036585564866528938nt_int(F,G)) = comp_i4330622015127193916nt_int(F,product_fst_int_int) ).

% fst_comp_map_prod
tff(fact_4729_fst__comp__apfst,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_fst_int_int,produc2355237562570658114nt_int(F)) = comp_i4330622015127193916nt_int(F,product_fst_int_int) ).

% fst_comp_apfst
tff(fact_4730_snd__comp__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] : comp_P3428181540658402820nteger(produc6174133586879617921nteger,aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F)) = comp_C1043215000170864188nteger(F,produc6174133586879617921nteger) ).

% snd_comp_apsnd
tff(fact_4731_snd__comp__apsnd,axiom,
    ! [F: fun_set_nat_set_nat] : comp_P2930521774885306735et_nat(produc8586169260539613262et_nat,produc3286488329990770625t_unit(F)) = comp_s5696514601250091763et_nat(F,produc8586169260539613262et_nat) ).

% snd_comp_apsnd
tff(fact_4732_snd__comp__apsnd,axiom,
    ! [F: fun_int_int] : comp_P5356232565778667509nt_int(product_snd_int_int,produc4463282112584876420nt_int(F)) = comp_i4330622015127193916nt_int(F,product_snd_int_int) ).

% snd_comp_apsnd
tff(fact_4733_apfst__apsnd,axiom,
    ! [F: fun_he2331061685413951235t_unit,G: fun_set_nat_set_nat,X: produc3658429121746597890et_nat] : aa_Pro3679490825741336130et_nat(produc4172003873565521026et_nat(F),aa_Pro3679490825741336130et_nat(produc3286488329990770625t_unit(G),X)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(F,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set_nat_set_nat(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apfst_apsnd
tff(fact_4734_apfst__apsnd,axiom,
    ! [F: fun_he1033112309995878576it_nat,G: fun_set_nat_nat,X: produc3658429121746597890et_nat] : produc8233931525780029747at_nat(F,produc8442695800884449271t_unit(G,X)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_hea5848002036693495265it_nat(F,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set_nat_nat(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apfst_apsnd
tff(fact_4735_apfst__apsnd,axiom,
    ! [F: fun_he6078633327341457676it_int,G: fun_set_nat_int,X: produc3658429121746597890et_nat] : produc7422730317258219nt_int(F,produc7345612820240450715t_unit(G,X)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_hea5845511566184444989it_int(F,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set_nat_int(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apfst_apsnd
tff(fact_4736_apfst__apsnd,axiom,
    ! [F: fun_he61746726229534885_nat_o,G: fun_se1339536933736954245et_nat,X: produc3658429121746597890et_nat] : produc435124687986648096et_nat(F,produc7441988914112407621t_unit(G,X)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_hea3621874372154836126_nat_o(F,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set3979848162462698438et_nat(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apfst_apsnd
tff(fact_4737_apfst__apsnd,axiom,
    ! [F: fun_he61746726229534885_nat_o,G: fun_se6555980291115086755et_nat,X: produc3658429121746597890et_nat] : produc720410928018758206et_nat(F,produc4267314560102770471t_unit(G,X)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_hea3621874372154836126_nat_o(F,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set4535993045952421604et_nat(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apfst_apsnd
tff(fact_4738_apfst__apsnd,axiom,
    ! [F: fun_in7572449478177386490t_unit,G: fun_int_set_nat,X: product_prod_int_int] : produc7086134074927288299et_nat(F,produc1253519502038700766at_int(G,X)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_int3094347076292422883t_unit(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_set_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apfst_apsnd
tff(fact_4739_apfst__apsnd,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,X: product_prod_int_int] : produc1358374321178653834at_nat(F,produc3463928400683821864at_int(G,X)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apfst_apsnd
tff(fact_4740_apfst__apsnd,axiom,
    ! [F: fun_int_int,G: fun_int_int,X: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(G),X)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_int(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apfst_apsnd
tff(fact_4741_apfst__apsnd,axiom,
    ! [F: fun_in1927439276604979228_nat_o,G: fun_in4666789911842550827et_nat,X: product_prod_int_int] : produc2858799776146633481et_nat(F,produc9121118195053341018at_int(G,X)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_int7568080844962427909_nat_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int9033694857635096172et_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apfst_apsnd
tff(fact_4742_apfst__apsnd,axiom,
    ! [F: fun_in1927439276604979228_nat_o,G: fun_in7989804389987109705et_nat,X: product_prod_int_int] : produc3800006316662135975et_nat(F,produc568365411330259192at_int(G,X)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_int7568080844962427909_nat_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int6920457780091180682et_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apfst_apsnd
tff(fact_4743_apsnd__apfst,axiom,
    ! [F: fun_set_nat_set_nat,G: fun_he2331061685413951235t_unit,X: produc3658429121746597890et_nat] : aa_Pro3679490825741336130et_nat(produc3286488329990770625t_unit(F),aa_Pro3679490825741336130et_nat(produc4172003873565521026et_nat(G),X)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(G,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set_nat_set_nat(F,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apsnd_apfst
tff(fact_4744_apsnd__apfst,axiom,
    ! [F: fun_set_nat_nat,G: fun_he1033112309995878576it_nat,X: produc3658429121746597890et_nat] : produc1747169720485888550at_nat(F,produc2974404638222180585et_nat(G,X)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_hea5848002036693495265it_nat(G,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set_nat_nat(F,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apsnd_apfst
tff(fact_4745_apsnd__apfst,axiom,
    ! [F: fun_set_nat_int,G: fun_he6078633327341457676it_int,X: produc3658429121746597890et_nat] : produc2744032961877892830nt_int(F,produc3346058198459025221et_nat(G,X)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_hea5845511566184444989it_int(G,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set_nat_int(F,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apsnd_apfst
tff(fact_4746_apsnd__apfst,axiom,
    ! [F: fun_se1339536933736954245et_nat,G: fun_he61746726229534885_nat_o,X: produc3658429121746597890et_nat] : produc6368439482561100391_nat_o(F,produc2357191563691442084et_nat(G,X)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_hea3621874372154836126_nat_o(G,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set3979848162462698438et_nat(F,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apsnd_apfst
tff(fact_4747_apsnd__apfst,axiom,
    ! [F: fun_se6555980291115086755et_nat,G: fun_he61746726229534885_nat_o,X: produc3658429121746597890et_nat] : produc907333889771390025_nat_o(F,produc2357191563691442084et_nat(G,X)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_hea3621874372154836126_nat_o(G,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))),aa_set4535993045952421604et_nat(F,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,X))) ).

% apsnd_apfst
tff(fact_4748_apsnd__apfst,axiom,
    ! [F: fun_int_set_nat,G: fun_in7572449478177386490t_unit,X: product_prod_int_int] : produc2122153897363312795t_unit(F,produc1158806734383251089it_int(G,X)) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_int3094347076292422883t_unit(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_set_nat(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apsnd_apfst
tff(fact_4749_apsnd__apfst,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,X: product_prod_int_int] : produc3466418871192872140at_nat(F,produc1355883850669603558at_int(G,X)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_int_nat(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_nat(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apsnd_apfst
tff(fact_4750_apsnd__apfst,axiom,
    ! [F: fun_int_int,G: fun_int_int,X: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(G),X)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int_int(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apsnd_apfst
tff(fact_4751_apsnd__apfst,axiom,
    ! [F: fun_in4666789911842550827et_nat,G: fun_in1927439276604979228_nat_o,X: product_prod_int_int] : produc1318474093340237377_nat_o(F,produc2698142598960073523_o_int(G,X)) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_int7568080844962427909_nat_o(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int9033694857635096172et_nat(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apsnd_apfst
tff(fact_4752_apsnd__apfst,axiom,
    ! [F: fun_in7989804389987109705et_nat,G: fun_in1927439276604979228_nat_o,X: product_prod_int_int] : produc8027331940323167011_nat_o(F,produc2698142598960073523_o_int(G,X)) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_int7568080844962427909_nat_o(G,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),aa_int6920457780091180682et_nat(F,aa_Pro6799258434465214121nt_int(product_snd_int_int,X))) ).

% apsnd_apfst
tff(fact_4753_apsnd__apfst__commute,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger,P2: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc3195571438000033749nteger,G),P2)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc3195571438000033749nteger,G),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),P2)) ).

% apsnd_apfst_commute
tff(fact_4754_fstI,axiom,
    ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
      ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Y),Z) )
     => ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X) = Y ) ) ).

% fstI
tff(fact_4755_fstI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
      ( ( X = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z) )
     => ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X) = Y ) ) ).

% fstI
tff(fact_4756_fstI,axiom,
    ! [X: produc8799533983703829792et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc3658429121746597890et_nat] :
      ( ( X = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(Y),Z) )
     => ( aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,X) = Y ) ) ).

% fstI
tff(fact_4757_fstI,axiom,
    ! [X: produc3112576180687729086et_nat,Y: fun_Pr6460944699146269935_nat_o,Z: produc8799533983703829792et_nat] :
      ( ( X = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(Y),Z) )
     => ( aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,X) = Y ) ) ).

% fstI
tff(fact_4758_fstI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Z) )
     => ( aa_Pro6799258434465214121nt_int(product_fst_int_int,X) = Y ) ) ).

% fstI
tff(fact_4759_fst__eqD,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: heap_e7401611519738050253t_unit] :
      ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y)) = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
tff(fact_4760_fst__eqD,axiom,
    ! [X: nat,Y: nat,A2: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)) = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
tff(fact_4761_fst__eqD,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y)) = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
tff(fact_4762_fst__eqD,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y)) = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
tff(fact_4763_fst__eqD,axiom,
    ! [X: int,Y: int,A2: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y)) = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
tff(fact_4764_fst__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] : aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X1),X22)) = X1 ).

% fst_conv
tff(fact_4765_fst__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = X1 ).

% fst_conv
tff(fact_4766_fst__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc3658429121746597890et_nat] : aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X1),X22)) = X1 ).

% fst_conv
tff(fact_4767_fst__conv,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X22: produc8799533983703829792et_nat] : aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X1),X22)) = X1 ).

% fst_conv
tff(fact_4768_fst__conv,axiom,
    ! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X1),X22)) = X1 ).

% fst_conv
tff(fact_4769_eq__fst__iff,axiom,
    ! [A2: heap_e7401611519738050253t_unit,P2: produc3658429121746597890et_nat] :
      ( ( A2 = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2) )
    <=> ? [B6: set_nat] : P2 = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B6) ) ).

% eq_fst_iff
tff(fact_4770_eq__fst__iff,axiom,
    ! [A2: nat,P2: product_prod_nat_nat] :
      ( ( A2 = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2) )
    <=> ? [B6: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B6) ) ).

% eq_fst_iff
tff(fact_4771_eq__fst__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,P2: produc8799533983703829792et_nat] :
      ( ( A2 = aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,P2) )
    <=> ? [B6: produc3658429121746597890et_nat] : P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B6) ) ).

% eq_fst_iff
tff(fact_4772_eq__fst__iff,axiom,
    ! [A2: fun_Pr6460944699146269935_nat_o,P2: produc3112576180687729086et_nat] :
      ( ( A2 = aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,P2) )
    <=> ? [B6: produc8799533983703829792et_nat] : P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B6) ) ).

% eq_fst_iff
tff(fact_4773_eq__fst__iff,axiom,
    ! [A2: int,P2: product_prod_int_int] :
      ( ( A2 = aa_Pro6799258434465214121nt_int(product_fst_int_int,P2) )
    <=> ? [B6: int] : P2 = aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B6) ) ).

% eq_fst_iff
tff(fact_4774_fstE,axiom,
    ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: fun_he8413527710729349838unit_o] :
      ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(A2),B) )
     => ( aa_hea7994211679605476167unit_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,X))
       => aa_hea7994211679605476167unit_o(P,A2) ) ) ).

% fstE
tff(fact_4775_fstE,axiom,
    ! [X: product_prod_nat_nat,A2: nat,B: nat,P: fun_nat_o] :
      ( ( X = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B) )
     => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X))
       => aa_nat_o(P,A2) ) ) ).

% fstE
tff(fact_4776_fstE,axiom,
    ! [X: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,P: fun_fu5135969500543783852at_o_o] :
      ( ( X = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(A2),B) )
     => ( aa_fun1675921005893230629at_o_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,X))
       => aa_fun1675921005893230629at_o_o(P,A2) ) ) ).

% fstE
tff(fact_4777_fstE,axiom,
    ! [X: produc3112576180687729086et_nat,A2: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,P: fun_fu5135969500543783852at_o_o] :
      ( ( X = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(A2),B) )
     => ( aa_fun1675921005893230629at_o_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,X))
       => aa_fun1675921005893230629at_o_o(P,A2) ) ) ).

% fstE
tff(fact_4778_fstE,axiom,
    ! [X: product_prod_int_int,A2: int,B: int,P: fun_int_o] :
      ( ( X = aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B) )
     => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,X))
       => aa_int_o(P,A2) ) ) ).

% fstE
tff(fact_4779_prod__eq__iff,axiom,
    ! [S2: produc3658429121746597890et_nat,T4: produc3658429121746597890et_nat] :
      ( ( S2 = T4 )
    <=> ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,S2) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,T4) )
        & ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,S2) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,T4) ) ) ) ).

% prod_eq_iff
tff(fact_4780_prod__eq__iff,axiom,
    ! [S2: product_prod_int_int,T4: product_prod_int_int] :
      ( ( S2 = T4 )
    <=> ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,S2) = aa_Pro6799258434465214121nt_int(product_fst_int_int,T4) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,S2) = aa_Pro6799258434465214121nt_int(product_snd_int_int,T4) ) ) ) ).

% prod_eq_iff
tff(fact_4781_prod__eqI,axiom,
    ! [P2: produc3658429121746597890et_nat,Q3: produc3658429121746597890et_nat] :
      ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Q3) )
     => ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Q3) )
       => ( P2 = Q3 ) ) ) ).

% prod_eqI
tff(fact_4782_prod__eqI,axiom,
    ! [P2: product_prod_int_int,Q3: product_prod_int_int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,P2) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Q3) )
     => ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Q3) )
       => ( P2 = Q3 ) ) ) ).

% prod_eqI
tff(fact_4783_prod_Oexpand,axiom,
    ! [Prod: produc3658429121746597890et_nat,Prod2: produc3658429121746597890et_nat] :
      ( ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod2) )
        & ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod) = aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_4784_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
      ( ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod2) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_4785_All__prod__contract,axiom,
    ! [P: fun_he5746958392356810273_nat_o] :
      ( ! [A5: heap_e7401611519738050253t_unit,X_12: set_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A5),X_12)
    <=> ! [Z2: produc3658429121746597890et_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Z2)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Z2)) ) ).

% All_prod_contract
tff(fact_4786_All__prod__contract,axiom,
    ! [P: fun_int_fun_int_o] :
      ( ! [A5: int,X_12: int] : aa_int_o(aa_int_fun_int_o(P,A5),X_12)
    <=> ! [Z2: product_prod_int_int] : aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Z2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Z2)) ) ).

% All_prod_contract
tff(fact_4787_Ex__prod__contract,axiom,
    ! [P: fun_he5746958392356810273_nat_o] :
      ( ? [A5: heap_e7401611519738050253t_unit,X_12: set_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A5),X_12)
    <=> ? [Z2: produc3658429121746597890et_nat] : aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Z2)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Z2)) ) ).

% Ex_prod_contract
tff(fact_4788_Ex__prod__contract,axiom,
    ! [P: fun_int_fun_int_o] :
      ( ? [A5: int,X_12: int] : aa_int_o(aa_int_fun_int_o(P,A5),X_12)
    <=> ? [Z2: product_prod_int_int] : aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Z2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Z2)) ) ).

% Ex_prod_contract
tff(fact_4789_apsnd__compose,axiom,
    ! [F: fun_Co1561183069634336733nteger,G: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger] : produc5043883716779652157nteger(F,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),X)) = produc5043883716779652157nteger(comp_C7848664941760151970nteger(F,G),X) ).

% apsnd_compose
tff(fact_4790_apsnd__compose,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger,X: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),X)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,comp_C7449957260575251196nteger(F,G)),X) ).

% apsnd_compose
tff(fact_4791_refl__on__INTER,axiom,
    ! [S: set_nat,A: fun_nat_set_nat,R2: fun_na7044673412300895210at_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),S)
         => refl_on_nat(aa_nat_set_nat(A,X3),aa_nat7062706319057040595at_nat(R2,X3)) )
     => refl_on_nat(comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(A),S)),comple1930127330929219688at_nat(image_7178329752028323786at_nat(R2,S))) ) ).

% refl_on_INTER
tff(fact_4792_cInf__eq__non__empty,axiom,
    ! [X5: set_o,A2: $o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X5)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A2)),(X3)) )
       => ( ! [Y2: $o] :
              ( ! [X4: $o] :
                  ( aa_set_o_o(member_o((X4)),X5)
                 => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Y2)),(X4)) )
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Y2)),(A2)) )
         => ( aa_set_o_o(complete_Inf_Inf_o,X5)
          <=> (A2) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4793_cInf__eq__non__empty,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( X5 != bot_bo8632255151191677115_nat_o )
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),X5)
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A2),X3) )
       => ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
              ( ! [X4: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),X5)
                 => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,Y2),X4) )
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,Y2),A2) )
         => ( comple6093339926577406454_nat_o(X5) = A2 ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4794_cInf__eq__non__empty,axiom,
    ! [X5: set_filter_nat,A2: filter_nat] :
      ( ( X5 != bot_bo498966703094740906er_nat )
     => ( ! [X3: filter_nat] :
            ( member_filter_nat(X3,X5)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),X3) )
       => ( ! [Y2: filter_nat] :
              ( ! [X4: filter_nat] :
                  ( member_filter_nat(X4,X5)
                 => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y2),X4) )
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Y2),A2) )
         => ( comple6612786467464916249er_nat(X5) = A2 ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4795_cInf__eq__non__empty,axiom,
    ! [X5: set_int,A2: int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),X3) )
       => ( ! [Y2: int] :
              ( ! [X4: int] :
                  ( aa_set_int_o(member_int(X4),X5)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X4) )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),A2) )
         => ( complete_Inf_Inf_int(X5) = A2 ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4796_cInf__eq__non__empty,axiom,
    ! [X5: set_nat,A2: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),X3) )
       => ( ! [Y2: nat] :
              ( ! [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),X5)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X4) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),A2) )
         => ( aa_set_nat_nat(complete_Inf_Inf_nat,X5) = A2 ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4797_cInf__eq__non__empty,axiom,
    ! [X5: set_set_nat,A2: set_nat] :
      ( ( X5 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,X5)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),X3) )
       => ( ! [Y2: set_nat] :
              ( ! [X4: set_nat] :
                  ( member_set_nat(X4,X5)
                 => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),X4) )
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),A2) )
         => ( comple7806235888213564991et_nat(X5) = A2 ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4798_cInf__eq__non__empty,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A2),X3) )
       => ( ! [Y2: product_unit] :
              ( ! [X4: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X4),X5)
                 => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Y2),X4) )
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Y2),A2) )
         => ( comple2584293577114468500t_unit(X5) = A2 ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4799_cInf__greatest,axiom,
    ! [X5: set_o,Z: $o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X5)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Z)),(X3)) )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Z)),aa_set_o_o(complete_Inf_Inf_o,X5)) ) ) ).

% cInf_greatest
tff(fact_4800_cInf__greatest,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,Z: fun_Pr6460944699146269935_nat_o] :
      ( ( X5 != bot_bo8632255151191677115_nat_o )
     => ( ! [X3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),X5)
           => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,Z),X3) )
       => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,Z),comple6093339926577406454_nat_o(X5)) ) ) ).

% cInf_greatest
tff(fact_4801_cInf__greatest,axiom,
    ! [X5: set_filter_nat,Z: filter_nat] :
      ( ( X5 != bot_bo498966703094740906er_nat )
     => ( ! [X3: filter_nat] :
            ( member_filter_nat(X3,X5)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Z),X3) )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,Z),comple6612786467464916249er_nat(X5)) ) ) ).

% cInf_greatest
tff(fact_4802_cInf__greatest,axiom,
    ! [X5: set_int,Z: int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X3) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),complete_Inf_Inf_int(X5)) ) ) ).

% cInf_greatest
tff(fact_4803_cInf__greatest,axiom,
    ! [X5: set_nat,Z: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X3) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_set_nat_nat(complete_Inf_Inf_nat,X5)) ) ) ).

% cInf_greatest
tff(fact_4804_cInf__greatest,axiom,
    ! [X5: set_set_nat,Z: set_nat] :
      ( ( X5 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,X5)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),X3) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),comple7806235888213564991et_nat(X5)) ) ) ).

% cInf_greatest
tff(fact_4805_cInf__greatest,axiom,
    ! [X5: set_Product_unit,Z: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Z),X3) )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,Z),comple2584293577114468500t_unit(X5)) ) ) ).

% cInf_greatest
tff(fact_4806_Inf__less__eq,axiom,
    ! [A: set_o,U: $o] :
      ( ! [V3: $o] :
          ( aa_set_o_o(member_o((V3)),A)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(V3)),(U)) )
     => ( ( A != bot_bot_set_o )
       => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Inf_Inf_o,A)),(U)) ) ) ).

% Inf_less_eq
tff(fact_4807_Inf__less__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o,U: fun_Pr6460944699146269935_nat_o] :
      ( ! [V3: fun_Pr6460944699146269935_nat_o] :
          ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(V3),A)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,V3),U) )
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple6093339926577406454_nat_o(A)),U) ) ) ).

% Inf_less_eq
tff(fact_4808_Inf__less__eq,axiom,
    ! [A: set_filter_nat,U: filter_nat] :
      ( ! [V3: filter_nat] :
          ( member_filter_nat(V3,A)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,V3),U) )
     => ( ( A != bot_bo498966703094740906er_nat )
       => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(A)),U) ) ) ).

% Inf_less_eq
tff(fact_4809_Inf__less__eq,axiom,
    ! [A: set_set_nat,U: set_nat] :
      ( ! [V3: set_nat] :
          ( member_set_nat(V3,A)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,V3),U) )
     => ( ( A != bot_bot_set_set_nat )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(A)),U) ) ) ).

% Inf_less_eq
tff(fact_4810_Inf__less__eq,axiom,
    ! [A: set_Product_unit,U: product_unit] :
      ( ! [V3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(V3),A)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,V3),U) )
     => ( ( A != bot_bo3957492148770167129t_unit )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(A)),U) ) ) ).

% Inf_less_eq
tff(fact_4811_cInf__lessD,axiom,
    ! [X5: set_int,Z: int] :
      ( ( X5 != bot_bot_set_int )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,complete_Inf_Inf_int(X5)),Z)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),X5)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z) ) ) ) ).

% cInf_lessD
tff(fact_4812_cInf__lessD,axiom,
    ! [X5: set_nat,Z: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(complete_Inf_Inf_nat,X5)),Z)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X5)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z) ) ) ) ).

% cInf_lessD
tff(fact_4813_cInf__lessD,axiom,
    ! [X5: set_Product_unit,Z: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,comple2584293577114468500t_unit(X5)),Z)
       => ? [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),X5)
            & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X3),Z) ) ) ) ).

% cInf_lessD
tff(fact_4814_INF__eq__const,axiom,
    ! [I: set_Product_unit,F: fun_Pr1628945989756059814et_nat,X: set_nat] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => ( aa_Pro6064029160526250199et_nat(F,I3) = X ) )
       => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4815_INF__eq__const,axiom,
    ! [I: set_o,F: fun_o_set_nat,X: set_nat] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => ( aa_o_set_nat(F,(I3)) = X ) )
       => ( comple7806235888213564991et_nat(image_o_set_nat(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4816_INF__eq__const,axiom,
    ! [I: set_nat,F: fun_nat_set_nat,X: set_nat] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => ( aa_nat_set_nat(F,I3) = X ) )
       => ( comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4817_INF__eq__const,axiom,
    ! [I: set_int,F: fun_int_set_nat,X: set_nat] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => ( aa_int_set_nat(F,I3) = X ) )
       => ( comple7806235888213564991et_nat(image_int_set_nat(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4818_INF__eq__const,axiom,
    ! [I: set_fu3555266933746557007_nat_o,F: fun_fu2098174926252251144et_nat,X: set_nat] :
      ( ( I != bot_bo8632255151191677115_nat_o )
     => ( ! [I3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I3),I)
           => ( aa_fun7872747175131507769et_nat(F,I3) = X ) )
       => ( comple7806235888213564991et_nat(image_7230913979079189040et_nat(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4819_INF__eq__const,axiom,
    ! [I: set_Product_unit,F: fun_Pr2191293291710260995t_unit,X: product_unit] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => ( aa_Pro8294289767544744956t_unit(F,I3) = X ) )
       => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4820_INF__eq__const,axiom,
    ! [I: set_o,F: fun_o_Product_unit,X: product_unit] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => ( aa_o_Product_unit(F,(I3)) = X ) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit2(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4821_INF__eq__const,axiom,
    ! [I: set_nat,F: fun_nat_Product_unit,X: product_unit] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => ( aa_nat_Product_unit(F,I3) = X ) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4822_INF__eq__const,axiom,
    ! [I: set_int,F: fun_int_Product_unit,X: product_unit] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => ( aa_int_Product_unit(F,I3) = X ) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4823_INF__eq__const,axiom,
    ! [I: set_fu3555266933746557007_nat_o,F: fun_fu3096152755035260449t_unit,X: product_unit] :
      ( ( I != bot_bo8632255151191677115_nat_o )
     => ( ! [I3: fun_Pr6460944699146269935_nat_o] :
            ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(I3),I)
           => ( aa_fun8210042732970293018t_unit(F,I3) = X ) )
       => ( comple2584293577114468500t_unit(image_5224052122049783779t_unit(F,I)) = X ) ) ) ).

% INF_eq_const
tff(fact_4824_card__1__singletonE,axiom,
    ! [A: set_Product_unit] :
      ( ( finite410649719033368117t_unit(A) = one_one_nat )
     => ~ ! [X3: product_unit] : A != aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X3),bot_bo3957492148770167129t_unit) ) ).

% card_1_singletonE
tff(fact_4825_card__1__singletonE,axiom,
    ! [A: set_o] :
      ( ( finite_card_o(A) = one_one_nat )
     => ~ ! [X3: $o] : A != aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X3)),bot_bot_set_o) ) ).

% card_1_singletonE
tff(fact_4826_card__1__singletonE,axiom,
    ! [A: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A) = one_one_nat )
     => ~ ! [X3: nat] : A != aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X3),bot_bot_set_nat) ) ).

% card_1_singletonE
tff(fact_4827_card__1__singletonE,axiom,
    ! [A: set_int] :
      ( ( finite_card_int(A) = one_one_nat )
     => ~ ! [X3: int] : A != aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X3),bot_bot_set_int) ) ).

% card_1_singletonE
tff(fact_4828_card__1__singletonE,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ( finite5795495896937551127_nat_o(A) = one_one_nat )
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o] : A != aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X3),bot_bo8632255151191677115_nat_o) ) ).

% card_1_singletonE
tff(fact_4829_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_nat_fun_nat_o,X: nat,Y: nat,A2: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,X),Y)
     => ( ( A2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y) )
       => aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4830_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_fu8419763957888848259_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,A2: produc8799533983703829792et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,X),Y)
     => ( ( A2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y) )
       => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,A2)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4831_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_fu2611741329380808165_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,A2: produc3112576180687729086et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,X),Y)
     => ( ( A2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y) )
       => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,A2)),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4832_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_he5746958392356810273_nat_o,X: heap_e7401611519738050253t_unit,Y: set_nat,A2: produc3658429121746597890et_nat] :
      ( aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,X),Y)
     => ( ( A2 = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y) )
       => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,A2)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4833_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: fun_int_fun_int_o,X: int,Y: int,A2: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,X),Y)
     => ( ( A2 = aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y) )
       => aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,A2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4834_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] : Prod = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_4835_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8799533983703829792et_nat] : Prod = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,Prod)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_4836_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3112576180687729086et_nat] : Prod = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,Prod)),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_4837_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3658429121746597890et_nat] : Prod = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Prod)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_4838_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% prod.exhaust_sel
tff(fact_4839_surjective__pairing,axiom,
    ! [T4: product_prod_nat_nat] : T4 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,T4)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,T4)) ).

% surjective_pairing
tff(fact_4840_surjective__pairing,axiom,
    ! [T4: produc8799533983703829792et_nat] : T4 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,T4)),aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,T4)) ).

% surjective_pairing
tff(fact_4841_surjective__pairing,axiom,
    ! [T4: produc3112576180687729086et_nat] : T4 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,T4)),aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,T4)) ).

% surjective_pairing
tff(fact_4842_surjective__pairing,axiom,
    ! [T4: produc3658429121746597890et_nat] : T4 = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,T4)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,T4)) ).

% surjective_pairing
tff(fact_4843_surjective__pairing,axiom,
    ! [T4: product_prod_int_int] : T4 = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,T4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,T4)) ).

% surjective_pairing
tff(fact_4844_card__insert__le,axiom,
    ! [A: set_nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A)),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A))) ).

% card_insert_le
tff(fact_4845_card__insert__le,axiom,
    ! [A: set_int,X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A)),finite_card_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A))) ).

% card_insert_le
tff(fact_4846_card__insert__le,axiom,
    ! [A: set_o,X: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A)),finite_card_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))) ).

% card_insert_le
tff(fact_4847_card__insert__le,axiom,
    ! [A: set_Product_unit,X: product_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A)),finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A))) ).

% card_insert_le
tff(fact_4848_in__fst__imageE,axiom,
    ! [X: heap_e7401611519738050253t_unit,S: set_Pr3948176798113811640et_nat] :
      ( member242389114213480694t_unit(X,image_6824039242672219652t_unit(produc1824681642469235216et_nat,S))
     => ~ ! [Y2: set_nat] : ~ member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y2),S) ) ).

% in_fst_imageE
tff(fact_4849_in__fst__imageE,axiom,
    ! [X: nat,S: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(member_nat(X),image_2486076414777270412at_nat(product_fst_nat_nat,S))
     => ~ ! [Y2: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y2),S) ) ).

% in_fst_imageE
tff(fact_4850_in__fst__imageE,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_Pr1312406793078187350et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),image_4444162529858428168_nat_o(produc5325900173150335662et_nat,S))
     => ~ ! [Y2: produc3658429121746597890et_nat] : ~ member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y2),S) ) ).

% in_fst_imageE
tff(fact_4851_in__fst__imageE,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,S: set_Pr4040394263813967220et_nat] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),image_8400968272122294250_nat_o(produc4078497350495916492et_nat,S))
     => ~ ! [Y2: produc8799533983703829792et_nat] : ~ member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y2),S) ) ).

% in_fst_imageE
tff(fact_4852_in__fst__imageE,axiom,
    ! [X: int,S: set_Pr958786334691620121nt_int] :
      ( aa_set_int_o(member_int(X),image_5042161079198086560nt_int(product_fst_int_int,S))
     => ~ ! [Y2: int] : ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y2),S) ) ).

% in_fst_imageE
tff(fact_4853_Inter__subset,axiom,
    ! [A: set_set_nat,B4: set_nat] :
      ( ! [X7: set_nat] :
          ( member_set_nat(X7,A)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X7),B4) )
     => ( ( A != bot_bot_set_set_nat )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(A)),B4) ) ) ).

% Inter_subset
tff(fact_4854_Inter__empty,axiom,
    comple7216939928495492468t_unit(bot_bo6284674557190688313t_unit) = top_to1996260823553986621t_unit ).

% Inter_empty
tff(fact_4855_Inter__empty,axiom,
    comple3063163877087187839_set_o(bot_bot_set_set_o) = top_top_set_o ).

% Inter_empty
tff(fact_4856_Inter__empty,axiom,
    comple3628384868704368283et_int(bot_bot_set_set_int) = top_top_set_int ).

% Inter_empty
tff(fact_4857_Inter__empty,axiom,
    comple7806235888213564991et_nat(bot_bot_set_set_nat) = top_top_set_nat ).

% Inter_empty
tff(fact_4858_fst__eq__Domain,axiom,
    ! [R: set_Pr958786334691620121nt_int] : image_5042161079198086560nt_int(product_fst_int_int,R) = domain_int_int(R) ).

% fst_eq_Domain
tff(fact_4859_Domain__fst,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : domain_int_int(R2) = image_5042161079198086560nt_int(product_fst_int_int,R2) ).

% Domain_fst
tff(fact_4860_card__partition,axiom,
    ! [C5: set_set_Code_integer,K3: nat] :
      ( finite6931041176100689706nteger(C5)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(C5))
       => ( ! [C2: set_Code_integer] :
              ( member1834298987299537450nteger(C2,C5)
             => ( finite4902975817058060853nteger(C2) = K3 ) )
         => ( ! [C1: set_Code_integer,C22: set_Code_integer] :
                ( member1834298987299537450nteger(C1,C5)
               => ( member1834298987299537450nteger(C22,C5)
                 => ( ( C1 != C22 )
                   => ( inf_in1364745209274528805nteger(C1,C22) = bot_bo3990330152332043303nteger ) ) ) )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),finite1340570857867686763nteger(C5)) = finite4902975817058060853nteger(comple739944243200306918nteger(C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4861_card__partition,axiom,
    ! [C5: set_set_Product_unit,K3: nat] :
      ( finite1772178364199683094t_unit(C5)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,comple3349607820504768346t_unit(C5))
       => ( ! [C2: set_Product_unit] :
              ( member5877623283571906838t_unit(C2,C5)
             => ( finite410649719033368117t_unit(C2) = K3 ) )
         => ( ! [C1: set_Product_unit,C22: set_Product_unit] :
                ( member5877623283571906838t_unit(C1,C5)
               => ( member5877623283571906838t_unit(C22,C5)
                 => ( ( C1 != C22 )
                   => ( inf_in4660618365625256667t_unit(C1,C22) = bot_bo3957492148770167129t_unit ) ) ) )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),finite22775419188893461t_unit(C5)) = finite410649719033368117t_unit(comple3349607820504768346t_unit(C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4862_card__partition,axiom,
    ! [C5: set_set_o,K3: nat] :
      ( finite_finite_set_o(C5)
     => ( aa_set_o_o(finite_finite_o,comple90263536869209701_set_o(C5))
       => ( ! [C2: set_o] :
              ( member_set_o(C2,C5)
             => ( finite_card_o(C2) = K3 ) )
         => ( ! [C1: set_o,C22: set_o] :
                ( member_set_o(C1,C5)
               => ( member_set_o(C22,C5)
                 => ( ( C1 != C22 )
                   => ( inf_inf_set_o(C1,C22) = bot_bot_set_o ) ) ) )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),finite_card_set_o(C5)) = finite_card_o(comple90263536869209701_set_o(C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4863_card__partition,axiom,
    ! [C5: set_set_nat,K3: nat] :
      ( finite1152437895449049373et_nat(C5)
     => ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(C5))
       => ( ! [C2: set_nat] :
              ( member_set_nat(C2,C5)
             => ( aa_set_nat_nat(finite_card_nat,C2) = K3 ) )
         => ( ! [C1: set_nat,C22: set_nat] :
                ( member_set_nat(C1,C5)
               => ( member_set_nat(C22,C5)
                 => ( ( C1 != C22 )
                   => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C1),C22) = bot_bot_set_nat ) ) ) )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),finite_card_set_nat(C5)) = aa_set_nat_nat(finite_card_nat,comple7399068483239264473et_nat(C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4864_card__partition,axiom,
    ! [C5: set_set_int,K3: nat] :
      ( finite6197958912794628473et_int(C5)
     => ( aa_set_int_o(finite_finite_int,comple3221217463730067765et_int(C5))
       => ( ! [C2: set_int] :
              ( member_set_int(C2,C5)
             => ( finite_card_int(C2) = K3 ) )
         => ( ! [C1: set_int,C22: set_int] :
                ( member_set_int(C1,C5)
               => ( member_set_int(C22,C5)
                 => ( ( C1 != C22 )
                   => ( inf_inf_set_int(C1,C22) = bot_bot_set_int ) ) ) )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),finite_card_set_int(C5)) = finite_card_int(comple3221217463730067765et_int(C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4865_card__partition,axiom,
    ! [C5: set_se6690607124073039791_nat_o,K3: nat] :
      ( finite9028524065462436600_nat_o(C5)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,comple4665849098253521980_nat_o(C5))
       => ( ! [C2: set_fu3555266933746557007_nat_o] :
              ( member361801372707117048_nat_o(C2,C5)
             => ( finite5795495896937551127_nat_o(C2) = K3 ) )
         => ( ! [C1: set_fu3555266933746557007_nat_o,C22: set_fu3555266933746557007_nat_o] :
                ( member361801372707117048_nat_o(C1,C5)
               => ( member361801372707117048_nat_o(C22,C5)
                 => ( ( C1 != C22 )
                   => ( inf_in3376754634429691197_nat_o(C1,C22) = bot_bo8632255151191677115_nat_o ) ) ) )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),finite6008746153568837751_nat_o(C5)) = finite5795495896937551127_nat_o(comple4665849098253521980_nat_o(C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4866_Inter__UNIV,axiom,
    comple7216939928495492468t_unit(top_to1767297665138865437t_unit) = bot_bo3957492148770167129t_unit ).

% Inter_UNIV
tff(fact_4867_Inter__UNIV,axiom,
    comple3063163877087187839_set_o(top_top_set_set_o) = bot_bot_set_o ).

% Inter_UNIV
tff(fact_4868_Inter__UNIV,axiom,
    comple3628384868704368283et_int(top_top_set_set_int) = bot_bot_set_int ).

% Inter_UNIV
tff(fact_4869_Inter__UNIV,axiom,
    comple5146519332901335382_nat_o(top_to7216145672142822655_nat_o) = bot_bo8632255151191677115_nat_o ).

% Inter_UNIV
tff(fact_4870_Inter__UNIV,axiom,
    comple7806235888213564991et_nat(top_top_set_set_nat) = bot_bot_set_nat ).

% Inter_UNIV
tff(fact_4871_inv__on__f__range,axiom,
    ! [Y: code_integer,F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(Y),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(inv_on4869000451854154612nteger(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4872_inv__on__f__range,axiom,
    ! [Y: code_integer,F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(Y),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => aa_set_nat_o(member_nat(aa_Code_integer_nat(inv_on1369624593536199399nteger(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4873_inv__on__f__range,axiom,
    ! [Y: nat,F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(Y),aa_set_nat_set_nat(image_nat_nat2(F),A))
     => aa_set_nat_o(member_nat(aa_nat_nat(inv_on_nat_nat(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4874_inv__on__f__range,axiom,
    ! [Y: nat,F: fun_int_nat,A: set_int] :
      ( aa_set_nat_o(member_nat(Y),image_int_nat2(F,A))
     => aa_set_int_o(member_int(aa_nat_int(inv_on_int_nat(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4875_inv__on__f__range,axiom,
    ! [Y: nat,F: fun_o_nat,A: set_o] :
      ( aa_set_nat_o(member_nat(Y),image_o_nat2(F,A))
     => aa_set_o_o(member_o(aa_nat_o(inv_on_o_nat(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4876_inv__on__f__range,axiom,
    ! [Y: nat,F: fun_Product_unit_nat,A: set_Product_unit] :
      ( aa_set_nat_o(member_nat(Y),image_875570014554754200it_nat(F,A))
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_nat_Product_unit(inv_on7727615724625388965it_nat(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4877_inv__on__f__range,axiom,
    ! [Y: int,F: fun_nat_int,A: set_nat] :
      ( aa_set_int_o(member_int(Y),image_nat_int2(F,A))
     => aa_set_nat_o(member_nat(aa_int_nat(inv_on_nat_int(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4878_inv__on__f__range,axiom,
    ! [Y: int,F: fun_int_int,A: set_int] :
      ( aa_set_int_o(member_int(Y),image_int_int2(F,A))
     => aa_set_int_o(member_int(aa_int_int(inv_on_int_int(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4879_inv__on__f__range,axiom,
    ! [Y: int,F: fun_o_int,A: set_o] :
      ( aa_set_int_o(member_int(Y),image_o_int2(F,A))
     => aa_set_o_o(member_o(aa_int_o(inv_on_o_int(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4880_inv__on__f__range,axiom,
    ! [Y: int,F: fun_Product_unit_int,A: set_Product_unit] :
      ( aa_set_int_o(member_int(Y),image_873079544045703924it_int(F,A))
     => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(inv_on7725125254116338689it_int(F,A),Y)),A) ) ).

% inv_on_f_range
tff(fact_4881_f__inv__on__f,axiom,
    ! [Y: set_nat,F: fun_nat_set_nat,A: set_nat] :
      ( member_set_nat(Y,aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
     => ( aa_nat_set_nat(F,aa_set_nat_nat(inv_on_nat_set_nat(F,A),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_4882_f__inv__on__f,axiom,
    ! [Y: code_integer,F: fun_Co8499238673658658103nteger,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(Y),aa_set8622560600277679972nteger(image_4470545334726330049nteger(F),A))
     => ( aa_Cod1783974652533396728nteger(F,aa_Cod1783974652533396728nteger(inv_on4869000451854154612nteger(F,A),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_4883_f__inv__on__f,axiom,
    ! [Y: code_integer,F: fun_nat_Code_integer,A: set_nat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(Y),aa_set7438092518518805847nteger(image_1215581382706833972nteger(F),A))
     => ( aa_nat_Code_integer(F,aa_Code_integer_nat(inv_on1369624593536199399nteger(F,A),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_4884_f__inv__on__f,axiom,
    ! [Y: nat,F: fun_nat_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(Y),aa_set_nat_set_nat(image_nat_nat2(F),A))
     => ( aa_nat_nat(F,aa_nat_nat(inv_on_nat_nat(F,A),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_4885_cINF__greatest,axiom,
    ! [A: set_Product_unit,M4: int,F: fun_Product_unit_int] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),aa_Product_unit_int(F,X3)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),complete_Inf_Inf_int(image_873079544045703924it_int(F,A))) ) ) ).

% cINF_greatest
tff(fact_4886_cINF__greatest,axiom,
    ! [A: set_o,M4: int,F: fun_o_int] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),aa_o_int(F,(X3))) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),complete_Inf_Inf_int(image_o_int2(F,A))) ) ) ).

% cINF_greatest
tff(fact_4887_cINF__greatest,axiom,
    ! [A: set_nat,M4: int,F: fun_nat_int] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),aa_nat_int(F,X3)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),complete_Inf_Inf_int(image_nat_int2(F,A))) ) ) ).

% cINF_greatest
tff(fact_4888_cINF__greatest,axiom,
    ! [A: set_int,M4: int,F: fun_int_int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),aa_int_int(F,X3)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),complete_Inf_Inf_int(image_int_int2(F,A))) ) ) ).

% cINF_greatest
tff(fact_4889_cINF__greatest,axiom,
    ! [A: set_Product_unit,M4: nat,F: fun_Product_unit_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_Product_unit_nat(F,X3)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A))) ) ) ).

% cINF_greatest
tff(fact_4890_cINF__greatest,axiom,
    ! [A: set_o,M4: nat,F: fun_o_nat] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_o_nat(F,(X3))) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,A))) ) ) ).

% cINF_greatest
tff(fact_4891_cINF__greatest,axiom,
    ! [A: set_nat,M4: nat,F: fun_nat_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_nat_nat(F,X3)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))) ) ) ).

% cINF_greatest
tff(fact_4892_cINF__greatest,axiom,
    ! [A: set_int,M4: nat,F: fun_int_nat] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_int_nat(F,X3)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,A))) ) ) ).

% cINF_greatest
tff(fact_4893_cINF__greatest,axiom,
    ! [A: set_Product_unit,M4: product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,M4),aa_Pro8294289767544744956t_unit(F,X3)) )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,M4),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A))) ) ) ).

% cINF_greatest
tff(fact_4894_cINF__greatest,axiom,
    ! [A: set_o,M4: product_unit,F: fun_o_Product_unit] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,M4),aa_o_Product_unit(F,(X3))) )
       => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,M4),comple2584293577114468500t_unit(image_o_Product_unit2(F,A))) ) ) ).

% cINF_greatest
tff(fact_4895_INF__eq__iff,axiom,
    ! [I: set_Product_unit,F: fun_Pr2191293291710260995t_unit,C: product_unit] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(F,I3)),C) )
       => ( ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,I)) = C )
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),I)
             => ( aa_Pro8294289767544744956t_unit(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4896_INF__eq__iff,axiom,
    ! [I: set_o,F: fun_o_Product_unit,C: product_unit] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_o_Product_unit(F,(I3))),C) )
       => ( ( comple2584293577114468500t_unit(image_o_Product_unit2(F,I)) = C )
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),I)
             => ( aa_o_Product_unit(F,(X2)) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4897_INF__eq__iff,axiom,
    ! [I: set_nat,F: fun_nat_Product_unit,C: product_unit] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_nat_Product_unit(F,I3)),C) )
       => ( ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,I)) = C )
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),I)
             => ( aa_nat_Product_unit(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4898_INF__eq__iff,axiom,
    ! [I: set_int,F: fun_int_Product_unit,C: product_unit] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_int_Product_unit(F,I3)),C) )
       => ( ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,I)) = C )
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),I)
             => ( aa_int_Product_unit(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4899_INF__eq__iff,axiom,
    ! [I: set_Product_unit,F: fun_Pr566567543205111936er_nat,C: filter_nat] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_Pro518797148967611313er_nat(F,I3)),C) )
       => ( ( comple6612786467464916249er_nat(image_1853478764106424488er_nat(F,I)) = C )
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),I)
             => ( aa_Pro518797148967611313er_nat(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4900_INF__eq__iff,axiom,
    ! [I: set_o,F: fun_o_filter_nat,C: filter_nat] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_o_filter_nat(F,(I3))),C) )
       => ( ( comple6612786467464916249er_nat(image_o_filter_nat(F,I)) = C )
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),I)
             => ( aa_o_filter_nat(F,(X2)) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4901_INF__eq__iff,axiom,
    ! [I: set_nat,F: fun_nat_filter_nat,C: filter_nat] :
      ( ( I != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_nat_filter_nat(F,I3)),C) )
       => ( ( comple6612786467464916249er_nat(image_nat_filter_nat(F,I)) = C )
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),I)
             => ( aa_nat_filter_nat(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4902_INF__eq__iff,axiom,
    ! [I: set_int,F: fun_int_filter_nat,C: filter_nat] :
      ( ( I != bot_bot_set_int )
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),I)
           => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,aa_int_filter_nat(F,I3)),C) )
       => ( ( comple6612786467464916249er_nat(image_int_filter_nat(F,I)) = C )
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),I)
             => ( aa_int_filter_nat(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4903_INF__eq__iff,axiom,
    ! [I: set_Product_unit,F: fun_Pr1628945989756059814et_nat,C: set_nat] :
      ( ( I != bot_bo3957492148770167129t_unit )
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_Pro6064029160526250199et_nat(F,I3)),C) )
       => ( ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,I)) = C )
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),I)
             => ( aa_Pro6064029160526250199et_nat(F,X2) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4904_INF__eq__iff,axiom,
    ! [I: set_o,F: fun_o_set_nat,C: set_nat] :
      ( ( I != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),I)
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_o_set_nat(F,(I3))),C) )
       => ( ( comple7806235888213564991et_nat(image_o_set_nat(F,I)) = C )
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),I)
             => ( aa_o_set_nat(F,(X2)) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4905_finite__less__Inf__iff,axiom,
    ! [X5: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( ( X5 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),complete_Inf_Inf_int(X5))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),X5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),X2) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4906_finite__less__Inf__iff,axiom,
    ! [X5: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( ( X5 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_set_nat_nat(complete_Inf_Inf_nat,X5))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),X5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),X2) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4907_finite__less__Inf__iff,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A2),comple2584293577114468500t_unit(X5))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),X5)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,A2),X2) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4908_Inf__le__Sup,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Inf_Inf_o,A)),aa_set_o_o(complete_Sup_Sup_o,A)) ) ).

% Inf_le_Sup
tff(fact_4909_Inf__le__Sup,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple6093339926577406454_nat_o(A)),comple7878897588548607196_nat_o(A)) ) ).

% Inf_le_Sup
tff(fact_4910_Inf__le__Sup,axiom,
    ! [A: set_filter_nat] :
      ( ( A != bot_bo498966703094740906er_nat )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(A)),comple2454491259644266675er_nat(A)) ) ).

% Inf_le_Sup
tff(fact_4911_Inf__le__Sup,axiom,
    ! [A: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(A)),comple7399068483239264473et_nat(A)) ) ).

% Inf_le_Sup
tff(fact_4912_Inf__le__Sup,axiom,
    ! [A: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(A)),comple4687483117567863418t_unit(A)) ) ).

% Inf_le_Sup
tff(fact_4913_finite__Inf__in,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y2: $o] :
              ( aa_set_o_o(member_o((X3)),A)
             => ( aa_set_o_o(member_o((Y2)),A)
               => aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(inf_inf_o,(X3)),(Y2))),A) ) )
         => aa_set_o_o(member_o(aa_set_o_o(complete_Inf_Inf_o,A)),A) ) ) ) ).

% finite_Inf_in
tff(fact_4914_finite__Inf__in,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( ! [X3: fun_Pr6460944699146269935_nat_o,Y2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X3),A)
             => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Y2),A)
               => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(X3),Y2)),A) ) )
         => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(comple6093339926577406454_nat_o(A)),A) ) ) ) ).

% finite_Inf_in
tff(fact_4915_finite__Inf__in,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( member_set_nat(X3,A)
             => ( member_set_nat(Y2,A)
               => member_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Y2),A) ) )
         => member_set_nat(comple7806235888213564991et_nat(A),A) ) ) ) ).

% finite_Inf_in
tff(fact_4916_finite__Inf__in,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit,Y2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
             => ( aa_set8780210009671629607unit_o(member_Product_unit(Y2),A)
               => aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X3),Y2)),A) ) )
         => aa_set8780210009671629607unit_o(member_Product_unit(comple2584293577114468500t_unit(A)),A) ) ) ) ).

% finite_Inf_in
tff(fact_4917_card__1__singletonI,axiom,
    ! [S: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( finite4902975817058060853nteger(S) = one_one_nat )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(X),S)
         => ( S = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger) ) ) ) ) ).

% card_1_singletonI
tff(fact_4918_card__1__singletonI,axiom,
    ! [S: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
     => ( ( finite410649719033368117t_unit(S) = one_one_nat )
       => ( aa_set8780210009671629607unit_o(member_Product_unit(X),S)
         => ( S = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit) ) ) ) ) ).

% card_1_singletonI
tff(fact_4919_card__1__singletonI,axiom,
    ! [S: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( finite_card_o(S) = one_one_nat )
       => ( aa_set_o_o(member_o((X)),S)
         => ( S = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o) ) ) ) ) ).

% card_1_singletonI
tff(fact_4920_card__1__singletonI,axiom,
    ! [S: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( aa_set_nat_nat(finite_card_nat,S) = one_one_nat )
       => ( aa_set_nat_o(member_nat(X),S)
         => ( S = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat) ) ) ) ) ).

% card_1_singletonI
tff(fact_4921_card__1__singletonI,axiom,
    ! [S: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( finite_card_int(S) = one_one_nat )
       => ( aa_set_int_o(member_int(X),S)
         => ( S = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int) ) ) ) ) ).

% card_1_singletonI
tff(fact_4922_card__1__singletonI,axiom,
    ! [S: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,S)
     => ( ( finite5795495896937551127_nat_o(S) = one_one_nat )
       => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),S)
         => ( S = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o) ) ) ) ) ).

% card_1_singletonI
tff(fact_4923_card__Diff__singleton__if,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_nat_nat(minus_minus_nat(finite410649719033368117t_unit(A)),one_one_nat),finite410649719033368117t_unit(A)) ).

% card_Diff_singleton_if
tff(fact_4924_card__Diff__singleton__if,axiom,
    ! [A: set_o,X: $o] :
      finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((X)),A),aa_nat_nat(minus_minus_nat(finite_card_o(A)),one_one_nat),finite_card_o(A)) ).

% card_Diff_singleton_if
tff(fact_4925_card__Diff__singleton__if,axiom,
    ! [A: set_int,X: int] :
      finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(X),A),aa_nat_nat(minus_minus_nat(finite_card_int(A)),one_one_nat),finite_card_int(A)) ).

% card_Diff_singleton_if
tff(fact_4926_card__Diff__singleton__if,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A),aa_nat_nat(minus_minus_nat(finite5795495896937551127_nat_o(A)),one_one_nat),finite5795495896937551127_nat_o(A)) ).

% card_Diff_singleton_if
tff(fact_4927_card__Diff__singleton__if,axiom,
    ! [A: set_nat,X: nat] :
      aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(X),A),aa_nat_nat(minus_minus_nat(aa_set_nat_nat(finite_card_nat,A)),one_one_nat),aa_set_nat_nat(finite_card_nat,A)) ).

% card_Diff_singleton_if
tff(fact_4928_card__Diff__singleton,axiom,
    ! [X: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
     => ( finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit))) = aa_nat_nat(minus_minus_nat(finite410649719033368117t_unit(A)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4929_card__Diff__singleton,axiom,
    ! [X: $o,A: set_o] :
      ( aa_set_o_o(member_o((X)),A)
     => ( finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))) = aa_nat_nat(minus_minus_nat(finite_card_o(A)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4930_card__Diff__singleton,axiom,
    ! [X: int,A: set_int] :
      ( aa_set_int_o(member_int(X),A)
     => ( finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int))) = aa_nat_nat(minus_minus_nat(finite_card_int(A)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4931_card__Diff__singleton,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
     => ( finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o))) = aa_nat_nat(minus_minus_nat(finite5795495896937551127_nat_o(A)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4932_card__Diff__singleton,axiom,
    ! [X: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(X),A)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat))) = aa_nat_nat(minus_minus_nat(aa_set_nat_nat(finite_card_nat,A)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4933_fst__image__mp,axiom,
    ! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
      ( ord_le4903435234951335757t_unit(image_6824039242672219652t_unit(produc1824681642469235216et_nat,A),B4)
     => ( member6260224972018164377et_nat(aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y),A)
       => member242389114213480694t_unit(X,B4) ) ) ).

% fst_image_mp
tff(fact_4934_fst__image__mp,axiom,
    ! [A: set_Pr1312406793078187350et_nat,B4: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_4444162529858428168_nat_o(produc5325900173150335662et_nat,A)),B4)
     => ( member4513763875858039863et_nat(aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4) ) ) ).

% fst_image_mp
tff(fact_4935_fst__image__mp,axiom,
    ! [A: set_Pr4040394263813967220et_nat,B4: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(image_8400968272122294250_nat_o(produc4078497350495916492et_nat,A)),B4)
     => ( member9174161442226468181et_nat(aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y),A)
       => aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),B4) ) ) ).

% fst_image_mp
tff(fact_4936_fst__image__mp,axiom,
    ! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_5042161079198086560nt_int(product_fst_int_int,A)),B4)
     => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),A)
       => aa_set_int_o(member_int(X),B4) ) ) ).

% fst_image_mp
tff(fact_4937_fst__image__mp,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(product_fst_nat_nat,A)),B4)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),A)
       => aa_set_nat_o(member_nat(X),B4) ) ) ).

% fst_image_mp
tff(fact_4938_fst__in__Field,axiom,
    ! [R: set_Pr958786334691620121nt_int] : aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,image_5042161079198086560nt_int(product_fst_int_int,R)),field_int(R)) ).

% fst_in_Field
tff(fact_4939_fst__in__Field,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_2486076414777270412at_nat(product_fst_nat_nat,R)),field_nat(R)) ).

% fst_in_Field
tff(fact_4940_Inf__fin__Inf,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(lattic4107685809792843317_fin_o,A)
        <=> aa_set_o_o(complete_Inf_Inf_o,A) ) ) ) ).

% Inf_fin_Inf
tff(fact_4941_Inf__fin__Inf,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( A != bot_bo8632255151191677115_nat_o )
       => ( lattic6089491107740071820_nat_o(A) = comple6093339926577406454_nat_o(A) ) ) ) ).

% Inf_fin_Inf
tff(fact_4942_Inf__fin__Inf,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ( lattic3014633134055518761et_nat(A) = comple7806235888213564991et_nat(A) ) ) ) ).

% Inf_fin_Inf
tff(fact_4943_Inf__fin__Inf,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic1263872656861969706t_unit(A) = comple2584293577114468500t_unit(A) ) ) ) ).

% Inf_fin_Inf
tff(fact_4944_cInf__eq__Inf__fin,axiom,
    ! [X5: set_o] :
      ( aa_set_o_o(finite_finite_o,X5)
     => ( ( X5 != bot_bot_set_o )
       => ( aa_set_o_o(complete_Inf_Inf_o,X5)
        <=> aa_set_o_o(lattic4107685809792843317_fin_o,X5) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4945_cInf__eq__Inf__fin,axiom,
    ! [X5: set_int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( ( X5 != bot_bot_set_int )
       => ( complete_Inf_Inf_int(X5) = lattic5235898064620869839in_int(X5) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4946_cInf__eq__Inf__fin,axiom,
    ! [X5: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,X5)
     => ( ( X5 != bot_bo8632255151191677115_nat_o )
       => ( comple6093339926577406454_nat_o(X5) = lattic6089491107740071820_nat_o(X5) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4947_cInf__eq__Inf__fin,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( ( X5 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,X5) = aa_set_nat_nat(lattic5238388535129920115in_nat,X5) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4948_cInf__eq__Inf__fin,axiom,
    ! [X5: set_set_nat] :
      ( finite1152437895449049373et_nat(X5)
     => ( ( X5 != bot_bot_set_set_nat )
       => ( comple7806235888213564991et_nat(X5) = lattic3014633134055518761et_nat(X5) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4949_cInf__eq__Inf__fin,axiom,
    ! [X5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ( comple2584293577114468500t_unit(X5) = lattic1263872656861969706t_unit(X5) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4950_prod__set__defs_I1_J,axiom,
    ! [X4: product_prod_int_int] : basic_fsts_int_int(X4) = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X4)),bot_bot_set_int) ).

% prod_set_defs(1)
tff(fact_4951_prod_Oswap__def,axiom,
    ! [P2: produc7348995623955488188t_unit] : aa_Pro2092192379717785596et_nat(produc1264744871843189710t_unit,P2) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(produc3550216979005444800t_unit(P2)),produc6012101397789842562t_unit(P2)) ).

% prod.swap_def
tff(fact_4952_prod_Oswap__def,axiom,
    ! [P2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(product_swap_nat_nat,P2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)) ).

% prod.swap_def
tff(fact_4953_prod_Oswap__def,axiom,
    ! [P2: produc2739566037923759970_nat_o] : aa_Pro1287330115173305408et_nat(produc6424530459701215220_nat_o,P2) = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(produc5282040756795020774_nat_o(P2)),produc5647884604762787496_nat_o(P2)) ).

% prod.swap_def
tff(fact_4954_prod_Oswap__def,axiom,
    ! [P2: produc6858501863712832324_nat_o] : aa_Pro7790764968675496000et_nat(produc1452984123473447766_nat_o,P2) = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(produc7941837674541168712_nat_o(P2)),produc2938390969378414602_nat_o(P2)) ).

% prod.swap_def
tff(fact_4955_prod_Oswap__def,axiom,
    ! [P2: produc3658429121746597890et_nat] : aa_Pro7370057327950226428t_unit(produc6300697153377358172et_nat,P2) = produc2471974422597015918t_unit(aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2)) ).

% prod.swap_def
tff(fact_4956_prod_Oswap__def,axiom,
    ! [P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(product_swap_int_int,P2) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)) ).

% prod.swap_def
tff(fact_4957_equiv__proj,axiom,
    ! [A: set_int,R: set_Pr958786334691620121nt_int,Z: product_prod_int_int] :
      ( equiv_equiv_int(A,R)
     => ( member5262025264175285858nt_int(Z,R)
       => ( comp_i5103216804979976198nt_int(equiv_proj_int_int(R),product_fst_int_int,Z) = comp_i5103216804979976198nt_int(equiv_proj_int_int(R),product_snd_int_int,Z) ) ) ) ).

% equiv_proj
tff(fact_4958_INF__le__SUP,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A))),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A))) ) ).

% INF_le_SUP
tff(fact_4959_INF__le__SUP,axiom,
    ! [A: set_o,F: fun_o_Product_unit] :
      ( ( A != bot_bot_set_o )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(image_o_Product_unit2(F,A))),comple4687483117567863418t_unit(image_o_Product_unit2(F,A))) ) ).

% INF_le_SUP
tff(fact_4960_INF__le__SUP,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit] :
      ( ( A != bot_bot_set_nat )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A))),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A))) ) ).

% INF_le_SUP
tff(fact_4961_INF__le__SUP,axiom,
    ! [A: set_int,F: fun_int_Product_unit] :
      ( ( A != bot_bot_set_int )
     => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A))),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A))) ) ).

% INF_le_SUP
tff(fact_4962_INF__le__SUP,axiom,
    ! [A: set_Product_unit,F: fun_Pr566567543205111936er_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(image_1853478764106424488er_nat(F,A))),comple2454491259644266675er_nat(image_1853478764106424488er_nat(F,A))) ) ).

% INF_le_SUP
tff(fact_4963_INF__le__SUP,axiom,
    ! [A: set_o,F: fun_o_filter_nat] :
      ( ( A != bot_bot_set_o )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(image_o_filter_nat(F,A))),comple2454491259644266675er_nat(image_o_filter_nat(F,A))) ) ).

% INF_le_SUP
tff(fact_4964_INF__le__SUP,axiom,
    ! [A: set_nat,F: fun_nat_filter_nat] :
      ( ( A != bot_bot_set_nat )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(image_nat_filter_nat(F,A))),comple2454491259644266675er_nat(image_nat_filter_nat(F,A))) ) ).

% INF_le_SUP
tff(fact_4965_INF__le__SUP,axiom,
    ! [A: set_int,F: fun_int_filter_nat] :
      ( ( A != bot_bot_set_int )
     => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(image_int_filter_nat(F,A))),comple2454491259644266675er_nat(image_int_filter_nat(F,A))) ) ).

% INF_le_SUP
tff(fact_4966_INF__le__SUP,axiom,
    ! [A: set_Product_unit,F: fun_Pr1628945989756059814et_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A))),comple7399068483239264473et_nat(image_8173849991827986510et_nat(F,A))) ) ).

% INF_le_SUP
tff(fact_4967_INF__le__SUP,axiom,
    ! [A: set_o,F: fun_o_set_nat] :
      ( ( A != bot_bot_set_o )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(image_o_set_nat(F,A))),comple7399068483239264473et_nat(image_o_set_nat(F,A))) ) ).

% INF_le_SUP
tff(fact_4968_card__Diff1__le,axiom,
    ! [A: set_Product_unit,X: product_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))),finite410649719033368117t_unit(A)) ).

% card_Diff1_le
tff(fact_4969_card__Diff1__le,axiom,
    ! [A: set_o,X: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))),finite_card_o(A)) ).

% card_Diff1_le
tff(fact_4970_card__Diff1__le,axiom,
    ! [A: set_int,X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))),finite_card_int(A)) ).

% card_Diff1_le
tff(fact_4971_card__Diff1__le,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))),finite5795495896937551127_nat_o(A)) ).

% card_Diff1_le
tff(fact_4972_card__Diff1__le,axiom,
    ! [A: set_nat,X: 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(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A)) ).

% card_Diff1_le
tff(fact_4973_exI__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,Y),X)
     => aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y))) ) ).

% exI_realizer
tff(fact_4974_exI__realizer,axiom,
    ! [P: fun_Pr5843349257357397163at_o_o,Y: produc3658429121746597890et_nat,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_Pro3588693547670092268at_o_o(P,Y),X)
     => aa_fun1675921005893230629at_o_o(aa_Pro3588693547670092268at_o_o(P,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y))),aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X),Y))) ) ).

% exI_realizer
tff(fact_4975_exI__realizer,axiom,
    ! [P: fun_Pr2647030239595059785at_o_o,Y: produc8799533983703829792et_nat,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aa_Pro1702148226396177290at_o_o(P,Y),X)
     => aa_fun1675921005893230629at_o_o(aa_Pro1702148226396177290at_o_o(P,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y))),aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X),Y))) ) ).

% exI_realizer
tff(fact_4976_exI__realizer,axiom,
    ! [P: fun_se9038067516210351697unit_o,Y: set_nat,X: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,Y),X)
     => aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y))),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X),Y))) ) ).

% exI_realizer
tff(fact_4977_exI__realizer,axiom,
    ! [P: fun_int_fun_int_o,Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(P,Y),X)
     => aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y))) ) ).

% exI_realizer
tff(fact_4978_conjI__realizer,axiom,
    ! [P: fun_nat_o,P2: nat,Q: fun_nat_o,Q3: nat] :
      ( aa_nat_o(P,P2)
     => ( aa_nat_o(Q,Q3)
       => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(P2),Q3)))
          & aa_nat_o(Q,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(P2),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_4979_conjI__realizer,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,P2: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,Q3: produc3658429121746597890et_nat] :
      ( aa_fun1675921005893230629at_o_o(P,P2)
     => ( aa_Pro653082453696054616_nat_o(Q,Q3)
       => ( aa_fun1675921005893230629at_o_o(P,aa_Pro8262445894397805713_nat_o(produc5325900173150335662et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(P2),Q3)))
          & aa_Pro653082453696054616_nat_o(Q,aa_Pro5541488491759012064et_nat(produc4960056325182568940et_nat,aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(P2),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_4980_conjI__realizer,axiom,
    ! [P: fun_fu5135969500543783852at_o_o,P2: fun_Pr6460944699146269935_nat_o,Q: fun_Pr2620521676128474833_nat_o,Q3: produc8799533983703829792et_nat] :
      ( aa_fun1675921005893230629at_o_o(P,P2)
     => ( aa_Pro5643009357358181178_nat_o(Q,Q3)
       => ( aa_fun1675921005893230629at_o_o(P,aa_Pro2837773213478468339_nat_o(produc4078497350495916492et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(P2),Q3)))
          & aa_Pro5643009357358181178_nat_o(Q,aa_Pro7798056222271281692et_nat(produc9081944055658670602et_nat,aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(P2),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_4981_conjI__realizer,axiom,
    ! [P: fun_he8413527710729349838unit_o,P2: heap_e7401611519738050253t_unit,Q: fun_set_nat_o,Q3: set_nat] :
      ( aa_hea7994211679605476167unit_o(P,P2)
     => ( aa_set_nat_o(Q,Q3)
       => ( aa_hea7994211679605476167unit_o(P,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(P2),Q3)))
          & aa_set_nat_o(Q,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(P2),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_4982_conjI__realizer,axiom,
    ! [P: fun_int_o,P2: int,Q: fun_int_o,Q3: int] :
      ( aa_int_o(P,P2)
     => ( aa_int_o(Q,Q3)
       => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(P2),Q3)))
          & aa_int_o(Q,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(product_Pair_int_int(P2),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_4983_apfst__convE,axiom,
    ! [Q3: produc3658429121746597890et_nat,F: fun_he2331061685413951235t_unit,P2: produc3658429121746597890et_nat] :
      ( ( Q3 = aa_Pro3679490825741336130et_nat(produc4172003873565521026et_nat(F),P2) )
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
            ( ( P2 = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(X3),Y2) )
           => ( Q3 != aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_hea2136485429539994620t_unit(F,X3)),Y2) ) ) ) ).

% apfst_convE
tff(fact_4984_apfst__convE,axiom,
    ! [Q3: product_prod_nat_nat,F: fun_nat_nat,P2: product_prod_nat_nat] :
      ( ( Q3 = produc986720760941809198at_nat(F,P2) )
     => ~ ! [X3: nat,Y2: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) )
           => ( Q3 != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,X3)),Y2) ) ) ) ).

% apfst_convE
tff(fact_4985_apfst__convE,axiom,
    ! [Q3: product_prod_int_int,F: fun_int_int,P2: product_prod_int_int] :
      ( ( Q3 = aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),P2) )
     => ~ ! [X3: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(product_Pair_int_int(X3),Y2) )
           => ( Q3 != aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(F,X3)),Y2) ) ) ) ).

% apfst_convE
tff(fact_4986_apfst__convE,axiom,
    ! [Q3: produc8799533983703829792et_nat,F: fun_fu8419763957888848259_nat_o,P2: produc8799533983703829792et_nat] :
      ( ( Q3 = aa_Pro5931322043027775230et_nat(produc1926458052860118910et_nat(F),P2) )
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
            ( ( P2 = aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(X3),Y2) )
           => ( Q3 != aa_Pro434340453207536480et_nat(produc6017135503889078682et_nat(aa_fun2638059832795814268_nat_o(F,X3)),Y2) ) ) ) ).

% apfst_convE
tff(fact_4987_apfst__convE,axiom,
    ! [Q3: produc3112576180687729086et_nat,F: fun_fu8419763957888848259_nat_o,P2: produc3112576180687729086et_nat] :
      ( ( Q3 = produc7064264152226348700et_nat(F,P2) )
     => ~ ! [X3: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
            ( ( P2 = aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(X3),Y2) )
           => ( Q3 != aa_Pro1979504083478021276et_nat(produc7291855603632349880et_nat(aa_fun2638059832795814268_nat_o(F,X3)),Y2) ) ) ) ).

% apfst_convE
tff(fact_4988_exE__realizer_H,axiom,
    ! [P: fun_se9038067516210351697unit_o,P2: produc3658429121746597890et_nat] :
      ( aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2)),aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2))
     => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] : ~ aa_hea7994211679605476167unit_o(aa_set6435858302011484818unit_o(P,Y2),X3) ) ).

% exE_realizer'
tff(fact_4989_exE__realizer_H,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ~ ! [X3: int,Y2: int] : ~ aa_int_o(aa_int_fun_int_o(P,Y2),X3) ) ).

% exE_realizer'
tff(fact_4990_snd__sndOp,axiom,
    ! [Bc: produc3658429121746597890et_nat,P: fun_he3009834522632631726unit_o,Q: fun_he5746958392356810273_nat_o] : aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Bc) = aa_Pro3416150927818442182et_nat(comp_P2930521774885306735et_nat(produc8586169260539613262et_nat,bNF_sn4039337771197282234et_nat(P,Q)),Bc) ).

% snd_sndOp
tff(fact_4991_snd__sndOp,axiom,
    ! [Bc: product_prod_int_int,P: fun_int_fun_int_o,Q: fun_int_fun_int_o] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Bc) = aa_Pro6799258434465214121nt_int(comp_P5356232565778667509nt_int(product_snd_int_int,bNF_sn1062102010912252026nt_int(P,Q)),Bc) ).

% snd_sndOp
tff(fact_4992_fst__fstOp,axiom,
    ! [Bc: product_prod_int_int,P: fun_int_fun_int_o,Q: fun_int_fun_int_o] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Bc) = aa_Pro6799258434465214121nt_int(comp_P5356232565778667509nt_int(product_fst_int_int,bNF_fs8167890477030535480nt_int(P,Q)),Bc) ).

% fst_fstOp
tff(fact_4993_card__Un__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( finite4902975817058060853nteger(sup_su848401254843788991nteger(A,B4)) = aa_nat_nat(plus_plus_nat(finite4902975817058060853nteger(A)),finite4902975817058060853nteger(B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4994_card__Un__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_nat_nat(plus_plus_nat(finite410649719033368117t_unit(A)),finite410649719033368117t_unit(B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4995_card__Un__disjoint,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
         => ( finite_card_o(aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) = aa_nat_nat(plus_plus_nat(finite_card_o(A)),finite_card_o(B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4996_card__Un__disjoint,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
         => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_nat_nat(plus_plus_nat(aa_set_nat_nat(finite_card_nat,A)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4997_card__Un__disjoint,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
         => ( finite_card_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_nat_nat(plus_plus_nat(finite_card_int(A)),finite_card_int(B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4998_card__Un__disjoint,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B4)
       => ( ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o )
         => ( finite5795495896937551127_nat_o(aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) = aa_nat_nat(plus_plus_nat(finite5795495896937551127_nat_o(A)),finite5795495896937551127_nat_o(B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4999_card__Suc__Diff1,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( aa_nat_nat(suc,finite4902975817058060853nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) = finite4902975817058060853nteger(A) ) ) ) ).

% card_Suc_Diff1
tff(fact_5000_card__Suc__Diff1,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_nat_nat(suc,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) = finite410649719033368117t_unit(A) ) ) ) ).

% card_Suc_Diff1
tff(fact_5001_card__Suc__Diff1,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) = finite_card_o(A) ) ) ) ).

% card_Suc_Diff1
tff(fact_5002_card__Suc__Diff1,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) = finite_card_int(A) ) ) ) ).

% card_Suc_Diff1
tff(fact_5003_card__Suc__Diff1,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))) = finite5795495896937551127_nat_o(A) ) ) ) ).

% card_Suc_Diff1
tff(fact_5004_card__Suc__Diff1,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) = aa_set_nat_nat(finite_card_nat,A) ) ) ) ).

% card_Suc_Diff1
tff(fact_5005_add__left__cancel,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_5006_add__left__cancel,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_5007_add__left__cancel,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(plus_plus_nat(A2),B) = aa_nat_nat(plus_plus_nat(A2),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_5008_add__left__cancel,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(A2),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_5009_add__right__cancel,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_5010_add__right__cancel,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_5011_add__right__cancel,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( aa_nat_nat(plus_plus_nat(B),A2) = aa_nat_nat(plus_plus_nat(C),A2) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_5012_add__right__cancel,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A2) = aa_int_int(plus_plus_int(C),A2) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_5013_add__le__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ).

% add_le_cancel_left
tff(fact_5014_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ).

% add_le_cancel_left
tff(fact_5015_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ).

% add_le_cancel_left
tff(fact_5016_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ).

% add_le_cancel_left
tff(fact_5017_add__le__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ).

% add_le_cancel_right
tff(fact_5018_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ).

% add_le_cancel_right
tff(fact_5019_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ).

% add_le_cancel_right
tff(fact_5020_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ).

% add_le_cancel_right
tff(fact_5021_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ).

% add_less_cancel_left
tff(fact_5022_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% add_less_cancel_left
tff(fact_5023_add__less__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ).

% add_less_cancel_left
tff(fact_5024_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% add_less_cancel_left
tff(fact_5025_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ).

% add_less_cancel_right
tff(fact_5026_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% add_less_cancel_right
tff(fact_5027_add__less__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ).

% add_less_cancel_right
tff(fact_5028_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% add_less_cancel_right
tff(fact_5029_add__diff__cancel,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),B) = A2 ).

% add_diff_cancel
tff(fact_5030_add__diff__cancel,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),B) = A2 ).

% add_diff_cancel
tff(fact_5031_add__diff__cancel,axiom,
    ! [A2: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B)),B) = A2 ).

% add_diff_cancel
tff(fact_5032_diff__add__cancel,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),B) = A2 ).

% diff_add_cancel
tff(fact_5033_diff__add__cancel,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A2),B)),B) = A2 ).

% diff_add_cancel
tff(fact_5034_diff__add__cancel,axiom,
    ! [A2: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A2),B)),B) = A2 ).

% diff_add_cancel
tff(fact_5035_add__diff__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) ).

% add_diff_cancel_left
tff(fact_5036_add__diff__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(minus_minus_rat(A2),B) ).

% add_diff_cancel_left
tff(fact_5037_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B)) = aa_nat_nat(minus_minus_nat(A2),B) ).

% add_diff_cancel_left
tff(fact_5038_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B)) = aa_int_int(minus_minus_int(A2),B) ).

% add_diff_cancel_left
tff(fact_5039_add__diff__cancel__left_H,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),A2) = B ).

% add_diff_cancel_left'
tff(fact_5040_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),A2) = B ).

% add_diff_cancel_left'
tff(fact_5041_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(A2),B)),A2) = B ).

% add_diff_cancel_left'
tff(fact_5042_add__diff__cancel__left_H,axiom,
    ! [A2: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B)),A2) = B ).

% add_diff_cancel_left'
tff(fact_5043_add__diff__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) ).

% add_diff_cancel_right
tff(fact_5044_add__diff__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(minus_minus_rat(A2),B) ).

% add_diff_cancel_right
tff(fact_5045_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C)) = aa_nat_nat(minus_minus_nat(A2),B) ).

% add_diff_cancel_right
tff(fact_5046_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(A2),B) ).

% add_diff_cancel_right
tff(fact_5047_add__diff__cancel__right_H,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),B) = A2 ).

% add_diff_cancel_right'
tff(fact_5048_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),B) = A2 ).

% add_diff_cancel_right'
tff(fact_5049_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(A2),B)),B) = A2 ).

% add_diff_cancel_right'
tff(fact_5050_add__diff__cancel__right_H,axiom,
    ! [A2: int,B: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B)),B) = A2 ).

% add_diff_cancel_right'
tff(fact_5051_card__insert__disjoint,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( finite5795495896937551127_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = aa_nat_nat(suc,finite5795495896937551127_nat_o(A)) ) ) ) ).

% card_insert_disjoint
tff(fact_5052_card__insert__disjoint,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( finite_card_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_nat_nat(suc,finite_card_int(A)) ) ) ) ).

% card_insert_disjoint
tff(fact_5053_card__insert__disjoint,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( finite_card_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_nat_nat(suc,finite_card_o(A)) ) ) ) ).

% card_insert_disjoint
tff(fact_5054_card__insert__disjoint,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_nat_nat(suc,finite410649719033368117t_unit(A)) ) ) ) ).

% card_insert_disjoint
tff(fact_5055_card__insert__disjoint,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A)) ) ) ) ).

% card_insert_disjoint
tff(fact_5056_card__insert__disjoint,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( finite4902975817058060853nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_nat_nat(suc,finite4902975817058060853nteger(A)) ) ) ) ).

% card_insert_disjoint
tff(fact_5057_Suc__diff,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M4)
       => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(N2),M4)) = aa_nat_nat(minus_minus_nat(N2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_5058_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( K3 = L ) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_5059_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K3 = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_5060_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K3 = L ) )
     => ( aa_nat_nat(plus_plus_nat(I2),K3) = aa_nat_nat(plus_plus_nat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_5061_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( ( I2 = J )
        & ( K3 = L ) )
     => ( aa_int_int(plus_plus_int(I2),K3) = aa_int_int(plus_plus_int(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_5062_group__cancel_Oadd1,axiom,
    ! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),A2) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)) ) ) ).

% group_cancel.add1
tff(fact_5063_group__cancel_Oadd1,axiom,
    ! [A: rat,K3: rat,A2: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),A2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)) ) ) ).

% group_cancel.add1
tff(fact_5064_group__cancel_Oadd1,axiom,
    ! [A: nat,K3: nat,A2: nat,B: nat] :
      ( ( A = aa_nat_nat(plus_plus_nat(K3),A2) )
     => ( aa_nat_nat(plus_plus_nat(A),B) = aa_nat_nat(plus_plus_nat(K3),aa_nat_nat(plus_plus_nat(A2),B)) ) ) ).

% group_cancel.add1
tff(fact_5065_group__cancel_Oadd1,axiom,
    ! [A: int,K3: int,A2: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(K3),A2) )
     => ( aa_int_int(plus_plus_int(A),B) = aa_int_int(plus_plus_int(K3),aa_int_int(plus_plus_int(A2),B)) ) ) ).

% group_cancel.add1
tff(fact_5066_group__cancel_Oadd2,axiom,
    ! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
      ( ( B4 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),B) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)) ) ) ).

% group_cancel.add2
tff(fact_5067_group__cancel_Oadd2,axiom,
    ! [B4: rat,K3: rat,B: rat,A2: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)) ) ) ).

% group_cancel.add2
tff(fact_5068_group__cancel_Oadd2,axiom,
    ! [B4: nat,K3: nat,B: nat,A2: nat] :
      ( ( B4 = aa_nat_nat(plus_plus_nat(K3),B) )
     => ( aa_nat_nat(plus_plus_nat(A2),B4) = aa_nat_nat(plus_plus_nat(K3),aa_nat_nat(plus_plus_nat(A2),B)) ) ) ).

% group_cancel.add2
tff(fact_5069_group__cancel_Oadd2,axiom,
    ! [B4: int,K3: int,B: int,A2: int] :
      ( ( B4 = aa_int_int(plus_plus_int(K3),B) )
     => ( aa_int_int(plus_plus_int(A2),B4) = aa_int_int(plus_plus_int(K3),aa_int_int(plus_plus_int(A2),B)) ) ) ).

% group_cancel.add2
tff(fact_5070_add_Oassoc,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% add.assoc
tff(fact_5071_add_Oassoc,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_5072_add_Oassoc,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(A2),B)),C) = aa_nat_nat(plus_plus_nat(A2),aa_nat_nat(plus_plus_nat(B),C)) ).

% add.assoc
tff(fact_5073_add_Oassoc,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A2),B)),C) = aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(B),C)) ).

% add.assoc
tff(fact_5074_add_Oleft__cancel,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_5075_add_Oleft__cancel,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_5076_add_Oleft__cancel,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(A2),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_5077_add_Oright__cancel,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_5078_add_Oright__cancel,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_5079_add_Oright__cancel,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A2) = aa_int_int(plus_plus_int(C),A2) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_5080_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_5081_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_5082_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: nat,B: nat] : aa_nat_nat(plus_plus_nat(A2),B) = aa_nat_nat(plus_plus_nat(B),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_5083_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: int,B: int] : aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(B),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_5084_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_5085_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_5086_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A2: nat,C: nat] : aa_nat_nat(plus_plus_nat(B),aa_nat_nat(plus_plus_nat(A2),C)) = aa_nat_nat(plus_plus_nat(A2),aa_nat_nat(plus_plus_nat(B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_5087_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A2: int,C: int] : aa_int_int(plus_plus_int(B),aa_int_int(plus_plus_int(A2),C)) = aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_5088_add__left__imp__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_5089_add__left__imp__eq,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_5090_add__left__imp__eq,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(plus_plus_nat(A2),B) = aa_nat_nat(plus_plus_nat(A2),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_5091_add__left__imp__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(A2),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_5092_add__right__imp__eq,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_5093_add__right__imp__eq,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_5094_add__right__imp__eq,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( aa_nat_nat(plus_plus_nat(B),A2) = aa_nat_nat(plus_plus_nat(C),A2) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_5095_add__right__imp__eq,axiom,
    ! [B: int,A2: int,C: int] :
      ( ( aa_int_int(plus_plus_int(B),A2) = aa_int_int(plus_plus_int(C),A2) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_5096_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),J)
        & ( K3 = L ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_5097_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & ( K3 = 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),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_5098_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & ( K3 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_5099_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & ( K3 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_5100_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K3),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_5101_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K3),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),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_5102_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I2 = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_5103_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( ( I2 = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K3),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_5104_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K3),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_5105_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K3),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),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_5106_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_5107_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K3),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_5108_add__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_mono
tff(fact_5109_add__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_mono
tff(fact_5110_add__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),D2)) ) ) ).

% add_mono
tff(fact_5111_add__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),D2)) ) ) ).

% add_mono
tff(fact_5112_add__left__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% add_left_mono
tff(fact_5113_add__left__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_5114_add__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B)) ) ).

% add_left_mono
tff(fact_5115_add__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B)) ) ).

% add_left_mono
tff(fact_5116_less__eqE,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ~ ! [C2: nat] : B != aa_nat_nat(plus_plus_nat(A2),C2) ) ).

% less_eqE
tff(fact_5117_add__right__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ).

% add_right_mono
tff(fact_5118_add__right__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_5119_add__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C)) ) ).

% add_right_mono
tff(fact_5120_add__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C)) ) ).

% add_right_mono
tff(fact_5121_le__iff__add,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
    <=> ? [C4: nat] : B = aa_nat_nat(plus_plus_nat(A2),C4) ) ).

% le_iff_add
tff(fact_5122_add__le__imp__le__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ).

% add_le_imp_le_left
tff(fact_5123_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ).

% add_le_imp_le_left
tff(fact_5124_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ).

% add_le_imp_le_left
tff(fact_5125_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ).

% add_le_imp_le_left
tff(fact_5126_add__le__imp__le__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ).

% add_le_imp_le_right
tff(fact_5127_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ).

% add_le_imp_le_right
tff(fact_5128_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ).

% add_le_imp_le_right
tff(fact_5129_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ).

% add_le_imp_le_right
tff(fact_5130_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_5131_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K3),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_5132_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I2),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K3),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_5133_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K3),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_5134_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I2 = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_5135_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( ( I2 = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K3),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_5136_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K3),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_5137_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K3),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_5138_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & ( K3 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_5139_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & ( K3 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_5140_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I2),J)
        & ( K3 = L ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_5141_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & ( K3 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_5142_add__strict__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),D2)) ) ) ).

% add_strict_mono
tff(fact_5143_add__strict__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),D2)) ) ) ).

% add_strict_mono
tff(fact_5144_add__strict__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_strict_mono
tff(fact_5145_add__strict__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_5146_add__strict__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B)) ) ).

% add_strict_left_mono
tff(fact_5147_add__strict__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B)) ) ).

% add_strict_left_mono
tff(fact_5148_add__strict__left__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% add_strict_left_mono
tff(fact_5149_add__strict__left__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_5150_add__strict__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C)) ) ).

% add_strict_right_mono
tff(fact_5151_add__strict__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C)) ) ).

% add_strict_right_mono
tff(fact_5152_add__strict__right__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ).

% add_strict_right_mono
tff(fact_5153_add__strict__right__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_5154_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(C),A2)),aa_nat_nat(plus_plus_nat(C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ).

% add_less_imp_less_left
tff(fact_5155_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C),A2)),aa_int_int(plus_plus_int(C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% add_less_imp_less_left
tff(fact_5156_add__less__imp__less__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ).

% add_less_imp_less_left
tff(fact_5157_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% add_less_imp_less_left
tff(fact_5158_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ).

% add_less_imp_less_right
tff(fact_5159_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% add_less_imp_less_right
tff(fact_5160_add__less__imp__less__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ).

% add_less_imp_less_right
tff(fact_5161_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% add_less_imp_less_right
tff(fact_5162_combine__common__factor,axiom,
    ! [A2: code_integer,E2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),E2)),C) ).

% combine_common_factor
tff(fact_5163_combine__common__factor,axiom,
    ! [A2: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),E2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),E2)),C) ).

% combine_common_factor
tff(fact_5164_combine__common__factor,axiom,
    ! [A2: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),E2)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A2),B)),E2)),C) ).

% combine_common_factor
tff(fact_5165_combine__common__factor,axiom,
    ! [A2: int,E2: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(A2),B)),E2)),C) ).

% combine_common_factor
tff(fact_5166_distrib__right,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% distrib_right
tff(fact_5167_distrib__right,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_5168_distrib__right,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A2),B)),C) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_5169_distrib__right,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(A2),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_5170_distrib__left,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) ).

% distrib_left
tff(fact_5171_distrib__left,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% distrib_left
tff(fact_5172_distrib__left,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(plus_plus_nat(B),C)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) ).

% distrib_left
tff(fact_5173_distrib__left,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% distrib_left
tff(fact_5174_comm__semiring__class_Odistrib,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% comm_semiring_class.distrib
tff(fact_5175_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_5176_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A2),B)),C) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_5177_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(A2),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_5178_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_5179_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_5180_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(plus_plus_int(B),C)) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_5181_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_5182_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_5183_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(A2),B)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_5184_group__cancel_Osub1,axiom,
    ! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),A2) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)) ) ) ).

% group_cancel.sub1
tff(fact_5185_group__cancel_Osub1,axiom,
    ! [A: rat,K3: rat,A2: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),A2) )
     => ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),aa_rat_rat(minus_minus_rat(A2),B)) ) ) ).

% group_cancel.sub1
tff(fact_5186_group__cancel_Osub1,axiom,
    ! [A: int,K3: int,A2: int,B: int] :
      ( ( A = aa_int_int(plus_plus_int(K3),A2) )
     => ( aa_int_int(minus_minus_int(A),B) = aa_int_int(plus_plus_int(K3),aa_int_int(minus_minus_int(A2),B)) ) ) ).

% group_cancel.sub1
tff(fact_5187_diff__eq__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) = C )
    <=> ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) ) ) ).

% diff_eq_eq
tff(fact_5188_diff__eq__eq,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B) = C )
    <=> ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_5189_diff__eq__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( aa_int_int(minus_minus_int(A2),B) = C )
    <=> ( A2 = aa_int_int(plus_plus_int(C),B) ) ) ).

% diff_eq_eq
tff(fact_5190_eq__diff__eq,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = C ) ) ).

% eq_diff_eq
tff(fact_5191_eq__diff__eq,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( ( A2 = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = C ) ) ).

% eq_diff_eq
tff(fact_5192_eq__diff__eq,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( A2 = aa_int_int(minus_minus_int(C),B) )
    <=> ( aa_int_int(plus_plus_int(A2),B) = C ) ) ).

% eq_diff_eq
tff(fact_5193_add__diff__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) ).

% add_diff_eq
tff(fact_5194_add__diff__eq,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) ).

% add_diff_eq
tff(fact_5195_add__diff__eq,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(plus_plus_int(A2),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B)),C) ).

% add_diff_eq
tff(fact_5196_diff__diff__eq2,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),B) ).

% diff_diff_eq2
tff(fact_5197_diff__diff__eq2,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B) ).

% diff_diff_eq2
tff(fact_5198_diff__diff__eq2,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(minus_minus_int(A2),aa_int_int(minus_minus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C)),B) ).

% diff_diff_eq2
tff(fact_5199_diff__add__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),B) ).

% diff_add_eq
tff(fact_5200_diff__add__eq,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A2),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B) ).

% diff_add_eq
tff(fact_5201_diff__add__eq,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A2),B)),C) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C)),B) ).

% diff_add_eq
tff(fact_5202_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_5203_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_5204_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(minus_minus_int(A2),aa_int_int(plus_plus_int(B),C)) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_5205_add__implies__diff,axiom,
    ! [C: code_integer,B: code_integer,A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) = A2 )
     => ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) ) ) ).

% add_implies_diff
tff(fact_5206_add__implies__diff,axiom,
    ! [C: rat,B: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A2 )
     => ( C = aa_rat_rat(minus_minus_rat(A2),B) ) ) ).

% add_implies_diff
tff(fact_5207_add__implies__diff,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( ( aa_nat_nat(plus_plus_nat(C),B) = A2 )
     => ( C = aa_nat_nat(minus_minus_nat(A2),B) ) ) ).

% add_implies_diff
tff(fact_5208_add__implies__diff,axiom,
    ! [C: int,B: int,A2: int] :
      ( ( aa_int_int(plus_plus_int(C),B) = A2 )
     => ( C = aa_int_int(minus_minus_int(A2),B) ) ) ).

% add_implies_diff
tff(fact_5209_diff__diff__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% diff_diff_eq
tff(fact_5210_diff__diff__eq,axiom,
    ! [A2: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),B)),C) = aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_5211_diff__diff__eq,axiom,
    ! [A2: nat,B: nat,C: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(A2),B)),C) = aa_nat_nat(minus_minus_nat(A2),aa_nat_nat(plus_plus_nat(B),C)) ).

% diff_diff_eq
tff(fact_5212_diff__diff__eq,axiom,
    ! [A2: int,B: int,C: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),B)),C) = aa_int_int(minus_minus_int(A2),aa_int_int(plus_plus_int(B),C)) ).

% diff_diff_eq
tff(fact_5213_mlex__bound,axiom,
    ! [A2: nat,A: nat,B: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N)) ) ) ).

% mlex_bound
tff(fact_5214_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B: nat,N: nat,B3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N)),B)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N)),B3)) ) ) ) ).

% mlex_fst_decrI
tff(fact_5215_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B: nat,B3: nat,N: nat] :
      ( ( A2 = A4 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N)),B)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N)),B3)) ) ) ).

% mlex_snd_decrI
tff(fact_5216_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B: nat,B3: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N)),B)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N)),B3)) ) ) ).

% mlex_leI
tff(fact_5217_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K4)),K4) ) ).

% Inf_nat_def1
tff(fact_5218_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,X),Y)),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Y),Z)) ).

% max_add_distrib_left
tff(fact_5219_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(ord_max_rat(X),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_5220_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(plus_plus_nat(X),Z)),aa_nat_nat(plus_plus_nat(Y),Z)) ).

% max_add_distrib_left
tff(fact_5221_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(plus_plus_int(X),Z)),aa_int_int(plus_plus_int(Y),Z)) ).

% max_add_distrib_left
tff(fact_5222_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,Y),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Z)) ).

% max_add_distrib_right
tff(fact_5223_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(ord_max_rat(Y),Z)) = aa_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),Z)) ).

% max_add_distrib_right
tff(fact_5224_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(plus_plus_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(plus_plus_nat(X),Y)),aa_nat_nat(plus_plus_nat(X),Z)) ).

% max_add_distrib_right
tff(fact_5225_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(plus_plus_int(X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(plus_plus_int(X),Y)),aa_int_int(plus_plus_int(X),Z)) ).

% max_add_distrib_right
tff(fact_5226_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I2),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K3),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_5227_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K3),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_5228_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_5229_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K3),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_5230_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I2),J)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K3),L) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_5231_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K3: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K3),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_5232_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(I2),K3)),aa_nat_nat(plus_plus_nat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_5233_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K3: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K3),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I2),K3)),aa_int_int(plus_plus_int(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_5234_add__le__less__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_5235_add__le__less__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_5236_add__le__less__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),D2)) ) ) ).

% add_le_less_mono
tff(fact_5237_add__le__less__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),D2)) ) ) ).

% add_le_less_mono
tff(fact_5238_add__less__le__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_5239_add__less__le__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_5240_add__less__le__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),C)),aa_nat_nat(plus_plus_nat(B),D2)) ) ) ).

% add_less_le_mono
tff(fact_5241_add__less__le__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C)),aa_int_int(plus_plus_int(B),D2)) ) ) ).

% add_less_le_mono
tff(fact_5242_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
       => ( ( aa_nat_nat(minus_minus_nat(B),A2) = C )
        <=> ( B = aa_nat_nat(plus_plus_nat(C),A2) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_5243_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(plus_plus_nat(A2),aa_nat_nat(minus_minus_nat(B),A2)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_5244_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(minus_minus_nat(C),aa_nat_nat(minus_minus_nat(B),A2)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(C),A2)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_5245_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(B),C)),A2) = aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(B),A2)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_5246_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(B),A2)),C) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(B),C)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_5247_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(C),B)),A2) = aa_nat_nat(plus_plus_nat(C),aa_nat_nat(minus_minus_nat(B),A2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_5248_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(plus_plus_nat(C),aa_nat_nat(minus_minus_nat(B),A2)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(C),B)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_5249_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(minus_minus_nat(B),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(C),A2)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_5250_le__add__diff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(B),C)),A2)) ) ).

% le_add_diff
tff(fact_5251_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(B),A2)),A2) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_5252_le__diff__eq,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) ) ).

% le_diff_eq
tff(fact_5253_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) ) ).

% le_diff_eq
tff(fact_5254_le__diff__eq,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),B)),C) ) ).

% le_diff_eq
tff(fact_5255_diff__le__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),C)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% diff_le_eq
tff(fact_5256_diff__le__eq,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_5257_diff__le__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(plus_plus_int(C),B)) ) ).

% diff_le_eq
tff(fact_5258_less__add__one,axiom,
    ! [A2: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A2),plus_p4538020629002901425atural(A2,one_one_Code_natural)) ).

% less_add_one
tff(fact_5259_less__add__one,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(plus_plus_nat(A2),one_one_nat)) ).

% less_add_one
tff(fact_5260_less__add__one,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(A2),one_one_int)) ).

% less_add_one
tff(fact_5261_less__add__one,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),one_one_Code_integer)) ).

% less_add_one
tff(fact_5262_less__add__one,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat)) ).

% less_add_one
tff(fact_5263_add__mono1,axiom,
    ! [A2: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A2),B)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,plus_p4538020629002901425atural(A2,one_one_Code_natural)),plus_p4538020629002901425atural(B,one_one_Code_natural)) ) ).

% add_mono1
tff(fact_5264_add__mono1,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),one_one_nat)),aa_nat_nat(plus_plus_nat(B),one_one_nat)) ) ).

% add_mono1
tff(fact_5265_add__mono1,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),one_one_int)),aa_int_int(plus_plus_int(B),one_one_int)) ) ).

% add_mono1
tff(fact_5266_add__mono1,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),one_one_Code_integer)) ) ).

% add_mono1
tff(fact_5267_add__mono1,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_5268_diff__less__eq,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(C),B)) ) ).

% diff_less_eq
tff(fact_5269_diff__less__eq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),C)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% diff_less_eq
tff(fact_5270_diff__less__eq,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_5271_less__diff__eq,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(minus_minus_int(C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B)),C) ) ).

% less_diff_eq
tff(fact_5272_less__diff__eq,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),C) ) ).

% less_diff_eq
tff(fact_5273_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)),C) ) ).

% less_diff_eq
tff(fact_5274_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X),Y)) ).

% square_diff_square_factored
tff(fact_5275_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_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(minus_minus_rat(X),Y)) ).

% square_diff_square_factored
tff(fact_5276_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_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(plus_plus_int(X),Y)),aa_int_int(minus_minus_int(X),Y)) ).

% square_diff_square_factored
tff(fact_5277_eq__add__iff2,axiom,
    ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2) )
    <=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A2)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_5278_eq__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: 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,A2),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A2)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_5279_eq__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B: int,D2: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2) )
    <=> ( C = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(B),A2)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_5280_eq__add__iff1,axiom,
    ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_5281_eq__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: 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,A2),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),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(minus_minus_rat(A2),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_5282_eq__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B: int,D2: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(A2),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_5283_Suc__n__minus__m__eq,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M4)
       => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(N2),M4)) = aa_nat_nat(minus_minus_nat(N2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_5284_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(suc,A2)) )
    <=> ( B = aa_nat_nat(suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_5285_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A2)) )
    <=> ( B = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_5286_wf__no__infinite__down__chainE,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: fun_nat_nat] :
      ( wf_nat(R2)
     => ~ ! [K: nat] : member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F,aa_nat_nat(suc,K))),aa_nat_nat(F,K)),R2) ) ).

% wf_no_infinite_down_chainE
tff(fact_5287_wf__no__infinite__down__chainE,axiom,
    ! [R2: set_Pr958786334691620121nt_int,F: fun_nat_int] :
      ( wf_int(R2)
     => ~ ! [K: nat] : member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F,aa_nat_nat(suc,K))),aa_nat_int(F,K)),R2) ) ).

% wf_no_infinite_down_chainE
tff(fact_5288_wf__iff__no__infinite__down__chain,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( wf_nat(R2)
    <=> ~ ? [F4: fun_nat_nat] :
          ! [I4: nat] : member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(F4,aa_nat_nat(suc,I4))),aa_nat_nat(F4,I4)),R2) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_5289_wf__iff__no__infinite__down__chain,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( wf_int(R2)
    <=> ~ ? [F4: fun_nat_int] :
          ! [I4: nat] : member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(F4,aa_nat_nat(suc,I4))),aa_nat_int(F4,I4)),R2) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_5290_Inf__INT__eq,axiom,
    ! [S: set_fu8039672399046406370at_o_o,X4: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(comple8649786507288350949at_o_o(S),X4)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),comple5146519332901335382_nat_o(image_401909237675433180_nat_o(collec6407654713642994138_nat_o,S))) ) ).

% Inf_INT_eq
tff(fact_5291_Inf__INT__eq,axiom,
    ! [S: set_fun_int_o,X4: int] :
      ( aa_int_o(comple7892175240008826188_int_o(S),X4)
    <=> aa_set_int_o(member_int(X4),comple3628384868704368283et_int(image_3849661908695225314et_int(collect_int,S))) ) ).

% Inf_INT_eq
tff(fact_5292_Inf__INT__eq,axiom,
    ! [S: set_fun_o_o,X4: $o] :
      ( aa_o_o(comple7101583973641323004un_o_o(S),(X4))
    <=> aa_set_o_o(member_o((X4)),comple3063163877087187839_set_o(image_fun_o_o_set_o(collect_o,S))) ) ).

% Inf_INT_eq
tff(fact_5293_Inf__INT__eq,axiom,
    ! [S: set_fu7570441001188634436unit_o,X4: product_unit] :
      ( aa_Product_unit_o(comple6120340423619789639unit_o(S),X4)
    <=> aa_set8780210009671629607unit_o(member_Product_unit(X4),comple7216939928495492468t_unit(image_2610332678118349720t_unit(collect_Product_unit,S))) ) ).

% Inf_INT_eq
tff(fact_5294_Inf__INT__eq,axiom,
    ! [S: set_fun_nat_o,X4: nat] :
      ( aa_nat_o(comple489845548586738344_nat_o(S),X4)
    <=> aa_set_nat_o(member_nat(X4),comple7806235888213564991et_nat(image_4516309135669182690et_nat(collect_nat,S))) ) ).

% Inf_INT_eq
tff(fact_5295_le__add__iff2,axiom,
    ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A2)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_5296_le__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: 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,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A2)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_5297_le__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(B),A2)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_5298_le__add__iff1,axiom,
    ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_5299_le__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: 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,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),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(minus_minus_rat(A2),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_5300_le__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(A2),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_5301_less__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(B),A2)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_5302_less__add__iff2,axiom,
    ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A2)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_5303_less__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: 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,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A2)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_5304_less__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(A2),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_5305_less__add__iff1,axiom,
    ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_5306_less__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: 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,A2),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),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(minus_minus_rat(A2),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_5307_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_5308_square__diff__one__factored,axiom,
    ! [X: rat] : aa_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(minus_minus_rat(X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_5309_square__diff__one__factored,axiom,
    ! [X: int] : aa_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(plus_plus_int(X),one_one_int)),aa_int_int(minus_minus_int(X),one_one_int)) ).

% square_diff_one_factored
tff(fact_5310_card__Suc__eq__finite,axiom,
    ! [A: set_fu3555266933746557007_nat_o,K3: nat] :
      ( ( finite5795495896937551127_nat_o(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: fun_Pr6460944699146269935_nat_o,B9: set_fu3555266933746557007_nat_o] :
          ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B6),B9) )
          & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B6),B9)
          & ( finite5795495896937551127_nat_o(B9) = K3 )
          & aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B9) ) ) ).

% card_Suc_eq_finite
tff(fact_5311_card__Suc__eq__finite,axiom,
    ! [A: set_int,K3: nat] :
      ( ( finite_card_int(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: int,B9: set_int] :
          ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B6),B9) )
          & ~ aa_set_int_o(member_int(B6),B9)
          & ( finite_card_int(B9) = K3 )
          & aa_set_int_o(finite_finite_int,B9) ) ) ).

% card_Suc_eq_finite
tff(fact_5312_card__Suc__eq__finite,axiom,
    ! [A: set_o,K3: nat] :
      ( ( finite_card_o(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: $o,B9: set_o] :
          ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B6)),B9) )
          & ~ aa_set_o_o(member_o((B6)),B9)
          & ( finite_card_o(B9) = K3 )
          & aa_set_o_o(finite_finite_o,B9) ) ) ).

% card_Suc_eq_finite
tff(fact_5313_card__Suc__eq__finite,axiom,
    ! [A: set_Product_unit,K3: nat] :
      ( ( finite410649719033368117t_unit(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: product_unit,B9: set_Product_unit] :
          ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B6),B9) )
          & ~ aa_set8780210009671629607unit_o(member_Product_unit(B6),B9)
          & ( finite410649719033368117t_unit(B9) = K3 )
          & aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B9) ) ) ).

% card_Suc_eq_finite
tff(fact_5314_card__Suc__eq__finite,axiom,
    ! [A: set_nat,K3: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: nat,B9: set_nat] :
          ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B6),B9) )
          & ~ aa_set_nat_o(member_nat(B6),B9)
          & ( aa_set_nat_nat(finite_card_nat,B9) = K3 )
          & aa_set_nat_o(finite_finite_nat,B9) ) ) ).

% card_Suc_eq_finite
tff(fact_5315_card__Suc__eq__finite,axiom,
    ! [A: set_Code_integer,K3: nat] :
      ( ( finite4902975817058060853nteger(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: code_integer,B9: set_Code_integer] :
          ( ( A = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,B6),B9) )
          & ~ aa_set5643387332196575111eger_o(member_Code_integer(B6),B9)
          & ( finite4902975817058060853nteger(B9) = K3 )
          & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B9) ) ) ).

% card_Suc_eq_finite
tff(fact_5316_card__insert__if,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( finite5795495896937551127_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A),finite5795495896937551127_nat_o(A),aa_nat_nat(suc,finite5795495896937551127_nat_o(A))) ) ) ).

% card_insert_if
tff(fact_5317_card__insert__if,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( finite_card_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(aa_set_int_o(member_int(X),A),finite_card_int(A),aa_nat_nat(suc,finite_card_int(A))) ) ) ).

% card_insert_if
tff(fact_5318_card__insert__if,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( finite_card_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = $ite(aa_set_o_o(member_o((X)),A),finite_card_o(A),aa_nat_nat(suc,finite_card_o(A))) ) ) ).

% card_insert_if
tff(fact_5319_card__insert__if,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),finite410649719033368117t_unit(A),aa_nat_nat(suc,finite410649719033368117t_unit(A))) ) ) ).

% card_insert_if
tff(fact_5320_card__insert__if,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_o(member_nat(X),A),aa_set_nat_nat(finite_card_nat,A),aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A))) ) ) ).

% card_insert_if
tff(fact_5321_card__insert__if,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( finite4902975817058060853nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A),finite4902975817058060853nteger(A),aa_nat_nat(suc,finite4902975817058060853nteger(A))) ) ) ).

% card_insert_if
tff(fact_5322_card__le__Suc__iff,axiom,
    ! [N2: nat,A: set_fu3555266933746557007_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),finite5795495896937551127_nat_o(A))
    <=> ? [A5: fun_Pr6460944699146269935_nat_o,B9: set_fu3555266933746557007_nat_o] :
          ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A5),B9) )
          & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A5),B9)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite5795495896937551127_nat_o(B9))
          & aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,B9) ) ) ).

% card_le_Suc_iff
tff(fact_5323_card__le__Suc__iff,axiom,
    ! [N2: nat,A: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),finite_card_int(A))
    <=> ? [A5: int,B9: set_int] :
          ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A5),B9) )
          & ~ aa_set_int_o(member_int(A5),B9)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_int(B9))
          & aa_set_int_o(finite_finite_int,B9) ) ) ).

% card_le_Suc_iff
tff(fact_5324_card__le__Suc__iff,axiom,
    ! [N2: nat,A: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),finite_card_o(A))
    <=> ? [A5: $o,B9: set_o] :
          ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A5)),B9) )
          & ~ aa_set_o_o(member_o((A5)),B9)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite_card_o(B9))
          & aa_set_o_o(finite_finite_o,B9) ) ) ).

% card_le_Suc_iff
tff(fact_5325_card__le__Suc__iff,axiom,
    ! [N2: nat,A: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),finite410649719033368117t_unit(A))
    <=> ? [A5: product_unit,B9: set_Product_unit] :
          ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A5),B9) )
          & ~ aa_set8780210009671629607unit_o(member_Product_unit(A5),B9)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite410649719033368117t_unit(B9))
          & aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B9) ) ) ).

% card_le_Suc_iff
tff(fact_5326_card__le__Suc__iff,axiom,
    ! [N2: nat,A: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_set_nat_nat(finite_card_nat,A))
    <=> ? [A5: nat,B9: set_nat] :
          ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A5),B9) )
          & ~ aa_set_nat_o(member_nat(A5),B9)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_set_nat_nat(finite_card_nat,B9))
          & aa_set_nat_o(finite_finite_nat,B9) ) ) ).

% card_le_Suc_iff
tff(fact_5327_card__le__Suc__iff,axiom,
    ! [N2: nat,A: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N2)),finite4902975817058060853nteger(A))
    <=> ? [A5: code_integer,B9: set_Code_integer] :
          ( ( A = aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A5),B9) )
          & ~ aa_set5643387332196575111eger_o(member_Code_integer(A5),B9)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),finite4902975817058060853nteger(B9))
          & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B9) ) ) ).

% card_le_Suc_iff
tff(fact_5328_card_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( finite4902975817058060853nteger(A) = aa_nat_nat(suc,finite4902975817058060853nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% card.remove
tff(fact_5329_card_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( finite410649719033368117t_unit(A) = aa_nat_nat(suc,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% card.remove
tff(fact_5330_card_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( finite_card_o(A) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ) ).

% card.remove
tff(fact_5331_card_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( finite_card_int(A) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) ) ) ) ).

% card.remove
tff(fact_5332_card_Oremove,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( finite5795495896937551127_nat_o(A) = aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))) ) ) ) ).

% card.remove
tff(fact_5333_card_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_nat(finite_card_nat,A) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) ) ) ) ).

% card.remove
tff(fact_5334_card_Oinsert__remove,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( finite4902975817058060853nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_nat_nat(suc,finite4902975817058060853nteger(minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% card.insert_remove
tff(fact_5335_card_Oinsert__remove,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_nat_nat(suc,finite410649719033368117t_unit(minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% card.insert_remove
tff(fact_5336_card_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( finite_card_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ).

% card.insert_remove
tff(fact_5337_card_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( finite_card_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) ) ) ).

% card.insert_remove
tff(fact_5338_card_Oinsert__remove,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( finite5795495896937551127_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = aa_nat_nat(suc,finite5795495896937551127_nat_o(minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)))) ) ) ).

% card.insert_remove
tff(fact_5339_card_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_5340_mult__Suc__right,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(suc,N2)) = aa_nat_nat(plus_plus_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) ).

% mult_Suc_right
tff(fact_5341_nat__mult__eq__1__iff,axiom,
    ! [M4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) = one_one_nat )
    <=> ( ( M4 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_5342_nat__1__eq__mult__iff,axiom,
    ! [M4: nat,N2: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) )
    <=> ( ( M4 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_5343_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M4)),N2) ) ) ).

% nat_less_add_iff1
tff(fact_5344_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N2)) ) ) ).

% nat_less_add_iff2
tff(fact_5345_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)) = aa_nat_nat(minus_minus_nat(M4),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N2)) ) ) ).

% nat_diff_add_eq2
tff(fact_5346_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M4)),N2) ) ) ).

% nat_diff_add_eq1
tff(fact_5347_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N2)) ) ) ).

% nat_le_add_iff2
tff(fact_5348_nat__mult__max__right,axiom,
    ! [M4: nat,N2: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),Q3)) ).

% nat_mult_max_right
tff(fact_5349_nat__mult__max__left,axiom,
    ! [M4: nat,N2: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M4),N2)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) ).

% nat_mult_max_left
tff(fact_5350_Suc__mult__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),M4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),N2) )
    <=> ( M4 = N2 ) ) ).

% Suc_mult_cancel1
tff(fact_5351_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),J)) ) ).

% mult_le_mono2
tff(fact_5352_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K3)) ) ).

% mult_le_mono1
tff(fact_5353_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K3: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),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),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_5354_le__square,axiom,
    ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),M4)) ).

% le_square
tff(fact_5355_le__cube,axiom,
    ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),M4))) ).

% le_cube
tff(fact_5356_add__mult__distrib,axiom,
    ! [M4: nat,N2: nat,K3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(M4),N2)),K3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3)) ).

% add_mult_distrib
tff(fact_5357_add__mult__distrib2,axiom,
    ! [K3: nat,M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),aa_nat_nat(plus_plus_nat(M4),N2)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)) ).

% add_mult_distrib2
tff(fact_5358_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K3: nat] : aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K3)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(I2),J)),U)),K3) ).

% left_add_mult_distrib
tff(fact_5359_diff__mult__distrib,axiom,
    ! [M4: nat,N2: nat,K3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(M4),N2)),K3) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3)) ).

% diff_mult_distrib
tff(fact_5360_diff__mult__distrib2,axiom,
    ! [K3: nat,M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),aa_nat_nat(minus_minus_nat(M4),N2)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)) ).

% diff_mult_distrib2
tff(fact_5361_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_5362_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_5363_Suc__mult__less__cancel1,axiom,
    ! [K3: nat,M4: 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,K3)),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ).

% Suc_mult_less_cancel1
tff(fact_5364_Suc__mult__le__cancel1,axiom,
    ! [K3: nat,M4: 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,K3)),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),N2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ).

% Suc_mult_le_cancel1
tff(fact_5365_mult__Suc,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M4)),N2) = aa_nat_nat(plus_plus_nat(N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) ).

% mult_Suc
tff(fact_5366_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2) )
      <=> ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M4) = N2 ) ) ) ).

% nat_eq_add_iff1
tff(fact_5367_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2) )
      <=> ( M4 = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I2)),U)),N2) ) ) ) ).

% nat_eq_add_iff2
tff(fact_5368_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),M4)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I2),J)),U)),M4)),N2) ) ) ).

% nat_le_add_iff1
tff(fact_5369_discrete,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),one_one_Code_integer)),B) ) ).

% discrete
tff(fact_5370_discrete,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),one_one_nat)),B) ) ).

% discrete
tff(fact_5371_discrete,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),one_one_int)),B) ) ).

% discrete
tff(fact_5372_fstOp__def,axiom,
    ! [P: fun_he5746958392356810273_nat_o,Q: fun_se2170385732985038782_nat_o,Ac: produc3658429121746597890et_nat] : aa_Pro3679490825741336130et_nat(bNF_fs3871268833404974121et_nat(P,Q),Ac) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Ac)),aa_set_nat_set_nat(bNF_pi4042512755101277815et_nat(P,Q,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Ac)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Ac))) ).

% fstOp_def
tff(fact_5373_fstOp__def,axiom,
    ! [P: fun_int_fun_int_o,Q: fun_int_fun_int_o,Ac: product_prod_int_int] : aa_Pro6970504127699319548nt_int(bNF_fs8167890477030535480nt_int(P,Q),Ac) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Ac)),aa_int_int(bNF_pi8853482479757052266nt_int(P,Q,aa_Pro6799258434465214121nt_int(product_fst_int_int,Ac)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Ac))) ).

% fstOp_def
tff(fact_5374_sndOp__def,axiom,
    ! [P: fun_he3009834522632631726unit_o,Q: fun_he5746958392356810273_nat_o,Ac: produc3658429121746597890et_nat] : aa_Pro3679490825741336130et_nat(bNF_sn4039337771197282234et_nat(P,Q),Ac) = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(bNF_pi3450177011487049898et_nat(P,Q,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,Ac),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Ac))),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,Ac)) ).

% sndOp_def
tff(fact_5375_sndOp__def,axiom,
    ! [P: fun_int_fun_int_o,Q: fun_int_fun_int_o,Ac: product_prod_int_int] : aa_Pro6970504127699319548nt_int(bNF_sn1062102010912252026nt_int(P,Q),Ac) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(bNF_pi8853482479757052266nt_int(P,Q,aa_Pro6799258434465214121nt_int(product_fst_int_int,Ac)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Ac))),aa_Pro6799258434465214121nt_int(product_snd_int_int,Ac)) ).

% sndOp_def
tff(fact_5376_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_nat,X: 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_set_nat_nat(finite_card_nat,Y)),aa_nat_nat(minus_minus_nat(N2),one_one_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_nat5279478944698590999et_nat(insert_nat,X),Y))),N2) ) ) ).

% card_insert_le_m1
tff(fact_5377_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_int,X: int] :
      ( 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,finite_card_int(Y)),aa_nat_nat(minus_minus_nat(N2),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),Y))),N2) ) ) ).

% card_insert_le_m1
tff(fact_5378_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_o,X: $o] :
      ( 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,finite_card_o(Y)),aa_nat_nat(minus_minus_nat(N2),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),Y))),N2) ) ) ).

% card_insert_le_m1
tff(fact_5379_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_Product_unit,X: product_unit] :
      ( 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,finite410649719033368117t_unit(Y)),aa_nat_nat(minus_minus_nat(N2),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),Y))),N2) ) ) ).

% card_insert_le_m1
tff(fact_5380_crossproduct__noteq,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A2 != B )
        & ( C != D2 ) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),D2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5381_crossproduct__noteq,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( ( ( A2 != B )
        & ( C != D2 ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),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,A2),D2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5382_crossproduct__noteq,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A2 != B )
        & ( C != D2 ) )
    <=> ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) != aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5383_crossproduct__noteq,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( ( ( A2 != B )
        & ( C != D2 ) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) != aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_5384_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5385_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5386_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5387_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_5388_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_5389_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_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_5390_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_5391_card__insert__disjoint_H,axiom,
    ! [A: set_fu3555266933746557007_nat_o,X: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X),A)
       => ( aa_nat_nat(minus_minus_nat(finite5795495896937551127_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A))),aa_nat_nat(suc,zero_zero_nat)) = finite5795495896937551127_nat_o(A) ) ) ) ).

% card_insert_disjoint'
tff(fact_5392_card__insert__disjoint_H,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( aa_nat_nat(minus_minus_nat(finite_card_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_int(A) ) ) ) ).

% card_insert_disjoint'
tff(fact_5393_card__insert__disjoint_H,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( aa_nat_nat(minus_minus_nat(finite_card_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_o(A) ) ) ) ).

% card_insert_disjoint'
tff(fact_5394_card__insert__disjoint_H,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_nat_nat(minus_minus_nat(finite410649719033368117t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A))),aa_nat_nat(suc,zero_zero_nat)) = finite410649719033368117t_unit(A) ) ) ) ).

% card_insert_disjoint'
tff(fact_5395_card__insert__disjoint_H,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( aa_nat_nat(minus_minus_nat(aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A))),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(finite_card_nat,A) ) ) ) ).

% card_insert_disjoint'
tff(fact_5396_card__insert__disjoint_H,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( aa_nat_nat(minus_minus_nat(finite4902975817058060853nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A))),aa_nat_nat(suc,zero_zero_nat)) = finite4902975817058060853nteger(A) ) ) ) ).

% card_insert_disjoint'
tff(fact_5397_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_5398_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_5399_add_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),zero_zero_rat) = A2 ).

% add.right_neutral
tff(fact_5400_add_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(plus_plus_nat(A2),zero_zero_nat) = A2 ).

% add.right_neutral
tff(fact_5401_add_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),zero_zero_int) = A2 ).

% add.right_neutral
tff(fact_5402_add_Oright__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),zero_z3403309356797280102nteger) = A2 ).

% add.right_neutral
tff(fact_5403_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_5404_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(plus_plus_int(A2),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_5405_double__zero__sym,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
tff(fact_5406_add__cancel__left__left,axiom,
    ! [B: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2) = A2 )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_5407_add__cancel__left__left,axiom,
    ! [B: nat,A2: nat] :
      ( ( aa_nat_nat(plus_plus_nat(B),A2) = A2 )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_5408_add__cancel__left__left,axiom,
    ! [B: int,A2: int] :
      ( ( aa_int_int(plus_plus_int(B),A2) = A2 )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_5409_add__cancel__left__left,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2) = A2 )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
tff(fact_5410_add__cancel__left__right,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = A2 )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_5411_add__cancel__left__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_nat_nat(plus_plus_nat(A2),B) = A2 )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_5412_add__cancel__left__right,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A2),B) = A2 )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_5413_add__cancel__left__right,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = A2 )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
tff(fact_5414_add__cancel__right__left,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_5415_add__cancel__right__left,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 = aa_nat_nat(plus_plus_nat(B),A2) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_5416_add__cancel__right__left,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(B),A2) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_5417_add__cancel__right__left,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
tff(fact_5418_add__cancel__right__right,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_5419_add__cancel__right__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 = aa_nat_nat(plus_plus_nat(A2),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_5420_add__cancel__right__right,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = aa_int_int(plus_plus_int(A2),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_5421_add__cancel__right__right,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
tff(fact_5422_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_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_5423_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(plus_plus_nat(X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_5424_add__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A2) = A2 ).

% add_0
tff(fact_5425_add__0,axiom,
    ! [A2: nat] : aa_nat_nat(plus_plus_nat(zero_zero_nat),A2) = A2 ).

% add_0
tff(fact_5426_add__0,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(zero_zero_int),A2) = A2 ).

% add_0
tff(fact_5427_add__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A2) = A2 ).

% add_0
tff(fact_5428_mult__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = B ) ) ) ).

% mult_cancel_right
tff(fact_5429_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B ) ) ) ).

% mult_cancel_right
tff(fact_5430_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B ) ) ) ).

% mult_cancel_right
tff(fact_5431_mult__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B ) ) ) ).

% mult_cancel_right
tff(fact_5432_mult__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = B ) ) ) ).

% mult_cancel_left
tff(fact_5433_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B ) ) ) ).

% mult_cancel_left
tff(fact_5434_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B ) ) ) ).

% mult_cancel_left
tff(fact_5435_mult__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B ) ) ) ).

% mult_cancel_left
tff(fact_5436_mult__eq__0__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) = zero_z3403309356797280102nteger )
    <=> ( ( A2 = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_5437_mult__eq__0__iff,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_5438_mult__eq__0__iff,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_5439_mult__eq__0__iff,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_5440_mult__zero__right,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_5441_mult__zero__right,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_5442_mult__zero__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_5443_mult__zero__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_5444_mult__zero__left,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A2) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_5445_mult__zero__left,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A2) = zero_zero_rat ).

% mult_zero_left
tff(fact_5446_mult__zero__left,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mult_zero_left
tff(fact_5447_mult__zero__left,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A2) = zero_zero_int ).

% mult_zero_left
tff(fact_5448_diff__self,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),A2) = zero_z3403309356797280102nteger ).

% diff_self
tff(fact_5449_diff__self,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),A2) = zero_zero_rat ).

% diff_self
tff(fact_5450_diff__self,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),A2) = zero_zero_int ).

% diff_self
tff(fact_5451_diff__0__right,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),zero_z3403309356797280102nteger) = A2 ).

% diff_0_right
tff(fact_5452_diff__0__right,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),zero_zero_rat) = A2 ).

% diff_0_right
tff(fact_5453_diff__0__right,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),zero_zero_int) = A2 ).

% diff_0_right
tff(fact_5454_zero__diff,axiom,
    ! [A2: nat] : aa_nat_nat(minus_minus_nat(zero_zero_nat),A2) = zero_zero_nat ).

% zero_diff
tff(fact_5455_diff__zero,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),zero_z3403309356797280102nteger) = A2 ).

% diff_zero
tff(fact_5456_diff__zero,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),zero_zero_rat) = A2 ).

% diff_zero
tff(fact_5457_diff__zero,axiom,
    ! [A2: nat] : aa_nat_nat(minus_minus_nat(A2),zero_zero_nat) = A2 ).

% diff_zero
tff(fact_5458_diff__zero,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),zero_zero_int) = A2 ).

% diff_zero
tff(fact_5459_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),A2) = zero_z3403309356797280102nteger ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_5460_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),A2) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_5461_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] : aa_nat_nat(minus_minus_nat(A2),A2) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_5462_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),A2) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_5463_mult__cancel2,axiom,
    ! [M4: nat,K3: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),K3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3) )
    <=> ( ( M4 = N2 )
        | ( K3 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_5464_mult__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2) )
    <=> ( ( M4 = N2 )
        | ( K3 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_5465_mult__0__right,axiom,
    ! [M4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_5466_mult__is__0,axiom,
    ! [M4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) = zero_zero_nat )
    <=> ( ( M4 = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_5467_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_5468_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_5469_dbl__dec__simps_I3_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,one_one_Code_integer) = one_one_Code_integer ).

% dbl_dec_simps(3)
tff(fact_5470_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_5471_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: 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,A2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_5472_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_5473_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),A2)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_5474_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_5475_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_5476_le__add__same__cancel2,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel2
tff(fact_5477_le__add__same__cancel2,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_5478_le__add__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(plus_plus_nat(B),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_5479_le__add__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(plus_plus_int(B),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_5480_le__add__same__cancel1,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel1
tff(fact_5481_le__add__same__cancel1,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_5482_le__add__same__cancel1,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(plus_plus_nat(A2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_5483_le__add__same__cancel1,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(plus_plus_int(A2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_5484_add__le__same__cancel2,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel2
tff(fact_5485_add__le__same__cancel2,axiom,
    ! [A2: 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,A2),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_5486_add__le__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_5487_add__le__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_5488_add__le__same__cancel1,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel1
tff(fact_5489_add__le__same__cancel1,axiom,
    ! [B: rat,A2: 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),A2)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_5490_add__le__same__cancel1,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(B),A2)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_5491_add__le__same__cancel1,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(B),A2)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_5492_diff__ge__0__iff__ge,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_5493_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_5494_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_5495_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_5496_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_5497_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_5498_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_5499_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_5500_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: 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,A2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_5501_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_5502_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),A2)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_5503_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_5504_less__add__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(plus_plus_nat(B),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_5505_less__add__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(B),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_5506_less__add__same__cancel2,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel2
tff(fact_5507_less__add__same__cancel2,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_5508_less__add__same__cancel1,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(plus_plus_nat(A2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_5509_less__add__same__cancel1,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(A2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_5510_less__add__same__cancel1,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel1
tff(fact_5511_less__add__same__cancel1,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_5512_add__less__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(A2),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_5513_add__less__same__cancel2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_5514_add__less__same__cancel2,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel2
tff(fact_5515_add__less__same__cancel2,axiom,
    ! [A2: 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,A2),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_5516_add__less__same__cancel1,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(plus_plus_nat(B),A2)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_5517_add__less__same__cancel1,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(B),A2)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_5518_add__less__same__cancel1,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A2)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel1
tff(fact_5519_add__less__same__cancel1,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A2)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_5520_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_5521_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_5522_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_5523_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_5524_diff__gt__0__iff__gt,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_5525_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_5526_diff__add__zero,axiom,
    ! [A2: nat,B: nat] : aa_nat_nat(minus_minus_nat(A2),aa_nat_nat(plus_plus_nat(A2),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_5527_mult__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_5528_mult__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_5529_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_5530_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_5531_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_5532_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_5533_mult__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_5534_mult__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_5535_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = C )
    <=> ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_5536_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_5537_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_5538_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_5539_image__add__0,axiom,
    ! [S: set_rat] : image_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),S) = S ).

% image_add_0
tff(fact_5540_image__add__0,axiom,
    ! [S: set_nat] : aa_set_nat_set_nat(image_nat_nat2(plus_plus_nat(zero_zero_nat)),S) = S ).

% image_add_0
tff(fact_5541_image__add__0,axiom,
    ! [S: set_int] : image_int_int2(plus_plus_int(zero_zero_int),S) = S ).

% image_add_0
tff(fact_5542_image__add__0,axiom,
    ! [S: set_Code_integer] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger)),S) = S ).

% image_add_0
tff(fact_5543_diff__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_5544_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_5545_diff__numeral__special_I9_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_5546_max__0__1_I2_J,axiom,
    ord_max_Code_natural(one_one_Code_natural,zero_z2226904508553997617atural) = one_one_Code_natural ).

% max_0_1(2)
tff(fact_5547_max__0__1_I2_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,one_one_Code_integer),zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_5548_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_5549_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_5550_max__0__1_I1_J,axiom,
    ord_max_Code_natural(zero_z2226904508553997617atural,one_one_Code_natural) = one_one_Code_natural ).

% max_0_1(1)
tff(fact_5551_max__0__1_I1_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(ord_max_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_5552_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_5553_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_5554_card_Oempty,axiom,
    finite410649719033368117t_unit(bot_bo3957492148770167129t_unit) = zero_zero_nat ).

% card.empty
tff(fact_5555_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_5556_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_5557_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_5558_card_Oempty,axiom,
    finite5795495896937551127_nat_o(bot_bo8632255151191677115_nat_o) = zero_zero_nat ).

% card.empty
tff(fact_5559_one__eq__mult__iff,axiom,
    ! [M4: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) )
    <=> ( ( M4 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_5560_mult__eq__1__iff,axiom,
    ! [M4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M4 = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_5561_nat__mult__less__cancel__disj,axiom,
    ! [K3: nat,M4: 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,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_5562_nat__0__less__mult__iff,axiom,
    ! [M4: 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,M4),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M4)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% nat_0_less_mult_iff
tff(fact_5563_mult__less__cancel2,axiom,
    ! [M4: nat,K3: 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,M4),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% mult_less_cancel2
tff(fact_5564_card__0__eq,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( finite4902975817058060853nteger(A) = zero_zero_nat )
      <=> ( A = bot_bo3990330152332043303nteger ) ) ) ).

% card_0_eq
tff(fact_5565_card__0__eq,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( finite410649719033368117t_unit(A) = zero_zero_nat )
      <=> ( A = bot_bo3957492148770167129t_unit ) ) ) ).

% card_0_eq
tff(fact_5566_card__0__eq,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( finite_card_o(A) = zero_zero_nat )
      <=> ( A = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_5567_card__0__eq,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( aa_set_nat_nat(finite_card_nat,A) = zero_zero_nat )
      <=> ( A = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_5568_card__0__eq,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( finite_card_int(A) = zero_zero_nat )
      <=> ( A = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_5569_card__0__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A)
     => ( ( finite5795495896937551127_nat_o(A) = zero_zero_nat )
      <=> ( A = bot_bo8632255151191677115_nat_o ) ) ) ).

% card_0_eq
tff(fact_5570_one__le__mult__iff,axiom,
    ! [M4: 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,M4),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M4)
        & 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_5571_mult__le__cancel2,axiom,
    ! [M4: nat,K3: 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,M4),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% mult_le_cancel2
tff(fact_5572_nat__mult__le__cancel__disj,axiom,
    ! [K3: nat,M4: 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,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_5573_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_5574_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_5575_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_5576_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
    <=> ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
tff(fact_5577_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_5578_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_5579_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_5580_gr__implies__not__zero,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2)
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_5581_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_5582_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_5583_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] : aa_nat_nat(plus_plus_nat(zero_zero_nat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_5584_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(zero_zero_int),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_5585_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_5586_add_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),zero_zero_rat) = A2 ).

% add.comm_neutral
tff(fact_5587_add_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(plus_plus_nat(A2),zero_zero_nat) = A2 ).

% add.comm_neutral
tff(fact_5588_add_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),zero_zero_int) = A2 ).

% add.comm_neutral
tff(fact_5589_add_Ocomm__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),zero_z3403309356797280102nteger) = A2 ).

% add.comm_neutral
tff(fact_5590_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A2) = A2 ).

% add.group_left_neutral
tff(fact_5591_add_Ogroup__left__neutral,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(zero_zero_int),A2) = A2 ).

% add.group_left_neutral
tff(fact_5592_add_Ogroup__left__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A2) = A2 ).

% add.group_left_neutral
tff(fact_5593_mult__right__cancel,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
      <=> ( A2 = B ) ) ) ).

% mult_right_cancel
tff(fact_5594_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A2 = B ) ) ) ).

% mult_right_cancel
tff(fact_5595_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A2 = B ) ) ) ).

% mult_right_cancel
tff(fact_5596_mult__right__cancel,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A2 = B ) ) ) ).

% mult_right_cancel
tff(fact_5597_mult__left__cancel,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
      <=> ( A2 = B ) ) ) ).

% mult_left_cancel
tff(fact_5598_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A2 = B ) ) ) ).

% mult_left_cancel
tff(fact_5599_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A2 = B ) ) ) ).

% mult_left_cancel
tff(fact_5600_mult__left__cancel,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A2 = B ) ) ) ).

% mult_left_cancel
tff(fact_5601_no__zero__divisors,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_5602_no__zero__divisors,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_5603_no__zero__divisors,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_5604_no__zero__divisors,axiom,
    ! [A2: int,B: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_5605_divisors__zero,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) = zero_z3403309356797280102nteger )
     => ( ( A2 = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_5606_divisors__zero,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_5607_divisors__zero,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_5608_divisors__zero,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_5609_mult__not__zero,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) != zero_z3403309356797280102nteger )
     => ( ( A2 != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_5610_mult__not__zero,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_5611_mult__not__zero,axiom,
    ! [A2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_5612_mult__not__zero,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_5613_zero__neq__one,axiom,
    zero_z2226904508553997617atural != one_one_Code_natural ).

% zero_neq_one
tff(fact_5614_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_5615_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_5616_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_5617_eq__iff__diff__eq__0,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) = zero_z3403309356797280102nteger ) ) ).

% eq_iff_diff_eq_0
tff(fact_5618_eq__iff__diff__eq__0,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 = B )
    <=> ( aa_rat_rat(minus_minus_rat(A2),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_5619_eq__iff__diff__eq__0,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = B )
    <=> ( aa_int_int(minus_minus_int(A2),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_5620_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_5621_nat__mult__eq__cancel__disj,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2) )
    <=> ( ( K3 = zero_zero_nat )
        | ( M4 = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_5622_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A2 = B )
          & ( C != D2 ) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),C)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5623_add__scale__eq__noteq,axiom,
    ! [R2: rat,A2: rat,B: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A2 = B )
          & ( C != D2 ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5624_add__scale__eq__noteq,axiom,
    ! [R2: nat,A2: nat,B: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A2 = B )
          & ( C != D2 ) )
       => ( aa_nat_nat(plus_plus_nat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(plus_plus_nat(B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5625_add__scale__eq__noteq,axiom,
    ! [R2: int,A2: int,B: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A2 = B )
          & ( C != D2 ) )
       => ( aa_int_int(plus_plus_int(A2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(plus_plus_int(B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5626_Sup__nat__def,axiom,
    ! [X5: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X5) = $ite(X5 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X5)) ).

% Sup_nat_def
tff(fact_5627_add__decreasing,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),B) ) ) ).

% add_decreasing
tff(fact_5628_add__decreasing,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B) ) ) ).

% add_decreasing
tff(fact_5629_add__decreasing,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),C)),B) ) ) ).

% add_decreasing
tff(fact_5630_add__decreasing,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C)),B) ) ) ).

% add_decreasing
tff(fact_5631_add__increasing,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)) ) ) ).

% add_increasing
tff(fact_5632_add__increasing,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) ) ) ).

% add_increasing
tff(fact_5633_add__increasing,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(plus_plus_nat(A2),C)) ) ) ).

% add_increasing
tff(fact_5634_add__increasing,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(plus_plus_int(A2),C)) ) ) ).

% add_increasing
tff(fact_5635_add__decreasing2,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)),B) ) ) ).

% add_decreasing2
tff(fact_5636_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B) ) ) ).

% add_decreasing2
tff(fact_5637_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(plus_plus_nat(A2),C)),B) ) ) ).

% add_decreasing2
tff(fact_5638_add__decreasing2,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C)),B) ) ) ).

% add_decreasing2
tff(fact_5639_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)) ) ) ).

% add_increasing2
tff(fact_5640_add__increasing2,axiom,
    ! [C: rat,B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) ) ) ).

% add_increasing2
tff(fact_5641_add__increasing2,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(plus_plus_nat(A2),C)) ) ) ).

% add_increasing2
tff(fact_5642_add__increasing2,axiom,
    ! [C: int,B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(plus_plus_int(A2),C)) ) ) ).

% add_increasing2
tff(fact_5643_add__nonneg__nonneg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_5644_add__nonneg__nonneg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( 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,A2),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_5645_add__nonneg__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( 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(plus_plus_nat(A2),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_5646_add__nonneg__nonneg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( 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(plus_plus_int(A2),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_5647_add__nonpos__nonpos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_nonpos
tff(fact_5648_add__nonpos__nonpos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_5649_add__nonpos__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),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(plus_plus_nat(A2),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_5650_add__nonpos__nonpos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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(plus_plus_int(A2),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_5651_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_5652_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_5653_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(plus_plus_nat(X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_5654_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(plus_plus_int(X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_5655_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_5656_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_5657_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(plus_plus_nat(X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_5658_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(plus_plus_int(X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_5659_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5660_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5661_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5662_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_5663_zero__le__mult__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_le_mult_iff
tff(fact_5664_zero__le__mult__iff,axiom,
    ! [A2: 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,A2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & 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,A2),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_5665_zero__le__mult__iff,axiom,
    ! [A2: 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,A2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & 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,A2),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_5666_mult__nonneg__nonpos2,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( 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),A2)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5667_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( 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),A2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5668_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( 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),A2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5669_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( 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),A2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_5670_mult__nonpos__nonneg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonpos_nonneg
tff(fact_5671_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_5672_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),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,A2),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_5673_mult__nonpos__nonneg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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,A2),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_5674_mult__nonneg__nonpos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos
tff(fact_5675_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( 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,A2),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_5676_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( 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,A2),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_5677_mult__nonneg__nonpos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( 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,A2),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_5678_mult__nonneg__nonneg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5679_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( 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,A2),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5680_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( 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,A2),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5681_mult__nonneg__nonneg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( 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,A2),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_5682_split__mult__neg__le,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ).

% split_mult_neg_le
tff(fact_5683_split__mult__neg__le,axiom,
    ! [A2: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & 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,A2),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,A2),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_5684_split__mult__neg__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
          & 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,A2),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,A2),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_5685_split__mult__neg__le,axiom,
    ! [A2: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & 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,A2),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,A2),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_5686_mult__le__0__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_le_0_iff
tff(fact_5687_mult__le__0__iff,axiom,
    ! [A2: 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,A2),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & 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,A2),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_5688_mult__le__0__iff,axiom,
    ! [A2: 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,A2),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & 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,A2),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_5689_mult__right__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono
tff(fact_5690_mult__right__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_5691_mult__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_5692_mult__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_5693_mult__right__mono__neg,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_5694_mult__right__mono__neg,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_5695_mult__right__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_5696_mult__left__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono
tff(fact_5697_mult__left__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_5698_mult__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_5699_mult__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_5700_mult__nonpos__nonpos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_5701_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_5702_mult__nonpos__nonpos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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,A2),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_5703_mult__left__mono__neg,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_5704_mult__left__mono__neg,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_5705_mult__left__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_5706_split__mult__pos__le,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)) ) ).

% split_mult_pos_le
tff(fact_5707_split__mult__pos__le,axiom,
    ! [A2: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & 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,A2),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,A2),B)) ) ).

% split_mult_pos_le
tff(fact_5708_split__mult__pos__le,axiom,
    ! [A2: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & 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,A2),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,A2),B)) ) ).

% split_mult_pos_le
tff(fact_5709_zero__le__square,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2)) ).

% zero_le_square
tff(fact_5710_zero__le__square,axiom,
    ! [A2: 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,A2),A2)) ).

% zero_le_square
tff(fact_5711_zero__le__square,axiom,
    ! [A2: 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,A2),A2)) ).

% zero_le_square
tff(fact_5712_mult__mono_H,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_5713_mult__mono_H,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_5714_mult__mono_H,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_5715_mult__mono_H,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_5716_mult__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_5717_mult__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),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),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_5718_mult__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),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),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_5719_mult__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),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),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_5720_not__one__le__zero,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),zero_z2226904508553997617atural) ).

% not_one_le_zero
tff(fact_5721_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_5722_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_5723_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_5724_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_5725_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_5726_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_5727_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_5728_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_5729_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_5730_zero__less__one__class_Ozero__le__one,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% zero_less_one_class.zero_le_one
tff(fact_5731_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_5732_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_5733_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_5734_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_5735_le__iff__diff__le__0,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),zero_z3403309356797280102nteger) ) ).

% le_iff_diff_le_0
tff(fact_5736_le__iff__diff__le__0,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_5737_le__iff__diff__le__0,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_5738_pos__add__strict,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(plus_plus_nat(A2),C)) ) ) ).

% pos_add_strict
tff(fact_5739_pos__add__strict,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A2),C)) ) ) ).

% pos_add_strict
tff(fact_5740_pos__add__strict,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)) ) ) ).

% pos_add_strict
tff(fact_5741_pos__add__strict,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) ) ) ).

% pos_add_strict
tff(fact_5742_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(plus_plus_nat(A2),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_5743_add__pos__pos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( 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(plus_plus_nat(A2),B)) ) ) ).

% add_pos_pos
tff(fact_5744_add__pos__pos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),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_int,zero_zero_int),aa_int_int(plus_plus_int(A2),B)) ) ) ).

% add_pos_pos
tff(fact_5745_add__pos__pos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)) ) ) ).

% add_pos_pos
tff(fact_5746_add__pos__pos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( 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,A2),B)) ) ) ).

% add_pos_pos
tff(fact_5747_add__neg__neg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),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(plus_plus_nat(A2),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_5748_add__neg__neg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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(plus_plus_int(A2),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_5749_add__neg__neg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_neg
tff(fact_5750_add__neg__neg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),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,A2),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_5751_mult__neg__neg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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,A2),B)) ) ) ).

% mult_neg_neg
tff(fact_5752_mult__neg__neg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),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,A2),B)) ) ) ).

% mult_neg_neg
tff(fact_5753_mult__neg__neg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),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,A2),B)) ) ) ).

% mult_neg_neg
tff(fact_5754_not__square__less__zero,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)),zero_zero_int) ).

% not_square_less_zero
tff(fact_5755_not__square__less__zero,axiom,
    ! [A2: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2)),zero_z3403309356797280102nteger) ).

% not_square_less_zero
tff(fact_5756_not__square__less__zero,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),A2)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_5757_mult__less__0__iff,axiom,
    ! [A2: 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,A2),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),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_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_5758_mult__less__0__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_less_0_iff
tff(fact_5759_mult__less__0__iff,axiom,
    ! [A2: 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,A2),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & 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,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_5760_mult__neg__pos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),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,A2),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_5761_mult__neg__pos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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,A2),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_5762_mult__neg__pos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_neg_pos
tff(fact_5763_mult__neg__pos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),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,A2),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_5764_mult__pos__neg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( 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,A2),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_5765_mult__pos__neg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),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_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_5766_mult__pos__neg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg
tff(fact_5767_mult__pos__neg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( 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,A2),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_5768_mult__pos__pos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( 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,A2),B)) ) ) ).

% mult_pos_pos
tff(fact_5769_mult__pos__pos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),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_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)) ) ) ).

% mult_pos_pos
tff(fact_5770_mult__pos__pos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)) ) ) ).

% mult_pos_pos
tff(fact_5771_mult__pos__pos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( 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,A2),B)) ) ) ).

% mult_pos_pos
tff(fact_5772_mult__pos__neg2,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( 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),A2)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_5773_mult__pos__neg2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),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_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A2)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_5774_mult__pos__neg2,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( 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),A2)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg2
tff(fact_5775_mult__pos__neg2,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( 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),A2)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_5776_zero__less__mult__iff,axiom,
    ! [A2: 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,A2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),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_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_5777_zero__less__mult__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_less_mult_iff
tff(fact_5778_zero__less__mult__iff,axiom,
    ! [A2: 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,A2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & 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,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_5779_zero__less__mult__pos,axiom,
    ! [A2: 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,A2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_5780_zero__less__mult__pos,axiom,
    ! [A2: 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,A2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_5781_zero__less__mult__pos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos
tff(fact_5782_zero__less__mult__pos,axiom,
    ! [A2: 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,A2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_5783_zero__less__mult__pos2,axiom,
    ! [B: nat,A2: 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),A2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_5784_zero__less__mult__pos2,axiom,
    ! [B: int,A2: 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),A2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_5785_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos2
tff(fact_5786_zero__less__mult__pos2,axiom,
    ! [B: rat,A2: 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),A2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_5787_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_5788_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_5789_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_5790_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_5791_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_5792_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_5793_mult__strict__left__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_5794_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_5795_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_5796_mult__strict__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_5797_mult__strict__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_5798_mult__strict__left__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_5799_mult__strict__left__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_5800_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_5801_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_5802_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_5803_mult__strict__right__mono__neg,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_5804_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_5805_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_5806_mult__strict__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_5807_mult__strict__right__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_5808_mult__strict__right__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_5809_mult__strict__right__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_5810_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_5811_mult__less__cancel__right__disj,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) )
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_5812_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_5813_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5814_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5815_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5816_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_5817_less__numeral__extra_I1_J,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% less_numeral_extra(1)
tff(fact_5818_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_5819_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_5820_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_5821_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_5822_zero__less__one,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% zero_less_one
tff(fact_5823_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_5824_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_5825_zero__less__one,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_5826_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_5827_not__one__less__zero,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),zero_z2226904508553997617atural) ).

% not_one_less_zero
tff(fact_5828_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_5829_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_5830_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_5831_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_5832_less__iff__diff__less__0,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_5833_less__iff__diff__less__0,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)),zero_z3403309356797280102nteger) ) ).

% less_iff_diff_less_0
tff(fact_5834_less__iff__diff__less__0,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_5835_inj__on__mult,axiom,
    ! [A2: code_integer,A: set_Code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2)),A) ) ).

% inj_on_mult
tff(fact_5836_inj__on__mult,axiom,
    ! [A2: rat,A: set_rat] :
      ( ( A2 != zero_zero_rat )
     => aa_set_rat_o(inj_on_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2)),A) ) ).

% inj_on_mult
tff(fact_5837_inj__on__mult,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( A2 != zero_zero_nat )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2)),A) ) ).

% inj_on_mult
tff(fact_5838_inj__on__mult,axiom,
    ! [A2: int,A: set_int] :
      ( ( A2 != zero_zero_int )
     => aa_set_int_o(inj_on_int_int(aa_int_fun_int_int(times_times_int,A2)),A) ) ).

% inj_on_mult
tff(fact_5839_nat__compl__induct,axiom,
    ! [P: fun_nat_o,N2: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N3)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N3)) )
       => aa_nat_o(P,N2) ) ) ).

% nat_compl_induct
tff(fact_5840_nat__compl__induct_H,axiom,
    ! [P: fun_nat_o,N2: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N3)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N3)) )
       => aa_nat_o(P,N2) ) ) ).

% nat_compl_induct'
tff(fact_5841_Suc__to__right,axiom,
    ! [N2: nat,M4: nat] :
      ( ( aa_nat_nat(suc,N2) = M4 )
     => ( N2 = aa_nat_nat(minus_minus_nat(M4),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_5842_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K3)) ) ) ).

% mult_less_mono1
tff(fact_5843_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),J)) ) ) ).

% mult_less_mono2
tff(fact_5844_nat__mult__eq__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2) )
      <=> ( M4 = N2 ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_5845_nat__mult__less__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% nat_mult_less_cancel1
tff(fact_5846_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_5847_mult__eq__self__implies__10,axiom,
    ! [M4: nat,N2: nat] :
      ( ( M4 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) )
     => ( ( N2 = one_one_nat )
        | ( M4 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_5848_add__strict__increasing2,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_5849_add__strict__increasing2,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_5850_add__strict__increasing2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(plus_plus_nat(A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_5851_add__strict__increasing2,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_5852_add__strict__increasing,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),C)) ) ) ).

% add_strict_increasing
tff(fact_5853_add__strict__increasing,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)) ) ) ).

% add_strict_increasing
tff(fact_5854_add__strict__increasing,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(plus_plus_nat(A2),C)) ) ) ).

% add_strict_increasing
tff(fact_5855_add__strict__increasing,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(plus_plus_int(A2),C)) ) ) ).

% add_strict_increasing
tff(fact_5856_add__pos__nonneg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)) ) ) ).

% add_pos_nonneg
tff(fact_5857_add__pos__nonneg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( 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,A2),B)) ) ) ).

% add_pos_nonneg
tff(fact_5858_add__pos__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( 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(plus_plus_nat(A2),B)) ) ) ).

% add_pos_nonneg
tff(fact_5859_add__pos__nonneg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( 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(plus_plus_int(A2),B)) ) ) ).

% add_pos_nonneg
tff(fact_5860_add__nonpos__neg,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_neg
tff(fact_5861_add__nonpos__neg,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_5862_add__nonpos__neg,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),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(plus_plus_nat(A2),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_5863_add__nonpos__neg,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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(plus_plus_int(A2),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_5864_add__nonneg__pos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( 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,A2),B)) ) ) ).

% add_nonneg_pos
tff(fact_5865_add__nonneg__pos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( 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,A2),B)) ) ) ).

% add_nonneg_pos
tff(fact_5866_add__nonneg__pos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( 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(plus_plus_nat(A2),B)) ) ) ).

% add_nonneg_pos
tff(fact_5867_add__nonneg__pos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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_int,zero_zero_int),aa_int_int(plus_plus_int(A2),B)) ) ) ).

% add_nonneg_pos
tff(fact_5868_add__neg__nonpos,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),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,A2),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_nonpos
tff(fact_5869_add__neg__nonpos,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),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,A2),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_5870_add__neg__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),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(plus_plus_nat(A2),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_5871_add__neg__nonpos,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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(plus_plus_int(A2),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_5872_mult__less__le__imp__less,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5873_mult__less__le__imp__less,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5874_mult__less__le__imp__less,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5875_mult__less__le__imp__less,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_5876_mult__le__less__imp__less,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5877_mult__le__less__imp__less,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5878_mult__le__less__imp__less,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5879_mult__le__less__imp__less,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_5880_mult__right__le__imp__le,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ) ).

% mult_right_le_imp_le
tff(fact_5881_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ) ).

% mult_right_le_imp_le
tff(fact_5882_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ) ).

% mult_right_le_imp_le
tff(fact_5883_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ) ).

% mult_right_le_imp_le
tff(fact_5884_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ) ).

% mult_left_le_imp_le
tff(fact_5885_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ) ).

% mult_left_le_imp_le
tff(fact_5886_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B) ) ) ).

% mult_left_le_imp_le
tff(fact_5887_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ) ).

% mult_left_le_imp_le
tff(fact_5888_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_5889_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_5890_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_5891_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_5892_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_5893_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_5894_mult__less__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_5895_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_5896_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_5897_mult__strict__mono_H,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5898_mult__strict__mono_H,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5899_mult__strict__mono_H,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5900_mult__strict__mono_H,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_5901_mult__right__less__imp__less,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ) ).

% mult_right_less_imp_less
tff(fact_5902_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ) ).

% mult_right_less_imp_less
tff(fact_5903_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ) ).

% mult_right_less_imp_less
tff(fact_5904_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ) ).

% mult_right_less_imp_less
tff(fact_5905_mult__less__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_5906_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_5907_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_5908_mult__strict__mono,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),D2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5909_mult__strict__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),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),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5910_mult__strict__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),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),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5911_mult__strict__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),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),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_5912_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ) ).

% mult_left_less_imp_less
tff(fact_5913_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ) ).

% mult_left_less_imp_less
tff(fact_5914_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ) ).

% mult_left_less_imp_less
tff(fact_5915_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ) ).

% mult_left_less_imp_less
tff(fact_5916_mult__le__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_5917_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_5918_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_5919_mult__le__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_5920_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_5921_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_5922_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_5923_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_5924_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(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_5925_mult__left__le,axiom,
    ! [C: code_natural,A2: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,C),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A2)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_5926_mult__left__le,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_5927_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_5928_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_5929_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_5930_mult__le__one,axiom,
    ! [A2: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A2),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),B)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,B),one_one_Code_natural)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B)),one_one_Code_natural) ) ) ) ).

% mult_le_one
tff(fact_5931_mult__le__one,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),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,A2),B)),one_one_Code_integer) ) ) ) ).

% mult_le_one
tff(fact_5932_mult__le__one,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_5933_mult__le__one,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),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,A2),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_5934_mult__le__one,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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,A2),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_5935_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_5936_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_5937_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_5938_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_5939_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_5940_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_5941_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_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_5942_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_5943_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_5944_zero__less__two,axiom,
    aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),plus_p4538020629002901425atural(one_one_Code_natural,one_one_Code_natural)) ).

% zero_less_two
tff(fact_5945_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(plus_plus_nat(one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_5946_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_5947_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_5948_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_5949_card__eq__0__iff,axiom,
    ! [A: set_Code_integer] :
      ( ( finite4902975817058060853nteger(A) = zero_zero_nat )
    <=> ( ( A = bot_bo3990330152332043303nteger )
        | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% card_eq_0_iff
tff(fact_5950_card__eq__0__iff,axiom,
    ! [A: set_Product_unit] :
      ( ( finite410649719033368117t_unit(A) = zero_zero_nat )
    <=> ( ( A = bot_bo3957492148770167129t_unit )
        | ~ aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A) ) ) ).

% card_eq_0_iff
tff(fact_5951_card__eq__0__iff,axiom,
    ! [A: set_o] :
      ( ( finite_card_o(A) = zero_zero_nat )
    <=> ( ( A = bot_bot_set_o )
        | ~ aa_set_o_o(finite_finite_o,A) ) ) ).

% card_eq_0_iff
tff(fact_5952_card__eq__0__iff,axiom,
    ! [A: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A) = zero_zero_nat )
    <=> ( ( A = bot_bot_set_nat )
        | ~ aa_set_nat_o(finite_finite_nat,A) ) ) ).

% card_eq_0_iff
tff(fact_5953_card__eq__0__iff,axiom,
    ! [A: set_int] :
      ( ( finite_card_int(A) = zero_zero_nat )
    <=> ( ( A = bot_bot_set_int )
        | ~ aa_set_int_o(finite_finite_int,A) ) ) ).

% card_eq_0_iff
tff(fact_5954_card__eq__0__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ( finite5795495896937551127_nat_o(A) = zero_zero_nat )
    <=> ( ( A = bot_bo8632255151191677115_nat_o )
        | ~ aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A) ) ) ).

% card_eq_0_iff
tff(fact_5955_one__less__mult,axiom,
    ! [N2: nat,M4: 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)),M4)
       => 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,M4),N2)) ) ) ).

% one_less_mult
tff(fact_5956_n__less__m__mult__n,axiom,
    ! [N2: nat,M4: 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)),M4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) ) ) ).

% n_less_m_mult_n
tff(fact_5957_n__less__n__mult__m,axiom,
    ! [N2: nat,M4: 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)),M4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M4)) ) ) ).

% n_less_n_mult_m
tff(fact_5958_nat__mult__le__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% nat_mult_le_cancel1
tff(fact_5959_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_5960_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_5961_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_5962_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_5963_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_5964_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_5965_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_5966_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_5967_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_5968_mult__le__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_5969_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_5970_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_5971_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_5972_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_5973_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_5974_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_5975_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_5976_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_5977_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_5978_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_5979_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_5980_mult__less__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),C)
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_5981_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_5982_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_5983_convex__bound__le,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),A2)
       => ( 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))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_5984_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A2)
       => ( 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))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_5985_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A2)
       => ( 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(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_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))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_5986_card__gt__0__iff,axiom,
    ! [A: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite4902975817058060853nteger(A))
    <=> ( ( A != bot_bo3990330152332043303nteger )
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% card_gt_0_iff
tff(fact_5987_card__gt__0__iff,axiom,
    ! [A: set_Product_unit] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A))
    <=> ( ( A != bot_bo3957492148770167129t_unit )
        & aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A) ) ) ).

% card_gt_0_iff
tff(fact_5988_card__gt__0__iff,axiom,
    ! [A: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A))
    <=> ( ( A != bot_bot_set_o )
        & aa_set_o_o(finite_finite_o,A) ) ) ).

% card_gt_0_iff
tff(fact_5989_card__gt__0__iff,axiom,
    ! [A: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A))
    <=> ( ( A != bot_bot_set_nat )
        & aa_set_nat_o(finite_finite_nat,A) ) ) ).

% card_gt_0_iff
tff(fact_5990_card__gt__0__iff,axiom,
    ! [A: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A))
    <=> ( ( A != bot_bot_set_int )
        & aa_set_int_o(finite_finite_int,A) ) ) ).

% card_gt_0_iff
tff(fact_5991_card__gt__0__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite5795495896937551127_nat_o(A))
    <=> ( ( A != bot_bo8632255151191677115_nat_o )
        & aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,A) ) ) ).

% card_gt_0_iff
tff(fact_5992_card__1__singleton__iff,axiom,
    ! [A: set_Product_unit] :
      ( ( finite410649719033368117t_unit(A) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X2: product_unit] : A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X2),bot_bo3957492148770167129t_unit) ) ).

% card_1_singleton_iff
tff(fact_5993_card__1__singleton__iff,axiom,
    ! [A: set_o] :
      ( ( finite_card_o(A) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X2: $o] : A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X2)),bot_bot_set_o) ) ).

% card_1_singleton_iff
tff(fact_5994_card__1__singleton__iff,axiom,
    ! [A: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X2: nat] : A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X2),bot_bot_set_nat) ) ).

% card_1_singleton_iff
tff(fact_5995_card__1__singleton__iff,axiom,
    ! [A: set_int] :
      ( ( finite_card_int(A) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X2: int] : A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X2),bot_bot_set_int) ) ).

% card_1_singleton_iff
tff(fact_5996_card__1__singleton__iff,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ( finite5795495896937551127_nat_o(A) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X2: fun_Pr6460944699146269935_nat_o] : A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X2),bot_bo8632255151191677115_nat_o) ) ).

% card_1_singleton_iff
tff(fact_5997_card__eq__SucD,axiom,
    ! [A: set_Product_unit,K3: nat] :
      ( ( finite410649719033368117t_unit(A) = aa_nat_nat(suc,K3) )
     => ? [B2: product_unit,B5: set_Product_unit] :
          ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B2),B5) )
          & ~ aa_set8780210009671629607unit_o(member_Product_unit(B2),B5)
          & ( finite410649719033368117t_unit(B5) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B5 = bot_bo3957492148770167129t_unit ) ) ) ) ).

% card_eq_SucD
tff(fact_5998_card__eq__SucD,axiom,
    ! [A: set_o,K3: nat] :
      ( ( finite_card_o(A) = aa_nat_nat(suc,K3) )
     => ? [B2: $o,B5: set_o] :
          ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B2)),B5) )
          & ~ aa_set_o_o(member_o((B2)),B5)
          & ( finite_card_o(B5) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B5 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
tff(fact_5999_card__eq__SucD,axiom,
    ! [A: set_nat,K3: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A) = aa_nat_nat(suc,K3) )
     => ? [B2: nat,B5: set_nat] :
          ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B2),B5) )
          & ~ aa_set_nat_o(member_nat(B2),B5)
          & ( aa_set_nat_nat(finite_card_nat,B5) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B5 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_6000_card__eq__SucD,axiom,
    ! [A: set_int,K3: nat] :
      ( ( finite_card_int(A) = aa_nat_nat(suc,K3) )
     => ? [B2: int,B5: set_int] :
          ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B2),B5) )
          & ~ aa_set_int_o(member_int(B2),B5)
          & ( finite_card_int(B5) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B5 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
tff(fact_6001_card__eq__SucD,axiom,
    ! [A: set_fu3555266933746557007_nat_o,K3: nat] :
      ( ( finite5795495896937551127_nat_o(A) = aa_nat_nat(suc,K3) )
     => ? [B2: fun_Pr6460944699146269935_nat_o,B5: set_fu3555266933746557007_nat_o] :
          ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B2),B5) )
          & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B2),B5)
          & ( finite5795495896937551127_nat_o(B5) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B5 = bot_bo8632255151191677115_nat_o ) ) ) ) ).

% card_eq_SucD
tff(fact_6002_card__Suc__eq,axiom,
    ! [A: set_Product_unit,K3: nat] :
      ( ( finite410649719033368117t_unit(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: product_unit,B9: set_Product_unit] :
          ( ( A = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B6),B9) )
          & ~ aa_set8780210009671629607unit_o(member_Product_unit(B6),B9)
          & ( finite410649719033368117t_unit(B9) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B9 = bot_bo3957492148770167129t_unit ) ) ) ) ).

% card_Suc_eq
tff(fact_6003_card__Suc__eq,axiom,
    ! [A: set_o,K3: nat] :
      ( ( finite_card_o(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: $o,B9: set_o] :
          ( ( A = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B6)),B9) )
          & ~ aa_set_o_o(member_o((B6)),B9)
          & ( finite_card_o(B9) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B9 = bot_bot_set_o ) ) ) ) ).

% card_Suc_eq
tff(fact_6004_card__Suc__eq,axiom,
    ! [A: set_nat,K3: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: nat,B9: set_nat] :
          ( ( A = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B6),B9) )
          & ~ aa_set_nat_o(member_nat(B6),B9)
          & ( aa_set_nat_nat(finite_card_nat,B9) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B9 = bot_bot_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_6005_card__Suc__eq,axiom,
    ! [A: set_int,K3: nat] :
      ( ( finite_card_int(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: int,B9: set_int] :
          ( ( A = aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B6),B9) )
          & ~ aa_set_int_o(member_int(B6),B9)
          & ( finite_card_int(B9) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B9 = bot_bot_set_int ) ) ) ) ).

% card_Suc_eq
tff(fact_6006_card__Suc__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o,K3: nat] :
      ( ( finite5795495896937551127_nat_o(A) = aa_nat_nat(suc,K3) )
    <=> ? [B6: fun_Pr6460944699146269935_nat_o,B9: set_fu3555266933746557007_nat_o] :
          ( ( A = aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B6),B9) )
          & ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B6),B9)
          & ( finite5795495896937551127_nat_o(B9) = K3 )
          & ( ( K3 = zero_zero_nat )
           => ( B9 = bot_bo8632255151191677115_nat_o ) ) ) ) ).

% card_Suc_eq
tff(fact_6007_nz__le__conv__less,axiom,
    ! [K3: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(K3),aa_nat_nat(suc,zero_zero_nat))),M4) ) ) ).

% nz_le_conv_less
tff(fact_6008_mult__eq__if,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) = $ite(M4 = zero_zero_nat,zero_zero_nat,aa_nat_nat(plus_plus_nat(N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(M4),one_one_nat)),N2))) ).

% mult_eq_if
tff(fact_6009_convex__bound__lt,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),A2)
       => ( 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))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_6010_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A2)
       => ( 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))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_6011_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A2)
       => ( 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(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_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))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_6012_inj__mult__left,axiom,
    ! [A2: code_integer] :
      ( aa_set5643387332196575111eger_o(inj_on1158064722259180589nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2)),top_to4645266643341252675nteger)
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% inj_mult_left
tff(fact_6013_inj__mult__left,axiom,
    ! [A2: rat] :
      ( aa_set_rat_o(inj_on_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2)),top_top_set_rat)
    <=> ( A2 != zero_zero_rat ) ) ).

% inj_mult_left
tff(fact_6014_inj__mult__left,axiom,
    ! [A2: int] :
      ( aa_set_int_o(inj_on_int_int(aa_int_fun_int_int(times_times_int,A2)),top_top_set_int)
    <=> ( A2 != zero_zero_int ) ) ).

% inj_mult_left
tff(fact_6015_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),X)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ).

% field_le_mult_one_interval
tff(fact_6016_size__prod__simp,axiom,
    ! [F: fun_he1033112309995878576it_nat,G: fun_set_nat_nat,P2: produc3658429121746597890et_nat] : aa_Pro8901340430247369232at_nat(basic_5567952622663070111et_nat(F,G),P2) = aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(aa_hea5848002036693495265it_nat(F,aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,P2))),aa_set_nat_nat(G,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,P2)))),aa_nat_nat(suc,zero_zero_nat)) ).

% size_prod_simp
tff(fact_6017_size__prod__simp,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,P2: product_prod_int_int] : basic_1872990034501187214nt_int(F,G,P2) = aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(aa_int_nat(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))),aa_int_nat(G,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)))),aa_nat_nat(suc,zero_zero_nat)) ).

% size_prod_simp
tff(fact_6018_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(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_6019_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_6020_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_6021_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_6022_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_6023_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(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_6024_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_6025_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_6026_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_6027_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_6028_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_6029_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_6030_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_6031_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_6032_divides__aux__eq,axiom,
    ! [Q3: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(Q3),R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_6033_divides__aux__def,axiom,
    ! [Qr: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,Qr)
    <=> ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Qr) = zero_zero_nat ) ) ).

% divides_aux_def
tff(fact_6034_divides__aux__def,axiom,
    ! [Qr: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,Qr)
    <=> ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Qr) = zero_zero_int ) ) ).

% divides_aux_def
tff(fact_6035_divides__aux__def,axiom,
    ! [Qr: produc8923325533196201883nteger] :
      ( unique5706413561485394159nteger(Qr)
    <=> ( aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Qr) = zero_z3403309356797280102nteger ) ) ).

% divides_aux_def
tff(fact_6036_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y) ) ) ).

% mult_less_iff1
tff(fact_6037_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Y) ) ) ).

% mult_less_iff1
tff(fact_6038_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y) ) ) ).

% mult_less_iff1
tff(fact_6039_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_6040_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_6041_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_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_6042_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_6043_dbl__inc__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% dbl_inc_simps(2)
tff(fact_6044_enumerate__Suc_H,axiom,
    ! [S: set_Product_unit,N2: nat] : aa_nat_Product_unit(infini7930543730640340914t_unit(S),aa_nat_nat(suc,N2)) = aa_nat_Product_unit(infini7930543730640340914t_unit(minus_6452836326544984404t_unit(S,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,aa_nat_Product_unit(infini7930543730640340914t_unit(S),zero_zero_nat)),bot_bo3957492148770167129t_unit))),N2) ).

% enumerate_Suc'
tff(fact_6045_enumerate__Suc_H,axiom,
    ! [S: set_nat,N2: nat] : aa_nat_nat(infini8530281810654367211te_nat(S),aa_nat_nat(suc,N2)) = aa_nat_nat(infini8530281810654367211te_nat(aa_set_nat_set_nat(minus_minus_set_nat(S),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_nat_nat(infini8530281810654367211te_nat(S),zero_zero_nat)),bot_bot_set_nat))),N2) ).

% enumerate_Suc'
tff(fact_6046_power__decreasing__iff,axiom,
    ! [B: code_natural,M4: nat,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),one_one_Code_natural)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(B),M4)),aa_nat_Code_natural(power_7079662738309270450atural(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4) ) ) ) ).

% power_decreasing_iff
tff(fact_6047_power__decreasing__iff,axiom,
    ! [B: code_integer,M4: 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),M4)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4) ) ) ) ).

% power_decreasing_iff
tff(fact_6048_power__decreasing__iff,axiom,
    ! [B: rat,M4: 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),M4)),aa_nat_rat(power_power_rat(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4) ) ) ) ).

% power_decreasing_iff
tff(fact_6049_power__decreasing__iff,axiom,
    ! [B: nat,M4: 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(power_power_nat(B),M4)),aa_nat_nat(power_power_nat(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4) ) ) ) ).

% power_decreasing_iff
tff(fact_6050_power__decreasing__iff,axiom,
    ! [B: int,M4: 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(power_power_int(B),M4)),aa_nat_int(power_power_int(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4) ) ) ) ).

% power_decreasing_iff
tff(fact_6051_dvd__partition,axiom,
    ! [C5: set_set_Code_integer,K3: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,comple739944243200306918nteger(C5))
     => ( ! [X3: set_Code_integer] :
            ( member1834298987299537450nteger(X3,C5)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite4902975817058060853nteger(X3)) )
       => ( ! [X3: set_Code_integer] :
              ( member1834298987299537450nteger(X3,C5)
             => ! [Xa4: set_Code_integer] :
                  ( member1834298987299537450nteger(Xa4,C5)
                 => ( ( X3 != Xa4 )
                   => ( inf_in1364745209274528805nteger(X3,Xa4) = bot_bo3990330152332043303nteger ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite4902975817058060853nteger(comple739944243200306918nteger(C5))) ) ) ) ).

% dvd_partition
tff(fact_6052_dvd__partition,axiom,
    ! [C5: set_set_Product_unit,K3: nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,comple3349607820504768346t_unit(C5))
     => ( ! [X3: set_Product_unit] :
            ( member5877623283571906838t_unit(X3,C5)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite410649719033368117t_unit(X3)) )
       => ( ! [X3: set_Product_unit] :
              ( member5877623283571906838t_unit(X3,C5)
             => ! [Xa4: set_Product_unit] :
                  ( member5877623283571906838t_unit(Xa4,C5)
                 => ( ( X3 != Xa4 )
                   => ( inf_in4660618365625256667t_unit(X3,Xa4) = bot_bo3957492148770167129t_unit ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite410649719033368117t_unit(comple3349607820504768346t_unit(C5))) ) ) ) ).

% dvd_partition
tff(fact_6053_dvd__partition,axiom,
    ! [C5: set_set_o,K3: nat] :
      ( aa_set_o_o(finite_finite_o,comple90263536869209701_set_o(C5))
     => ( ! [X3: set_o] :
            ( member_set_o(X3,C5)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite_card_o(X3)) )
       => ( ! [X3: set_o] :
              ( member_set_o(X3,C5)
             => ! [Xa4: set_o] :
                  ( member_set_o(Xa4,C5)
                 => ( ( X3 != Xa4 )
                   => ( inf_inf_set_o(X3,Xa4) = bot_bot_set_o ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite_card_o(comple90263536869209701_set_o(C5))) ) ) ) ).

% dvd_partition
tff(fact_6054_dvd__partition,axiom,
    ! [C5: set_set_nat,K3: nat] :
      ( aa_set_nat_o(finite_finite_nat,comple7399068483239264473et_nat(C5))
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,C5)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),aa_set_nat_nat(finite_card_nat,X3)) )
       => ( ! [X3: set_nat] :
              ( member_set_nat(X3,C5)
             => ! [Xa4: set_nat] :
                  ( member_set_nat(Xa4,C5)
                 => ( ( X3 != Xa4 )
                   => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Xa4) = bot_bot_set_nat ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),aa_set_nat_nat(finite_card_nat,comple7399068483239264473et_nat(C5))) ) ) ) ).

% dvd_partition
tff(fact_6055_dvd__partition,axiom,
    ! [C5: set_set_int,K3: nat] :
      ( aa_set_int_o(finite_finite_int,comple3221217463730067765et_int(C5))
     => ( ! [X3: set_int] :
            ( member_set_int(X3,C5)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite_card_int(X3)) )
       => ( ! [X3: set_int] :
              ( member_set_int(X3,C5)
             => ! [Xa4: set_int] :
                  ( member_set_int(Xa4,C5)
                 => ( ( X3 != Xa4 )
                   => ( inf_inf_set_int(X3,Xa4) = bot_bot_set_int ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite_card_int(comple3221217463730067765et_int(C5))) ) ) ) ).

% dvd_partition
tff(fact_6056_dvd__partition,axiom,
    ! [C5: set_se6690607124073039791_nat_o,K3: nat] :
      ( aa_set5091322549519198405at_o_o(finite4610005178611006360_nat_o,comple4665849098253521980_nat_o(C5))
     => ( ! [X3: set_fu3555266933746557007_nat_o] :
            ( member361801372707117048_nat_o(X3,C5)
           => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite5795495896937551127_nat_o(X3)) )
       => ( ! [X3: set_fu3555266933746557007_nat_o] :
              ( member361801372707117048_nat_o(X3,C5)
             => ! [Xa4: set_fu3555266933746557007_nat_o] :
                  ( member361801372707117048_nat_o(Xa4,C5)
                 => ( ( X3 != Xa4 )
                   => ( inf_in3376754634429691197_nat_o(X3,Xa4) = bot_bo8632255151191677115_nat_o ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K3),finite5795495896937551127_nat_o(comple4665849098253521980_nat_o(C5))) ) ) ) ).

% dvd_partition
tff(fact_6057_cInf__le__cSup,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_o_o(condit5488710616941104124bove_o,A)
       => ( aa_set_o_o(condit5413489452508810728elow_o,A)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Inf_Inf_o,A)),aa_set_o_o(complete_Sup_Sup_o,A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6058_cInf__le__cSup,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit1129181918374225235_nat_o,A)
       => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,A)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple6093339926577406454_nat_o(A)),comple7878897588548607196_nat_o(A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6059_cInf__le__cSup,axiom,
    ! [A: set_filter_nat] :
      ( ( A != bot_bo498966703094740906er_nat )
     => ( condit4029664723990530684er_nat(A)
       => ( condit3133331395780078992er_nat(A)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(A)),comple2454491259644266675er_nat(A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6060_cInf__le__cSup,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,A)
       => ( aa_set_int_o(condit1735850657277959132ow_int,A)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(A)),complete_Sup_Sup_int(A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6061_cInf__le__cSup,axiom,
    ! [A: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( condit5477540289124974626et_nat(A)
       => ( condit68592940725977398et_nat(A)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(A)),comple7399068483239264473et_nat(A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6062_cInf__le__cSup,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A)
       => ( aa_set_nat_o(condit1738341127787009408ow_nat,A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,A)),aa_set_nat_nat(complete_Sup_Sup_nat,A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6063_cInf__le__cSup,axiom,
    ! [A: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit2080475194861589617t_unit,A)
       => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,A)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(A)),comple4687483117567863418t_unit(A)) ) ) ) ).

% cInf_le_cSup
tff(fact_6064_sum__le__card__Max,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups7237345082560585321er_nat(F,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite4902975817058060853nteger(A)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_951025933927791156er_nat(F,A)))) ) ).

% sum_le_card_Max
tff(fact_6065_sum__le__card__Max,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,F),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(finite_card_nat,A)),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)))) ) ).

% sum_le_card_Max
tff(fact_6066_gbinomial__reduce__nat,axiom,
    ! [K3: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_rat(gbinomial_rat(A2),K3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),aa_nat_nat(minus_minus_nat(K3),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K3)) ) ) ).

% gbinomial_reduce_nat
tff(fact_6067_power__one,axiom,
    ! [N2: nat] : aa_nat_assn(power_power_assn(one_one_assn),N2) = one_one_assn ).

% power_one
tff(fact_6068_power__one,axiom,
    ! [N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N2) = one_one_Code_integer ).

% power_one
tff(fact_6069_power__one,axiom,
    ! [N2: nat] : aa_nat_Code_natural(power_7079662738309270450atural(one_one_Code_natural),N2) = one_one_Code_natural ).

% power_one
tff(fact_6070_power__one,axiom,
    ! [N2: nat] : aa_nat_int(power_power_int(one_one_int),N2) = one_one_int ).

% power_one
tff(fact_6071_power__one,axiom,
    ! [N2: nat] : aa_nat_nat(power_power_nat(one_one_nat),N2) = one_one_nat ).

% power_one
tff(fact_6072_bdd__below__empty,axiom,
    aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,bot_bo3957492148770167129t_unit) ).

% bdd_below_empty
tff(fact_6073_bdd__below__empty,axiom,
    aa_set_o_o(condit5413489452508810728elow_o,bot_bot_set_o) ).

% bdd_below_empty
tff(fact_6074_bdd__below__empty,axiom,
    aa_set_nat_o(condit1738341127787009408ow_nat,bot_bot_set_nat) ).

% bdd_below_empty
tff(fact_6075_bdd__below__empty,axiom,
    aa_set_int_o(condit1735850657277959132ow_int,bot_bot_set_int) ).

% bdd_below_empty
tff(fact_6076_bdd__below__empty,axiom,
    aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,bot_bo8632255151191677115_nat_o) ).

% bdd_below_empty
tff(fact_6077_bdd__below__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))
    <=> aa_set_nat_o(condit1738341127787009408ow_nat,A) ) ).

% bdd_below_insert
tff(fact_6078_bdd__below__insert,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))
    <=> aa_set_int_o(condit1735850657277959132ow_int,A) ) ).

% bdd_below_insert
tff(fact_6079_bdd__below__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(condit5413489452508810728elow_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
    <=> aa_set_o_o(condit5413489452508810728elow_o,A) ) ).

% bdd_below_insert
tff(fact_6080_bdd__below__insert,axiom,
    ! [A2: product_unit,A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))
    <=> aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,A) ) ).

% bdd_below_insert
tff(fact_6081_dvd__times__right__cancel__iff,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6082_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6083_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_6084_dvd__times__left__cancel__iff,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6085_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6086_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_6087_dvd__mult__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6088_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6089_dvd__mult__cancel__right,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_6090_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6091_dvd__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6092_dvd__mult__cancel__left,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_6093_dvd__add__times__triv__left__iff,axiom,
    ! [A2: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6094_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6095_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6096_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_6097_dvd__add__times__triv__right__iff,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6098_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6099_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(plus_plus_nat(B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6100_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_6101_unit__prod,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_6102_unit__prod,axiom,
    ! [A2: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B)),one_one_Code_natural) ) ) ).

% unit_prod
tff(fact_6103_unit__prod,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_6104_unit__prod,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_6105_power__inject__exp,axiom,
    ! [A2: code_natural,M4: nat,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => ( ( aa_nat_Code_natural(power_7079662738309270450atural(A2),M4) = aa_nat_Code_natural(power_7079662738309270450atural(A2),N2) )
      <=> ( M4 = N2 ) ) ) ).

% power_inject_exp
tff(fact_6106_power__inject__exp,axiom,
    ! [A2: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( ( aa_nat_nat(power_power_nat(A2),M4) = aa_nat_nat(power_power_nat(A2),N2) )
      <=> ( M4 = N2 ) ) ) ).

% power_inject_exp
tff(fact_6107_power__inject__exp,axiom,
    ! [A2: int,M4: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( ( aa_nat_int(power_power_int(A2),M4) = aa_nat_int(power_power_int(A2),N2) )
      <=> ( M4 = N2 ) ) ) ).

% power_inject_exp
tff(fact_6108_power__inject__exp,axiom,
    ! [A2: code_integer,M4: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),M4) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2) )
      <=> ( M4 = N2 ) ) ) ).

% power_inject_exp
tff(fact_6109_power__inject__exp,axiom,
    ! [A2: rat,M4: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( ( aa_nat_rat(power_power_rat(A2),M4) = aa_nat_rat(power_power_rat(A2),N2) )
      <=> ( M4 = N2 ) ) ) ).

% power_inject_exp
tff(fact_6110_nat__mult__dvd__cancel__disj,axiom,
    ! [K3: nat,M4: 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,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
    <=> ( ( K3 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M4),N2) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_6111_gbinomial__0_I1_J,axiom,
    ! [A2: nat] : aa_nat_nat(gbinomial_nat(A2),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_6112_gbinomial__0_I1_J,axiom,
    ! [A2: int] : aa_nat_int(gbinomial_int(A2),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_6113_gbinomial__0_I1_J,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(gbinom8545251970709558553nteger(A2),zero_zero_nat) = one_one_Code_integer ).

% gbinomial_0(1)
tff(fact_6114_gbinomial__0_I1_J,axiom,
    ! [A2: code_natural] : aa_nat_Code_natural(gbinom7368847122466276068atural(A2),zero_zero_nat) = one_one_Code_natural ).

% gbinomial_0(1)
tff(fact_6115_power__strict__increasing__iff,axiom,
    ! [B: code_natural,X: nat,Y: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(B),X)),aa_nat_Code_natural(power_7079662738309270450atural(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_6116_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(power_power_nat(B),X)),aa_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_6117_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(power_power_int(B),X)),aa_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_6118_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_6119_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_6120_power__strict__decreasing__iff,axiom,
    ! [B: code_natural,M4: nat,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,B),one_one_Code_natural)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(B),M4)),aa_nat_Code_natural(power_7079662738309270450atural(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M4) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_6121_power__strict__decreasing__iff,axiom,
    ! [B: nat,M4: 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(power_power_nat(B),M4)),aa_nat_nat(power_power_nat(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M4) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_6122_power__strict__decreasing__iff,axiom,
    ! [B: int,M4: 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(power_power_int(B),M4)),aa_nat_int(power_power_int(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M4) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_6123_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M4: 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),M4)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M4) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_6124_power__strict__decreasing__iff,axiom,
    ! [B: rat,M4: 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),M4)),aa_nat_rat(power_power_rat(B),N2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),M4) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_6125_power__increasing__iff,axiom,
    ! [B: code_natural,X: nat,Y: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(B),X)),aa_nat_Code_natural(power_7079662738309270450atural(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_6126_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_6127_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_6128_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(power_power_nat(B),X)),aa_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_6129_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(power_power_int(B),X)),aa_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_6130_is__unit__power__iff,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6131_is__unit__power__iff,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)),one_one_Code_natural)
    <=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6132_is__unit__power__iff,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),N2)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6133_is__unit__power__iff,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(power_power_nat(A2),N2)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6134_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N2: nat] :
      ( ( aa_assn_assn(times_times_assn(X),Y) = aa_assn_assn(times_times_assn(Y),X) )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(X),N2)),Y) = aa_assn_assn(times_times_assn(Y),aa_nat_assn(power_power_assn(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_6135_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N2)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_6136_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(X),N2)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(power_power_nat(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_6137_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N2)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),N2)) ) ) ).

% power_commuting_commutes
tff(fact_6138_power__mult__distrib,axiom,
    ! [A2: assn,B: assn,N2: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(times_times_assn(A2),B)),N2) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),N2)),aa_nat_assn(power_power_assn(B),N2)) ).

% power_mult_distrib
tff(fact_6139_power__mult__distrib,axiom,
    ! [A2: rat,B: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),N2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A2),N2)),aa_nat_rat(power_power_rat(B),N2)) ).

% power_mult_distrib
tff(fact_6140_power__mult__distrib,axiom,
    ! [A2: nat,B: nat,N2: nat] : aa_nat_nat(power_power_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(A2),N2)),aa_nat_nat(power_power_nat(B),N2)) ).

% power_mult_distrib
tff(fact_6141_power__mult__distrib,axiom,
    ! [A2: int,B: int,N2: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),N2)),aa_nat_int(power_power_int(B),N2)) ).

% power_mult_distrib
tff(fact_6142_power__commutes,axiom,
    ! [A2: assn,N2: nat] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),N2)),A2) = aa_assn_assn(times_times_assn(A2),aa_nat_assn(power_power_assn(A2),N2)) ).

% power_commutes
tff(fact_6143_power__commutes,axiom,
    ! [A2: rat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A2),N2)),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(power_power_rat(A2),N2)) ).

% power_commutes
tff(fact_6144_power__commutes,axiom,
    ! [A2: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(A2),N2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(power_power_nat(A2),N2)) ).

% power_commutes
tff(fact_6145_power__commutes,axiom,
    ! [A2: int,N2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),N2)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N2)) ).

% power_commutes
tff(fact_6146_dvd__triv__right,axiom,
    ! [A2: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),aa_assn_assn(times_times_assn(B),A2)) ).

% dvd_triv_right
tff(fact_6147_dvd__triv__right,axiom,
    ! [A2: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A2)) ).

% dvd_triv_right
tff(fact_6148_dvd__triv__right,axiom,
    ! [A2: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A2)) ).

% dvd_triv_right
tff(fact_6149_dvd__triv__right,axiom,
    ! [A2: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),A2)) ).

% dvd_triv_right
tff(fact_6150_dvd__mult__right,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(times_times_assn(A2),B)),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),C) ) ).

% dvd_mult_right
tff(fact_6151_dvd__mult__right,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_6152_dvd__mult__right,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_6153_dvd__mult__right,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_6154_mult__dvd__mono,axiom,
    ! [A2: assn,B: assn,C: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,C),D2)
       => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(times_times_assn(A2),C)),aa_assn_assn(times_times_assn(B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6155_mult__dvd__mono,axiom,
    ! [A2: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6156_mult__dvd__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6157_mult__dvd__mono,axiom,
    ! [A2: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6158_dvd__triv__left,axiom,
    ! [A2: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),aa_assn_assn(times_times_assn(A2),B)) ).

% dvd_triv_left
tff(fact_6159_dvd__triv__left,axiom,
    ! [A2: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)) ).

% dvd_triv_left
tff(fact_6160_dvd__triv__left,axiom,
    ! [A2: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)) ).

% dvd_triv_left
tff(fact_6161_dvd__triv__left,axiom,
    ! [A2: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)) ).

% dvd_triv_left
tff(fact_6162_dvd__mult__left,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,aa_assn_assn(times_times_assn(A2),B)),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),C) ) ).

% dvd_mult_left
tff(fact_6163_dvd__mult__left,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C) ) ).

% dvd_mult_left
tff(fact_6164_dvd__mult__left,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ).

% dvd_mult_left
tff(fact_6165_dvd__mult__left,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ).

% dvd_mult_left
tff(fact_6166_dvd__mult2,axiom,
    ! [A2: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),B)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),aa_assn_assn(times_times_assn(B),C)) ) ).

% dvd_mult2
tff(fact_6167_dvd__mult2,axiom,
    ! [A2: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult2
tff(fact_6168_dvd__mult2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_6169_dvd__mult2,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_6170_dvd__mult,axiom,
    ! [A2: assn,C: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),C)
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,A2),aa_assn_assn(times_times_assn(B),C)) ) ).

% dvd_mult
tff(fact_6171_dvd__mult,axiom,
    ! [A2: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult
tff(fact_6172_dvd__mult,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_6173_dvd__mult,axiom,
    ! [A2: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_6174_dvd__def,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A2)
    <=> ? [K5: assn] : A2 = aa_assn_assn(times_times_assn(B),K5) ) ).

% dvd_def
tff(fact_6175_dvd__def,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A2)
    <=> ? [K5: rat] : A2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K5) ) ).

% dvd_def
tff(fact_6176_dvd__def,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A2)
    <=> ? [K5: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K5) ) ).

% dvd_def
tff(fact_6177_dvd__def,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A2)
    <=> ? [K5: int] : A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),K5) ) ).

% dvd_def
tff(fact_6178_dvdI,axiom,
    ! [A2: assn,B: assn,K3: assn] :
      ( ( A2 = aa_assn_assn(times_times_assn(B),K3) )
     => aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A2) ) ).

% dvdI
tff(fact_6179_dvdI,axiom,
    ! [A2: rat,B: rat,K3: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K3) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A2) ) ).

% dvdI
tff(fact_6180_dvdI,axiom,
    ! [A2: nat,B: nat,K3: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A2) ) ).

% dvdI
tff(fact_6181_dvdI,axiom,
    ! [A2: int,B: int,K3: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A2) ) ).

% dvdI
tff(fact_6182_dvdE,axiom,
    ! [B: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,B),A2)
     => ~ ! [K: assn] : A2 != aa_assn_assn(times_times_assn(B),K) ) ).

% dvdE
tff(fact_6183_dvdE,axiom,
    ! [B: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A2)
     => ~ ! [K: rat] : A2 != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) ) ).

% dvdE
tff(fact_6184_dvdE,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A2)
     => ~ ! [K: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) ) ).

% dvdE
tff(fact_6185_dvdE,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A2)
     => ~ ! [K: int] : A2 != aa_int_int(aa_int_fun_int_int(times_times_int,B),K) ) ).

% dvdE
tff(fact_6186_dvd__unit__imp__unit,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_6187_dvd__unit__imp__unit,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_6188_dvd__unit__imp__unit,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_6189_dvd__unit__imp__unit,axiom,
    ! [A2: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),B)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural) ) ) ).

% dvd_unit_imp_unit
tff(fact_6190_unit__imp__dvd,axiom,
    ! [B: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A2) ) ).

% unit_imp_dvd
tff(fact_6191_unit__imp__dvd,axiom,
    ! [B: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A2) ) ).

% unit_imp_dvd
tff(fact_6192_unit__imp__dvd,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A2) ) ).

% unit_imp_dvd
tff(fact_6193_unit__imp__dvd,axiom,
    ! [B: code_natural,A2: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),A2) ) ).

% unit_imp_dvd
tff(fact_6194_one__dvd,axiom,
    ! [A2: assn] : aa_assn_o(aa_assn_fun_assn_o(dvd_dvd_assn,one_one_assn),A2) ).

% one_dvd
tff(fact_6195_one__dvd,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A2) ).

% one_dvd
tff(fact_6196_one__dvd,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A2) ).

% one_dvd
tff(fact_6197_one__dvd,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A2) ).

% one_dvd
tff(fact_6198_one__dvd,axiom,
    ! [A2: code_natural] : aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,one_one_Code_natural),A2) ).

% one_dvd
tff(fact_6199_power__mult,axiom,
    ! [A2: int,M4: nat,N2: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),M4)),N2) ).

% power_mult
tff(fact_6200_power__mult,axiom,
    ! [A2: nat,M4: nat,N2: nat] : aa_nat_nat(power_power_nat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) = aa_nat_nat(power_power_nat(aa_nat_nat(power_power_nat(A2),M4)),N2) ).

% power_mult
tff(fact_6201_dvd__power__iff,axiom,
    ! [X: code_natural,M4: nat,N2: nat] :
      ( ( X != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_nat_Code_natural(power_7079662738309270450atural(X),M4)),aa_nat_Code_natural(power_7079662738309270450atural(X),N2))
      <=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,X),one_one_Code_natural)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6202_dvd__power__iff,axiom,
    ! [X: code_integer,M4: nat,N2: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M4)),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,M4),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6203_dvd__power__iff,axiom,
    ! [X: int,M4: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),M4)),aa_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,M4),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6204_dvd__power__iff,axiom,
    ! [X: nat,M4: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(power_power_nat(X),M4)),aa_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,M4),N2) ) ) ) ).

% dvd_power_iff
tff(fact_6205_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_6206_dvd__power,axiom,
    ! [N2: nat,X: code_natural] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
        | ( X = one_one_Code_natural ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,X),aa_nat_Code_natural(power_7079662738309270450atural(X),N2)) ) ).

% dvd_power
tff(fact_6207_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(power_power_int(X),N2)) ) ).

% dvd_power
tff(fact_6208_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(power_power_nat(X),N2)) ) ).

% dvd_power
tff(fact_6209_gbinomial__Suc__Suc,axiom,
    ! [A2: rat,K3: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat)),aa_nat_nat(suc,K3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A2),K3)),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K3))) ).

% gbinomial_Suc_Suc
tff(fact_6210_one__le__power,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ).

% one_le_power
tff(fact_6211_one__le__power,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),A2)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)) ) ).

% one_le_power
tff(fact_6212_one__le__power,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N2)) ) ).

% one_le_power
tff(fact_6213_one__le__power,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(power_power_nat(A2),N2)) ) ).

% one_le_power
tff(fact_6214_one__le__power,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A2),N2)) ) ).

% one_le_power
tff(fact_6215_not__is__unit__0,axiom,
    ~ aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,zero_z2226904508553997617atural),one_one_Code_natural) ).

% not_is_unit_0
tff(fact_6216_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_6217_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_6218_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_6219_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N2)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_6220_left__right__inverse__power,axiom,
    ! [X: code_natural,Y: code_natural,N2: nat] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Y) = one_one_Code_natural )
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_nat_Code_natural(power_7079662738309270450atural(X),N2)),aa_nat_Code_natural(power_7079662738309270450atural(Y),N2)) = one_one_Code_natural ) ) ).

% left_right_inverse_power
tff(fact_6221_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N2: nat] :
      ( ( aa_assn_assn(times_times_assn(X),Y) = one_one_assn )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(X),N2)),aa_nat_assn(power_power_assn(Y),N2)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_6222_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(X),N2)),aa_nat_rat(power_power_rat(Y),N2)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_6223_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(X),N2)),aa_nat_nat(power_power_nat(Y),N2)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_6224_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N2)),aa_nat_int(power_power_int(Y),N2)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_6225_power__Suc2,axiom,
    ! [A2: assn,N2: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,N2)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),N2)),A2) ).

% power_Suc2
tff(fact_6226_power__Suc2,axiom,
    ! [A2: rat,N2: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A2),N2)),A2) ).

% power_Suc2
tff(fact_6227_power__Suc2,axiom,
    ! [A2: nat,N2: nat] : aa_nat_nat(power_power_nat(A2),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(A2),N2)),A2) ).

% power_Suc2
tff(fact_6228_power__Suc2,axiom,
    ! [A2: int,N2: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),N2)),A2) ).

% power_Suc2
tff(fact_6229_power__Suc,axiom,
    ! [A2: assn,N2: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,N2)) = aa_assn_assn(times_times_assn(A2),aa_nat_assn(power_power_assn(A2),N2)) ).

% power_Suc
tff(fact_6230_power__Suc,axiom,
    ! [A2: rat,N2: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(power_power_rat(A2),N2)) ).

% power_Suc
tff(fact_6231_power__Suc,axiom,
    ! [A2: nat,N2: nat] : aa_nat_nat(power_power_nat(A2),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(power_power_nat(A2),N2)) ).

% power_Suc
tff(fact_6232_power__Suc,axiom,
    ! [A2: int,N2: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N2)) ).

% power_Suc
tff(fact_6233_unit__mult__right__cancel,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6234_unit__mult__right__cancel,axiom,
    ! [A2: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
     => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A2) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A2) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6235_unit__mult__right__cancel,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6236_unit__mult__right__cancel,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_6237_unit__mult__left__cancel,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6238_unit__mult__left__cancel,axiom,
    ! [A2: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
     => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6239_unit__mult__left__cancel,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6240_unit__mult__left__cancel,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_6241_mult__unit__dvd__iff_H,axiom,
    ! [A2: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6242_mult__unit__dvd__iff_H,axiom,
    ! [A2: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B)),C)
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6243_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6244_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6245_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6246_dvd__mult__unit__iff_H,axiom,
    ! [B: code_natural,A2: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6247_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6248_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6249_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6250_mult__unit__dvd__iff,axiom,
    ! [B: code_natural,A2: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B)),C)
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6251_mult__unit__dvd__iff,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6252_mult__unit__dvd__iff,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_6253_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6254_dvd__mult__unit__iff,axiom,
    ! [B: code_natural,A2: code_natural,C: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))
      <=> aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6255_dvd__mult__unit__iff,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6256_dvd__mult__unit__iff,axiom,
    ! [B: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_6257_is__unit__mult__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_6258_is__unit__mult__iff,axiom,
    ! [A2: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),B)),one_one_Code_natural)
    <=> ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
        & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,B),one_one_Code_natural) ) ) ).

% is_unit_mult_iff
tff(fact_6259_is__unit__mult__iff,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_6260_is__unit__mult__iff,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_6261_power__0,axiom,
    ! [A2: assn] : aa_nat_assn(power_power_assn(A2),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_6262_power__0,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_6263_power__0,axiom,
    ! [A2: code_natural] : aa_nat_Code_natural(power_7079662738309270450atural(A2),zero_zero_nat) = one_one_Code_natural ).

% power_0
tff(fact_6264_power__0,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_6265_power__0,axiom,
    ! [A2: nat] : aa_nat_nat(power_power_nat(A2),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_6266_power__add,axiom,
    ! [A2: assn,M4: nat,N2: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(plus_plus_nat(M4),N2)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),M4)),aa_nat_assn(power_power_assn(A2),N2)) ).

% power_add
tff(fact_6267_power__add,axiom,
    ! [A2: rat,M4: nat,N2: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(plus_plus_nat(M4),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A2),M4)),aa_nat_rat(power_power_rat(A2),N2)) ).

% power_add
tff(fact_6268_power__add,axiom,
    ! [A2: nat,M4: nat,N2: nat] : aa_nat_nat(power_power_nat(A2),aa_nat_nat(plus_plus_nat(M4),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(A2),M4)),aa_nat_nat(power_power_nat(A2),N2)) ).

% power_add
tff(fact_6269_power__add,axiom,
    ! [A2: int,M4: nat,N2: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(plus_plus_nat(M4),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),M4)),aa_nat_int(power_power_int(A2),N2)) ).

% power_add
tff(fact_6270_gbinomial__addition__formula,axiom,
    ! [A2: rat,K3: nat] : aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K3)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),aa_nat_nat(suc,K3))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K3)) ).

% gbinomial_addition_formula
tff(fact_6271_power__le__one,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A2)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A2),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)),one_one_Code_natural) ) ) ).

% power_le_one
tff(fact_6272_power__le__one,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_6273_power__le__one,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N2)),one_one_rat) ) ) ).

% power_le_one
tff(fact_6274_power__le__one,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(A2),N2)),one_one_nat) ) ) ).

% power_le_one
tff(fact_6275_power__le__one,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N2)),one_one_int) ) ) ).

% power_le_one
tff(fact_6276_power__less__power__Suc,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2))) ) ).

% power_less_power_Suc
tff(fact_6277_power__less__power__Suc,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(A2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(power_power_nat(A2),N2))) ) ).

% power_less_power_Suc
tff(fact_6278_power__less__power__Suc,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N2))) ) ).

% power_less_power_Suc
tff(fact_6279_power__less__power__Suc,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2))) ) ).

% power_less_power_Suc
tff(fact_6280_power__less__power__Suc,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(power_power_rat(A2),N2))) ) ).

% power_less_power_Suc
tff(fact_6281_power__gt1__lemma,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2))) ) ).

% power_gt1_lemma
tff(fact_6282_power__gt1__lemma,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => 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,A2),aa_nat_nat(power_power_nat(A2),N2))) ) ).

% power_gt1_lemma
tff(fact_6283_power__gt1__lemma,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => 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,A2),aa_nat_int(power_power_int(A2),N2))) ) ).

% power_gt1_lemma
tff(fact_6284_power__gt1__lemma,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2))) ) ).

% power_gt1_lemma
tff(fact_6285_power__gt1__lemma,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => 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,A2),aa_nat_rat(power_power_rat(A2),N2))) ) ).

% power_gt1_lemma
tff(fact_6286_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X2: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X2))
    <=> ? [X2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X2),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6287_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X2: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X2))
    <=> ? [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X2),zero_zero_rat))
          & aa_rat_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6288_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X2: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X2))
    <=> ? [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(plus_plus_nat(X2),zero_zero_nat))
          & aa_nat_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6289_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X2: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X2))
    <=> ? [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(plus_plus_int(X2),zero_zero_int))
          & aa_int_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6290_power__0__left,axiom,
    ! [N2: nat] :
      aa_nat_Code_natural(power_7079662738309270450atural(zero_z2226904508553997617atural),N2) = $ite(N2 = zero_zero_nat,one_one_Code_natural,zero_z2226904508553997617atural) ).

% power_0_left
tff(fact_6291_power__0__left,axiom,
    ! [N2: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N2) = $ite(N2 = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_6292_power__0__left,axiom,
    ! [N2: nat] :
      aa_nat_int(power_power_int(zero_zero_int),N2) = $ite(N2 = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_6293_power__0__left,axiom,
    ! [N2: nat] :
      aa_nat_nat(power_power_nat(zero_zero_nat),N2) = $ite(N2 = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_6294_unit__dvdE,axiom,
    ! [A2: code_natural,B: code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(dvd_dvd_Code_natural,A2),one_one_Code_natural)
     => ~ ( ( A2 != zero_z2226904508553997617atural )
         => ! [C2: code_natural] : B != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),C2) ) ) ).

% unit_dvdE
tff(fact_6295_unit__dvdE,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C2) ) ) ).

% unit_dvdE
tff(fact_6296_unit__dvdE,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ~ ( ( A2 != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) ) ) ).

% unit_dvdE
tff(fact_6297_unit__dvdE,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ~ ( ( A2 != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2) ) ) ).

% unit_dvdE
tff(fact_6298_power__gt1,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A2),aa_nat_nat(suc,N2))) ) ).

% power_gt1
tff(fact_6299_power__gt1,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(power_power_nat(A2),aa_nat_nat(suc,N2))) ) ).

% power_gt1
tff(fact_6300_power__gt1,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N2))) ) ).

% power_gt1
tff(fact_6301_power__gt1,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N2))) ) ).

% power_gt1
tff(fact_6302_power__gt1,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N2))) ) ).

% power_gt1
tff(fact_6303_power__strict__increasing,axiom,
    ! [N2: nat,N: nat,A2: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)),aa_nat_Code_natural(power_7079662738309270450atural(A2),N)) ) ) ).

% power_strict_increasing
tff(fact_6304_power__strict__increasing,axiom,
    ! [N2: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(A2),N2)),aa_nat_nat(power_power_nat(A2),N)) ) ) ).

% power_strict_increasing
tff(fact_6305_power__strict__increasing,axiom,
    ! [N2: nat,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N2)),aa_nat_int(power_power_int(A2),N)) ) ) ).

% power_strict_increasing
tff(fact_6306_power__strict__increasing,axiom,
    ! [N2: nat,N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ).

% power_strict_increasing
tff(fact_6307_power__strict__increasing,axiom,
    ! [N2: nat,N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N2)),aa_nat_rat(power_power_rat(A2),N)) ) ) ).

% power_strict_increasing
tff(fact_6308_power__less__imp__less__exp,axiom,
    ! [A2: code_natural,M4: nat,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),M4)),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_6309_power__less__imp__less__exp,axiom,
    ! [A2: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(A2),M4)),aa_nat_nat(power_power_nat(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_6310_power__less__imp__less__exp,axiom,
    ! [A2: int,M4: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),M4)),aa_nat_int(power_power_int(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_6311_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M4: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M4)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_6312_power__less__imp__less__exp,axiom,
    ! [A2: rat,M4: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),M4)),aa_nat_rat(power_power_rat(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2) ) ) ).

% power_less_imp_less_exp
tff(fact_6313_power__increasing,axiom,
    ! [N2: nat,N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ).

% power_increasing
tff(fact_6314_power__increasing,axiom,
    ! [N2: nat,N: nat,A2: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),A2)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)),aa_nat_Code_natural(power_7079662738309270450atural(A2),N)) ) ) ).

% power_increasing
tff(fact_6315_power__increasing,axiom,
    ! [N2: nat,N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N2)),aa_nat_rat(power_power_rat(A2),N)) ) ) ).

% power_increasing
tff(fact_6316_power__increasing,axiom,
    ! [N2: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(A2),N2)),aa_nat_nat(power_power_nat(A2),N)) ) ) ).

% power_increasing
tff(fact_6317_power__increasing,axiom,
    ! [N2: nat,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N2)),aa_nat_int(power_power_int(A2),N)) ) ) ).

% power_increasing
tff(fact_6318_inf__period_I3_J,axiom,
    ! [D2: code_integer,D3: code_integer,T4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
     => ! [X4: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),T4))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))),T4)) ) ) ).

% inf_period(3)
tff(fact_6319_inf__period_I3_J,axiom,
    ! [D2: rat,D3: rat,T4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D3)
     => ! [X4: rat,K2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),T4))
        <=> 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(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))),T4)) ) ) ).

% inf_period(3)
tff(fact_6320_inf__period_I3_J,axiom,
    ! [D2: int,D3: int,T4: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X4: int,K2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),T4))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))),T4)) ) ) ).

% inf_period(3)
tff(fact_6321_inf__period_I4_J,axiom,
    ! [D2: code_integer,D3: code_integer,T4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),D3)
     => ! [X4: code_integer,K2: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),T4))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,X4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D3))),T4)) ) ) ).

% inf_period(4)
tff(fact_6322_inf__period_I4_J,axiom,
    ! [D2: rat,D3: rat,T4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D3)
     => ! [X4: rat,K2: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),T4))
        <=> ~ 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(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D3))),T4)) ) ) ).

% inf_period(4)
tff(fact_6323_inf__period_I4_J,axiom,
    ! [D2: int,D3: int,T4: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D3)
     => ! [X4: int,K2: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),T4))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D3))),T4)) ) ) ).

% inf_period(4)
tff(fact_6324_cInf__mono,axiom,
    ! [B4: set_o,A: set_o] :
      ( ( B4 != bot_bot_set_o )
     => ( aa_set_o_o(condit5413489452508810728elow_o,A)
       => ( ! [B2: $o] :
              ( aa_set_o_o(member_o((B2)),B4)
             => ? [X4: $o] :
                  ( aa_set_o_o(member_o((X4)),A)
                  & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X4)),(B2)) ) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Inf_Inf_o,A)),aa_set_o_o(complete_Inf_Inf_o,B4)) ) ) ) ).

% cInf_mono
tff(fact_6325_cInf__mono,axiom,
    ! [B4: set_fu3555266933746557007_nat_o,A: set_fu3555266933746557007_nat_o] :
      ( ( B4 != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,A)
       => ( ! [B2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B2),B4)
             => ? [X4: fun_Pr6460944699146269935_nat_o] :
                  ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X4),A)
                  & aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,X4),B2) ) )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple6093339926577406454_nat_o(A)),comple6093339926577406454_nat_o(B4)) ) ) ) ).

% cInf_mono
tff(fact_6326_cInf__mono,axiom,
    ! [B4: set_filter_nat,A: set_filter_nat] :
      ( ( B4 != bot_bo498966703094740906er_nat )
     => ( condit3133331395780078992er_nat(A)
       => ( ! [B2: filter_nat] :
              ( member_filter_nat(B2,B4)
             => ? [X4: filter_nat] :
                  ( member_filter_nat(X4,A)
                  & aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,X4),B2) ) )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(A)),comple6612786467464916249er_nat(B4)) ) ) ) ).

% cInf_mono
tff(fact_6327_cInf__mono,axiom,
    ! [B4: set_int,A: set_int] :
      ( ( B4 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,A)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),B4)
             => ? [X4: int] :
                  ( aa_set_int_o(member_int(X4),A)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),B2) ) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(A)),complete_Inf_Inf_int(B4)) ) ) ) ).

% cInf_mono
tff(fact_6328_cInf__mono,axiom,
    ! [B4: set_nat,A: set_nat] :
      ( ( B4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),B2) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,A)),aa_set_nat_nat(complete_Inf_Inf_nat,B4)) ) ) ) ).

% cInf_mono
tff(fact_6329_cInf__mono,axiom,
    ! [B4: set_set_nat,A: set_set_nat] :
      ( ( B4 != bot_bot_set_set_nat )
     => ( condit68592940725977398et_nat(A)
       => ( ! [B2: set_nat] :
              ( member_set_nat(B2,B4)
             => ? [X4: set_nat] :
                  ( member_set_nat(X4,A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),B2) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(A)),comple7806235888213564991et_nat(B4)) ) ) ) ).

% cInf_mono
tff(fact_6330_cInf__mono,axiom,
    ! [B4: set_Product_unit,A: set_Product_unit] :
      ( ( B4 != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,A)
       => ( ! [B2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(B2),B4)
             => ? [X4: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X4),A)
                  & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X4),B2) ) )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(A)),comple2584293577114468500t_unit(B4)) ) ) ) ).

% cInf_mono
tff(fact_6331_le__cInf__iff,axiom,
    ! [S: set_o,A2: $o] :
      ( ( S != bot_bot_set_o )
     => ( aa_set_o_o(condit5413489452508810728elow_o,S)
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A2)),aa_set_o_o(complete_Inf_Inf_o,S))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),S)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A2)),(X2)) ) ) ) ) ).

% le_cInf_iff
tff(fact_6332_le__cInf__iff,axiom,
    ! [S: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( S != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,S)
       => ( aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A2),comple6093339926577406454_nat_o(S))
        <=> ! [X2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(X2),S)
             => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,A2),X2) ) ) ) ) ).

% le_cInf_iff
tff(fact_6333_le__cInf__iff,axiom,
    ! [S: set_filter_nat,A2: filter_nat] :
      ( ( S != bot_bo498966703094740906er_nat )
     => ( condit3133331395780078992er_nat(S)
       => ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),comple6612786467464916249er_nat(S))
        <=> ! [X2: filter_nat] :
              ( member_filter_nat(X2,S)
             => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,A2),X2) ) ) ) ) ).

% le_cInf_iff
tff(fact_6334_le__cInf__iff,axiom,
    ! [S: set_int,A2: int] :
      ( ( S != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,S)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),complete_Inf_Inf_int(S))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),S)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),X2) ) ) ) ) ).

% le_cInf_iff
tff(fact_6335_le__cInf__iff,axiom,
    ! [S: set_nat,A2: nat] :
      ( ( S != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,S)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_set_nat_nat(complete_Inf_Inf_nat,S))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),S)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),X2) ) ) ) ) ).

% le_cInf_iff
tff(fact_6336_le__cInf__iff,axiom,
    ! [S: set_set_nat,A2: set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ( condit68592940725977398et_nat(S)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),comple7806235888213564991et_nat(S))
        <=> ! [X2: set_nat] :
              ( member_set_nat(X2,S)
             => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),X2) ) ) ) ) ).

% le_cInf_iff
tff(fact_6337_le__cInf__iff,axiom,
    ! [S: set_Product_unit,A2: product_unit] :
      ( ( S != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,S)
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A2),comple2584293577114468500t_unit(S))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),S)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A2),X2) ) ) ) ) ).

% le_cInf_iff
tff(fact_6338_cInf__less__iff,axiom,
    ! [X5: set_int,Y: int] :
      ( ( X5 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,X5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,complete_Inf_Inf_int(X5)),Y)
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),X5)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y) ) ) ) ) ).

% cInf_less_iff
tff(fact_6339_cInf__less__iff,axiom,
    ! [X5: set_nat,Y: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,X5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(complete_Inf_Inf_nat,X5)),Y)
        <=> ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),X5)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y) ) ) ) ) ).

% cInf_less_iff
tff(fact_6340_cInf__less__iff,axiom,
    ! [X5: set_Product_unit,Y: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,X5)
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,comple2584293577114468500t_unit(X5)),Y)
        <=> ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),X5)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X2),Y) ) ) ) ) ).

% cInf_less_iff
tff(fact_6341_dvd__mult__cancel,axiom,
    ! [K3: nat,M4: 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,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M4),N2) ) ) ).

% dvd_mult_cancel
tff(fact_6342_nat__mult__dvd__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M4),N2) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_6343_power__Suc__less,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A2)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A2),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A2),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2))),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)) ) ) ).

% power_Suc_less
tff(fact_6344_power__Suc__less,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),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,A2),aa_nat_nat(power_power_nat(A2),N2))),aa_nat_nat(power_power_nat(A2),N2)) ) ) ).

% power_Suc_less
tff(fact_6345_power__Suc__less,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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,A2),aa_nat_int(power_power_int(A2),N2))),aa_nat_int(power_power_int(A2),N2)) ) ) ).

% power_Suc_less
tff(fact_6346_power__Suc__less,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ) ).

% power_Suc_less
tff(fact_6347_power__Suc__less,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),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,A2),aa_nat_rat(power_power_rat(A2),N2))),aa_nat_rat(power_power_rat(A2),N2)) ) ) ).

% power_Suc_less
tff(fact_6348_power__Suc__le__self,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A2)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A2),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),aa_nat_nat(suc,N2))),A2) ) ) ).

% power_Suc_le_self
tff(fact_6349_power__Suc__le__self,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N2))),A2) ) ) ).

% power_Suc_le_self
tff(fact_6350_power__Suc__le__self,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N2))),A2) ) ) ).

% power_Suc_le_self
tff(fact_6351_power__Suc__le__self,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(A2),aa_nat_nat(suc,N2))),A2) ) ) ).

% power_Suc_le_self
tff(fact_6352_power__Suc__le__self,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N2))),A2) ) ) ).

% power_Suc_le_self
tff(fact_6353_power__Suc__less__one,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A2)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A2),one_one_Code_natural)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),aa_nat_nat(suc,N2))),one_one_Code_natural) ) ) ).

% power_Suc_less_one
tff(fact_6354_power__Suc__less__one,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(A2),aa_nat_nat(suc,N2))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_6355_power__Suc__less__one,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N2))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_6356_power__Suc__less__one,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N2))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_6357_power__Suc__less__one,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N2))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_6358_power__strict__decreasing,axiom,
    ! [N2: nat,N: nat,A2: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,zero_z2226904508553997617atural),A2)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,A2),one_one_Code_natural)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N)),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_6359_power__strict__decreasing,axiom,
    ! [N2: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(A2),N)),aa_nat_nat(power_power_nat(A2),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_6360_power__strict__decreasing,axiom,
    ! [N2: nat,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_6361_power__strict__decreasing,axiom,
    ! [N2: nat,N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_6362_power__strict__decreasing,axiom,
    ! [N2: nat,N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N2)) ) ) ) ).

% power_strict_decreasing
tff(fact_6363_power__decreasing,axiom,
    ! [N2: nat,N: nat,A2: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),A2)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A2),one_one_Code_natural)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),N)),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)) ) ) ) ).

% power_decreasing
tff(fact_6364_power__decreasing,axiom,
    ! [N2: nat,N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ) ) ).

% power_decreasing
tff(fact_6365_power__decreasing,axiom,
    ! [N2: nat,N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N2)) ) ) ) ).

% power_decreasing
tff(fact_6366_power__decreasing,axiom,
    ! [N2: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(A2),N)),aa_nat_nat(power_power_nat(A2),N2)) ) ) ) ).

% power_decreasing
tff(fact_6367_power__decreasing,axiom,
    ! [N2: nat,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),N2)) ) ) ) ).

% power_decreasing
tff(fact_6368_power__le__imp__le__exp,axiom,
    ! [A2: code_natural,M4: nat,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(power_7079662738309270450atural(A2),M4)),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_6369_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M4: nat,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M4)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_6370_power__le__imp__le__exp,axiom,
    ! [A2: rat,M4: nat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),M4)),aa_nat_rat(power_power_rat(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_6371_power__le__imp__le__exp,axiom,
    ! [A2: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(A2),M4)),aa_nat_nat(power_power_nat(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_6372_power__le__imp__le__exp,axiom,
    ! [A2: int,M4: nat,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),M4)),aa_nat_int(power_power_int(A2),N2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2) ) ) ).

% power_le_imp_le_exp
tff(fact_6373_self__le__power,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ) ).

% self_le_power
tff(fact_6374_self__le__power,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,A2),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)) ) ) ).

% self_le_power
tff(fact_6375_self__le__power,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
     => ( 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,A2),aa_nat_rat(power_power_rat(A2),N2)) ) ) ).

% self_le_power
tff(fact_6376_self__le__power,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
     => ( 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,A2),aa_nat_nat(power_power_nat(A2),N2)) ) ) ).

% self_le_power
tff(fact_6377_self__le__power,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
     => ( 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,A2),aa_nat_int(power_power_int(A2),N2)) ) ) ).

% self_le_power
tff(fact_6378_one__less__power,axiom,
    ! [A2: code_natural,N2: nat] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
       => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le5570908160329646204atural,one_one_Code_natural),aa_nat_Code_natural(power_7079662738309270450atural(A2),N2)) ) ) ).

% one_less_power
tff(fact_6379_one__less__power,axiom,
    ! [A2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( 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(power_power_nat(A2),N2)) ) ) ).

% one_less_power
tff(fact_6380_one__less__power,axiom,
    ! [A2: int,N2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( 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(power_power_int(A2),N2)) ) ) ).

% one_less_power
tff(fact_6381_one__less__power,axiom,
    ! [A2: code_integer,N2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( 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(A2),N2)) ) ) ).

% one_less_power
tff(fact_6382_one__less__power,axiom,
    ! [A2: rat,N2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( 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(A2),N2)) ) ) ).

% one_less_power
tff(fact_6383_cINF__mono,axiom,
    ! [B4: set_Product_unit,F: fun_nat_nat,A: set_nat,G: fun_Product_unit_nat] :
      ( ( B4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ! [M2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_Product_unit_nat(G,M2)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6384_cINF__mono,axiom,
    ! [B4: set_o,F: fun_nat_nat,A: set_nat,G: fun_o_nat] :
      ( ( B4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ! [M2: $o] :
              ( aa_set_o_o(member_o((M2)),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_o_nat(G,(M2))) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6385_cINF__mono,axiom,
    ! [B4: set_nat,F: fun_nat_nat,A: set_nat,G: fun_nat_nat] :
      ( ( B4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ! [M2: nat] :
              ( aa_set_nat_o(member_nat(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(G,M2)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))) ) ) ) ).

% cINF_mono
tff(fact_6386_cINF__mono,axiom,
    ! [B4: set_int,F: fun_nat_nat,A: set_nat,G: fun_int_nat] :
      ( ( B4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ! [M2: int] :
              ( aa_set_int_o(member_int(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_int_nat(G,M2)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6387_cINF__mono,axiom,
    ! [B4: set_fu3555266933746557007_nat_o,F: fun_nat_nat,A: set_nat,G: fun_fu3504661237974713170_o_nat] :
      ( ( B4 != bot_bo8632255151191677115_nat_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ! [M2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_fun767828674014743683_o_nat(G,M2)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_5746879016480857722_o_nat(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6388_cINF__mono,axiom,
    ! [B4: set_Product_unit,F: fun_nat_set_nat,A: set_nat,G: fun_Pr1628945989756059814et_nat] :
      ( ( B4 != bot_bo3957492148770167129t_unit )
     => ( condit68592940725977398et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
       => ( ! [M2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,X4)),aa_Pro6064029160526250199et_nat(G,M2)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))),comple7806235888213564991et_nat(image_8173849991827986510et_nat(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6389_cINF__mono,axiom,
    ! [B4: set_o,F: fun_nat_set_nat,A: set_nat,G: fun_o_set_nat] :
      ( ( B4 != bot_bot_set_o )
     => ( condit68592940725977398et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
       => ( ! [M2: $o] :
              ( aa_set_o_o(member_o((M2)),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,X4)),aa_o_set_nat(G,(M2))) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))),comple7806235888213564991et_nat(image_o_set_nat(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6390_cINF__mono,axiom,
    ! [B4: set_nat,F: fun_nat_set_nat,A: set_nat,G: fun_nat_set_nat] :
      ( ( B4 != bot_bot_set_nat )
     => ( condit68592940725977398et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
       => ( ! [M2: nat] :
              ( aa_set_nat_o(member_nat(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,X4)),aa_nat_set_nat(G,M2)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))),comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(G),B4))) ) ) ) ).

% cINF_mono
tff(fact_6391_cINF__mono,axiom,
    ! [B4: set_int,F: fun_nat_set_nat,A: set_nat,G: fun_int_set_nat] :
      ( ( B4 != bot_bot_set_int )
     => ( condit68592940725977398et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
       => ( ! [M2: int] :
              ( aa_set_int_o(member_int(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,X4)),aa_int_set_nat(G,M2)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))),comple7806235888213564991et_nat(image_int_set_nat(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6392_cINF__mono,axiom,
    ! [B4: set_fu3555266933746557007_nat_o,F: fun_nat_set_nat,A: set_nat,G: fun_fu2098174926252251144et_nat] :
      ( ( B4 != bot_bo8632255151191677115_nat_o )
     => ( condit68592940725977398et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))
       => ( ! [M2: fun_Pr6460944699146269935_nat_o] :
              ( aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(M2),B4)
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A)
                  & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(F,X4)),aa_fun7872747175131507769et_nat(G,M2)) ) )
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A))),comple7806235888213564991et_nat(image_7230913979079189040et_nat(G,B4))) ) ) ) ).

% cINF_mono
tff(fact_6393_le__cINF__iff,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_int,U: int] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),complete_Inf_Inf_int(image_873079544045703924it_int(F,A)))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_Product_unit_int(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6394_le__cINF__iff,axiom,
    ! [A: set_o,F: fun_o_int,U: int] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int2(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),complete_Inf_Inf_int(image_o_int2(F,A)))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_o_int(F,(X2))) ) ) ) ) ).

% le_cINF_iff
tff(fact_6395_le__cINF__iff,axiom,
    ! [A: set_nat,F: fun_nat_int,U: int] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_nat_int2(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),complete_Inf_Inf_int(image_nat_int2(F,A)))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_nat_int(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6396_le__cINF__iff,axiom,
    ! [A: set_int,F: fun_int_int,U: int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_int_int2(F,A))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),complete_Inf_Inf_int(image_int_int2(F,A)))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_int_int(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6397_le__cINF__iff,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,U: nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(F,A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A)))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_Product_unit_nat(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6398_le__cINF__iff,axiom,
    ! [A: set_o,F: fun_o_nat,U: nat] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat2(F,A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,A)))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_o_nat(F,(X2))) ) ) ) ) ).

% le_cINF_iff
tff(fact_6399_le__cINF__iff,axiom,
    ! [A: set_nat,F: fun_nat_nat,U: nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_nat_nat(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6400_le__cINF__iff,axiom,
    ! [A: set_int,F: fun_int_nat,U: nat] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_int_nat2(F,A))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,A)))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_int_nat(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6401_le__cINF__iff,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,U: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_405062704495631173t_unit(F,A))
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,U),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A)))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,U),aa_Pro8294289767544744956t_unit(F,X2)) ) ) ) ) ).

% le_cINF_iff
tff(fact_6402_le__cINF__iff,axiom,
    ! [A: set_o,F: fun_o_Product_unit,U: product_unit] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_o_Product_unit2(F,A))
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,U),comple2584293577114468500t_unit(image_o_Product_unit2(F,A)))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,U),aa_o_Product_unit(F,(X2))) ) ) ) ) ).

% le_cINF_iff
tff(fact_6403_cInf__superset__mono,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_o_o(condit5413489452508810728elow_o,B4)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(complete_Inf_Inf_o,B4)),aa_set_o_o(complete_Inf_Inf_o,A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6404_cInf__superset__mono,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,B4)
       => ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),B4)
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,comple6093339926577406454_nat_o(B4)),comple6093339926577406454_nat_o(A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6405_cInf__superset__mono,axiom,
    ! [A: set_filter_nat,B4: set_filter_nat] :
      ( ( A != bot_bo498966703094740906er_nat )
     => ( condit3133331395780078992er_nat(B4)
       => ( ord_le2426478655948331894er_nat(A,B4)
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,comple6612786467464916249er_nat(B4)),comple6612786467464916249er_nat(A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6406_cInf__superset__mono,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,B4)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(B4)),complete_Inf_Inf_int(A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6407_cInf__superset__mono,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,B4)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,B4)),aa_set_nat_nat(complete_Inf_Inf_nat,A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6408_cInf__superset__mono,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( condit68592940725977398et_nat(B4)
       => ( ord_le6893508408891458716et_nat(A,B4)
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,comple7806235888213564991et_nat(B4)),comple7806235888213564991et_nat(A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6409_cInf__superset__mono,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,B4)
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(B4)),comple2584293577114468500t_unit(A)) ) ) ) ).

% cInf_superset_mono
tff(fact_6410_cInf__insert,axiom,
    ! [X5: set_o,A2: $o] :
      ( ( X5 != bot_bot_set_o )
     => ( aa_set_o_o(condit5413489452508810728elow_o,X5)
       => ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),X5))
        <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,(A2)),aa_set_o_o(complete_Inf_Inf_o,X5)) ) ) ) ).

% cInf_insert
tff(fact_6411_cInf__insert,axiom,
    ! [X5: set_int,A2: int] :
      ( ( X5 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,X5)
       => ( complete_Inf_Inf_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),X5)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),complete_Inf_Inf_int(X5)) ) ) ) ).

% cInf_insert
tff(fact_6412_cInf__insert,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( ( X5 != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,X5)
       => ( comple6093339926577406454_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),X5)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(A2),comple6093339926577406454_nat_o(X5)) ) ) ) ).

% cInf_insert
tff(fact_6413_cInf__insert,axiom,
    ! [X5: set_nat,A2: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,X5)
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),X5)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),aa_set_nat_nat(complete_Inf_Inf_nat,X5)) ) ) ) ).

% cInf_insert
tff(fact_6414_cInf__insert,axiom,
    ! [X5: set_set_nat,A2: set_nat] :
      ( ( X5 != bot_bot_set_set_nat )
     => ( condit68592940725977398et_nat(X5)
       => ( comple7806235888213564991et_nat(insert_set_nat(A2,X5)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),comple7806235888213564991et_nat(X5)) ) ) ) ).

% cInf_insert
tff(fact_6415_cInf__insert,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( ( X5 != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,X5)
       => ( comple2584293577114468500t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),X5)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A2),comple2584293577114468500t_unit(X5)) ) ) ) ).

% cInf_insert
tff(fact_6416_cInf__insert__If,axiom,
    ! [X5: set_o,A2: $o] :
      ( aa_set_o_o(condit5413489452508810728elow_o,X5)
     => ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),X5))
      <=> $ite(X5 = bot_bot_set_o,(A2),aa_o_o(aa_o_fun_o_o(inf_inf_o,(A2)),aa_set_o_o(complete_Inf_Inf_o,X5))) ) ) ).

% cInf_insert_If
tff(fact_6417_cInf__insert__If,axiom,
    ! [X5: set_int,A2: int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,X5)
     => ( complete_Inf_Inf_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),X5)) = $ite(X5 = bot_bot_set_int,A2,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),complete_Inf_Inf_int(X5))) ) ) ).

% cInf_insert_If
tff(fact_6418_cInf__insert__If,axiom,
    ! [X5: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,X5)
     => ( comple6093339926577406454_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),X5)) = $ite(X5 = bot_bo8632255151191677115_nat_o,A2,aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(A2),comple6093339926577406454_nat_o(X5))) ) ) ).

% cInf_insert_If
tff(fact_6419_cInf__insert__If,axiom,
    ! [X5: set_nat,A2: nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,X5)
     => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),X5)) = $ite(X5 = bot_bot_set_nat,A2,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),aa_set_nat_nat(complete_Inf_Inf_nat,X5))) ) ) ).

% cInf_insert_If
tff(fact_6420_cInf__insert__If,axiom,
    ! [X5: set_set_nat,A2: set_nat] :
      ( condit68592940725977398et_nat(X5)
     => ( comple7806235888213564991et_nat(insert_set_nat(A2,X5)) = $ite(X5 = bot_bot_set_set_nat,A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),comple7806235888213564991et_nat(X5))) ) ) ).

% cInf_insert_If
tff(fact_6421_cInf__insert__If,axiom,
    ! [X5: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,X5)
     => ( comple2584293577114468500t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),X5)) = $ite(X5 = bot_bo3957492148770167129t_unit,A2,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,A2),comple2584293577114468500t_unit(X5))) ) ) ).

% cInf_insert_If
tff(fact_6422_cInf__union__distrib,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_o_o(condit5413489452508810728elow_o,A)
       => ( ( B4 != bot_bot_set_o )
         => ( aa_set_o_o(condit5413489452508810728elow_o,B4)
           => ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))
            <=> aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(complete_Inf_Inf_o,A)),aa_set_o_o(complete_Inf_Inf_o,B4)) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6423_cInf__union__distrib,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,A)
       => ( ( B4 != bot_bot_set_int )
         => ( aa_set_int_o(condit1735850657277959132ow_int,B4)
           => ( complete_Inf_Inf_int(aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,complete_Inf_Inf_int(A)),complete_Inf_Inf_int(B4)) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6424_cInf__union__distrib,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( A != bot_bo8632255151191677115_nat_o )
     => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,A)
       => ( ( B4 != bot_bo8632255151191677115_nat_o )
         => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,B4)
           => ( comple6093339926577406454_nat_o(aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,A),B4)) = aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(comple6093339926577406454_nat_o(A)),comple6093339926577406454_nat_o(B4)) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6425_cInf__union__distrib,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A)
       => ( ( B4 != bot_bot_set_nat )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,B4)
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,A)),aa_set_nat_nat(complete_Inf_Inf_nat,B4)) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6426_cInf__union__distrib,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( condit68592940725977398et_nat(A)
       => ( ( B4 != bot_bot_set_set_nat )
         => ( condit68592940725977398et_nat(B4)
           => ( comple7806235888213564991et_nat(sup_sup_set_set_nat(A,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,comple7806235888213564991et_nat(A)),comple7806235888213564991et_nat(B4)) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6427_cInf__union__distrib,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,A)
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,B4)
           => ( comple2584293577114468500t_unit(aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple2584293577114468500t_unit(A)),comple2584293577114468500t_unit(B4)) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6428_dbl__inc__def,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).

% dbl_inc_def
tff(fact_6429_dbl__inc__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_6430_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_6431_dvd__mult__cancel1,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)),M4)
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_6432_dvd__mult__cancel2,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M4)),M4)
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_6433_dvd__minus__add,axiom,
    ! [Q3: nat,N2: nat,R2: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M4))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M4),aa_nat_nat(minus_minus_nat(N2),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M4),aa_nat_nat(plus_plus_nat(N2),aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M4)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_6434_cINF__superset__mono,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_int,B4: set_Product_unit,F: fun_Product_unit_int] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(G,B4))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),B4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(G,X3)),aa_Product_unit_int(F,X3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(image_873079544045703924it_int(G,B4))),complete_Inf_Inf_int(image_873079544045703924it_int(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6435_cINF__superset__mono,axiom,
    ! [A: set_o,G: fun_o_int,B4: set_o,F: fun_o_int] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int2(G,B4))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),B4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(G,(X3))),aa_o_int(F,(X3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(image_o_int2(G,B4))),complete_Inf_Inf_int(image_o_int2(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6436_cINF__superset__mono,axiom,
    ! [A: set_int,G: fun_int_int,B4: set_int,F: fun_int_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_int_int2(G,B4))
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),B4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(G,X3)),aa_int_int(F,X3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(image_int_int2(G,B4))),complete_Inf_Inf_int(image_int_int2(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6437_cINF__superset__mono,axiom,
    ! [A: set_nat,G: fun_nat_int,B4: set_nat,F: fun_nat_int] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_nat_int2(G,B4))
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),B4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(G,X3)),aa_nat_int(F,X3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,complete_Inf_Inf_int(image_nat_int2(G,B4))),complete_Inf_Inf_int(image_nat_int2(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6438_cINF__superset__mono,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_nat,B4: set_Product_unit,F: fun_Product_unit_nat] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(G,B4))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),B4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(G,X3)),aa_Product_unit_nat(F,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(G,B4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6439_cINF__superset__mono,axiom,
    ! [A: set_o,G: fun_o_nat,B4: set_o,F: fun_o_nat] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat2(G,B4))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),B4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(G,(X3))),aa_o_nat(F,(X3))) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(G,B4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6440_cINF__superset__mono,axiom,
    ! [A: set_int,G: fun_int_nat,B4: set_int,F: fun_int_nat] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_int_nat2(G,B4))
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),B4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(G,B4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6441_cINF__superset__mono,axiom,
    ! [A: set_nat,G: fun_nat_nat,B4: set_nat,F: fun_nat_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),B4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(G),B4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6442_cINF__superset__mono,axiom,
    ! [A: set_Product_unit,G: fun_Pr2191293291710260995t_unit,B4: set_Product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_405062704495631173t_unit(G,B4))
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),B4)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(G,X3)),aa_Pro8294289767544744956t_unit(F,X3)) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(image_405062704495631173t_unit(G,B4))),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6443_cINF__superset__mono,axiom,
    ! [A: set_o,G: fun_o_Product_unit,B4: set_o,F: fun_o_Product_unit] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_o_Product_unit2(G,B4))
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),B4)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_o_Product_unit(G,(X3))),aa_o_Product_unit(F,(X3))) )
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,comple2584293577114468500t_unit(image_o_Product_unit2(G,B4))),comple2584293577114468500t_unit(image_o_Product_unit2(F,A))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6444_less__eq__cInf__inter,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(condit5413489452508810728elow_o,A)
     => ( aa_set_o_o(condit5413489452508810728elow_o,B4)
       => ( ( inf_inf_set_o(A,B4) != bot_bot_set_o )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_o_o(aa_o_fun_o_o(inf_inf_o,aa_set_o_o(complete_Inf_Inf_o,A)),aa_set_o_o(complete_Inf_Inf_o,B4))),aa_set_o_o(complete_Inf_Inf_o,inf_inf_set_o(A,B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6445_less__eq__cInf__inter,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,A)
     => ( aa_set5091322549519198405at_o_o(condit270854758665545535_nat_o,B4)
       => ( ( inf_in3376754634429691197_nat_o(A,B4) != bot_bo8632255151191677115_nat_o )
         => aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(ord_le5018287362838994063_nat_o,aa_fun2638059832795814268_nat_o(inf_in4424744116413743581_nat_o(comple6093339926577406454_nat_o(A)),comple6093339926577406454_nat_o(B4))),comple6093339926577406454_nat_o(inf_in3376754634429691197_nat_o(A,B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6446_less__eq__cInf__inter,axiom,
    ! [A: set_filter_nat,B4: set_filter_nat] :
      ( condit3133331395780078992er_nat(A)
     => ( condit3133331395780078992er_nat(B4)
       => ( ( inf_in7395780696554941096er_nat(A,B4) != bot_bo498966703094740906er_nat )
         => aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,inf_inf_filter_nat(comple6612786467464916249er_nat(A),comple6612786467464916249er_nat(B4))),comple6612786467464916249er_nat(inf_in7395780696554941096er_nat(A,B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6447_less__eq__cInf__inter,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,A)
     => ( aa_set_int_o(condit1735850657277959132ow_int,B4)
       => ( ( inf_inf_set_int(A,B4) != bot_bot_set_int )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,complete_Inf_Inf_int(A)),complete_Inf_Inf_int(B4))),complete_Inf_Inf_int(inf_inf_set_int(A,B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6448_less__eq__cInf__inter,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,A)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,A)),aa_set_nat_nat(complete_Inf_Inf_nat,B4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6449_less__eq__cInf__inter,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( condit68592940725977398et_nat(A)
     => ( condit68592940725977398et_nat(B4)
       => ( ( inf_inf_set_set_nat(A,B4) != bot_bot_set_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,comple7806235888213564991et_nat(A)),comple7806235888213564991et_nat(B4))),comple7806235888213564991et_nat(inf_inf_set_set_nat(A,B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6450_less__eq__cInf__inter,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,A)
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) != bot_bo3957492148770167129t_unit )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple2584293577114468500t_unit(A)),comple2584293577114468500t_unit(B4))),comple2584293577114468500t_unit(inf_in4660618365625256667t_unit(A,B4))) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6451_cINF__insert,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,A2: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(F,A))
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_Product_unit_nat(F,A2)),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6452_cINF__insert,axiom,
    ! [A: set_o,F: fun_o_nat,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat2(F,A))
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_o_nat(F,(A2))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6453_cINF__insert,axiom,
    ! [A: set_nat,F: fun_nat_nat,A2: nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(F,A2)),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))) ) ) ) ).

% cINF_insert
tff(fact_6454_cINF__insert,axiom,
    ! [A: set_int,F: fun_int_nat,A2: int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_int_nat2(F,A))
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_int_nat(F,A2)),aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6455_cINF__insert,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,A2: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_405062704495631173t_unit(F,A))
       => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(F,A2)),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6456_cINF__insert,axiom,
    ! [A: set_o,F: fun_o_Product_unit,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_o_Product_unit2(F,A))
       => ( comple2584293577114468500t_unit(image_o_Product_unit2(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_o_Product_unit(F,(A2))),comple2584293577114468500t_unit(image_o_Product_unit2(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6457_cINF__insert,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit,A2: nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_8730104196221521654t_unit(F,A))
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_nat_Product_unit(F,A2)),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6458_cINF__insert,axiom,
    ! [A: set_int,F: fun_int_Product_unit,A2: int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_5846526143423701914t_unit(F,A))
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_int_Product_unit(F,A2)),comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6459_cINF__insert,axiom,
    ! [A: set_Product_unit,F: fun_Pr1628945989756059814et_nat,A2: product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,A))
       => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_Pro6064029160526250199et_nat(F,A2)),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6460_cINF__insert,axiom,
    ! [A: set_o,F: fun_o_set_nat,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( condit68592940725977398et_nat(image_o_set_nat(F,A))
       => ( comple7806235888213564991et_nat(image_o_set_nat(F,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_o_set_nat(F,(A2))),comple7806235888213564991et_nat(image_o_set_nat(F,A))) ) ) ) ).

% cINF_insert
tff(fact_6461_power__eq__if,axiom,
    ! [P2: code_integer,M4: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(P2),M4) = $ite(M4 = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P2),aa_nat_Code_integer(power_8256067586552552935nteger(P2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))) ).

% power_eq_if
tff(fact_6462_power__eq__if,axiom,
    ! [P2: code_natural,M4: nat] :
      aa_nat_Code_natural(power_7079662738309270450atural(P2),M4) = $ite(M4 = zero_zero_nat,one_one_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,P2),aa_nat_Code_natural(power_7079662738309270450atural(P2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))) ).

% power_eq_if
tff(fact_6463_power__eq__if,axiom,
    ! [P2: assn,M4: nat] :
      aa_nat_assn(power_power_assn(P2),M4) = $ite(M4 = zero_zero_nat,one_one_assn,aa_assn_assn(times_times_assn(P2),aa_nat_assn(power_power_assn(P2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))) ).

% power_eq_if
tff(fact_6464_power__eq__if,axiom,
    ! [P2: rat,M4: nat] :
      aa_nat_rat(power_power_rat(P2),M4) = $ite(M4 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))) ).

% power_eq_if
tff(fact_6465_power__eq__if,axiom,
    ! [P2: nat,M4: nat] :
      aa_nat_nat(power_power_nat(P2),M4) = $ite(M4 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(power_power_nat(P2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))) ).

% power_eq_if
tff(fact_6466_power__eq__if,axiom,
    ! [P2: int,M4: nat] :
      aa_nat_int(power_power_int(P2),M4) = $ite(M4 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))) ).

% power_eq_if
tff(fact_6467_power__minus__mult,axiom,
    ! [N2: nat,A2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),aa_nat_nat(minus_minus_nat(N2),one_one_nat))),A2) = aa_nat_assn(power_power_assn(A2),N2) ) ) ).

% power_minus_mult
tff(fact_6468_power__minus__mult,axiom,
    ! [N2: nat,A2: 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(A2),aa_nat_nat(minus_minus_nat(N2),one_one_nat))),A2) = aa_nat_rat(power_power_rat(A2),N2) ) ) ).

% power_minus_mult
tff(fact_6469_power__minus__mult,axiom,
    ! [N2: nat,A2: 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(power_power_nat(A2),aa_nat_nat(minus_minus_nat(N2),one_one_nat))),A2) = aa_nat_nat(power_power_nat(A2),N2) ) ) ).

% power_minus_mult
tff(fact_6470_power__minus__mult,axiom,
    ! [N2: nat,A2: 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(power_power_int(A2),aa_nat_nat(minus_minus_nat(N2),one_one_nat))),A2) = aa_nat_int(power_power_int(A2),N2) ) ) ).

% power_minus_mult
tff(fact_6471_cINF__union,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(F,A))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(F,B4))
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6472_cINF__union,axiom,
    ! [A: set_o,F: fun_o_nat,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat2(F,A))
       => ( ( B4 != bot_bot_set_o )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat2(F,B4))
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat2(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6473_cINF__union,axiom,
    ! [A: set_nat,F: fun_nat_nat,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))
       => ( ( B4 != bot_bot_set_nat )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat2(F),B4))
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),A))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat2(F),B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6474_cINF__union,axiom,
    ! [A: set_int,F: fun_int_nat,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_int_nat2(F,A))
       => ( ( B4 != bot_bot_set_int )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_int_nat2(F,B4))
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,A))),aa_set_nat_nat(complete_Inf_Inf_nat,image_int_nat2(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6475_cINF__union,axiom,
    ! [A: set_Product_unit,F: fun_Pr2191293291710260995t_unit,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_405062704495631173t_unit(F,A))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_405062704495631173t_unit(F,B4))
           => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A))),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6476_cINF__union,axiom,
    ! [A: set_o,F: fun_o_Product_unit,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_o_Product_unit2(F,A))
       => ( ( B4 != bot_bot_set_o )
         => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_o_Product_unit2(F,B4))
           => ( comple2584293577114468500t_unit(image_o_Product_unit2(F,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple2584293577114468500t_unit(image_o_Product_unit2(F,A))),comple2584293577114468500t_unit(image_o_Product_unit2(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6477_cINF__union,axiom,
    ! [A: set_nat,F: fun_nat_Product_unit,B4: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_8730104196221521654t_unit(F,A))
       => ( ( B4 != bot_bot_set_nat )
         => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_8730104196221521654t_unit(F,B4))
           => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A))),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6478_cINF__union,axiom,
    ! [A: set_int,F: fun_int_Product_unit,B4: set_int] :
      ( ( A != bot_bot_set_int )
     => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_5846526143423701914t_unit(F,A))
       => ( ( B4 != bot_bot_set_int )
         => ( aa_set8780210009671629607unit_o(condit9161856309968323677t_unit,image_5846526143423701914t_unit(F,B4))
           => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4))) = aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A))),comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6479_cINF__union,axiom,
    ! [A: set_Product_unit,F: fun_Pr1628945989756059814et_nat,B4: set_Product_unit] :
      ( ( A != bot_bo3957492148770167129t_unit )
     => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,A))
       => ( ( B4 != bot_bo3957492148770167129t_unit )
         => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,B4))
           => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A))),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6480_cINF__union,axiom,
    ! [A: set_o,F: fun_o_set_nat,B4: set_o] :
      ( ( A != bot_bot_set_o )
     => ( condit68592940725977398et_nat(image_o_set_nat(F,A))
       => ( ( B4 != bot_bot_set_o )
         => ( condit68592940725977398et_nat(image_o_set_nat(F,B4))
           => ( comple7806235888213564991et_nat(image_o_set_nat(F,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4))) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,comple7806235888213564991et_nat(image_o_set_nat(F,A))),comple7806235888213564991et_nat(image_o_set_nat(F,B4))) ) ) ) ) ) ).

% cINF_union
tff(fact_6481_sum_Oinsert,axiom,
    ! [A: set_int,X: int,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(G,X)),groups7873554091576472773nteger(G,A)) ) ) ) ).

% sum.insert
tff(fact_6482_sum_Oinsert,axiom,
    ! [A: set_o,X: $o,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( groups4406642042086082107nteger(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(G,(X))),groups4406642042086082107nteger(G,A)) ) ) ) ).

% sum.insert
tff(fact_6483_sum_Oinsert,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups5393439102872367408nteger(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Pro1705297278780387758nteger(G,X)),groups5393439102872367408nteger(G,A)) ) ) ) ).

% sum.insert
tff(fact_6484_sum_Oinsert,axiom,
    ! [A: set_nat,X: nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( groups7501900531339628137nteger(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(G,X)),groups7501900531339628137nteger(G,A)) ) ) ) ).

% sum.insert
tff(fact_6485_sum_Oinsert,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups879477027807139574nteger(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(G,X)),groups879477027807139574nteger(G,A)) ) ) ) ).

% sum.insert
tff(fact_6486_sum_Oinsert,axiom,
    ! [A: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_6487_sum_Oinsert,axiom,
    ! [A: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X))),groups7872700643590313910_o_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_6488_sum_Oinsert,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups8131667117488461483it_rat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Product_unit_rat(G,X)),groups8131667117488461483it_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_6489_sum_Oinsert,axiom,
    ! [A: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X)),groups2906978787729119204at_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_6490_sum_Oinsert,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups6602215022474089585er_rat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(G,X)),groups6602215022474089585er_rat(G,A)) ) ) ) ).

% sum.insert
tff(fact_6491_sum_Oempty,axiom,
    ! [G: fun_Product_unit_nat] : groups8766797177574957219it_nat(G,bot_bo3957492148770167129t_unit) = zero_zero_nat ).

% sum.empty
tff(fact_6492_sum_Oempty,axiom,
    ! [G: fun_Product_unit_int] : groups8764306707065906943it_int(G,bot_bo3957492148770167129t_unit) = zero_zero_int ).

% sum.empty
tff(fact_6493_sum_Oempty,axiom,
    ! [G: fun_Pr1820225556744648189nteger] : groups5393439102872367408nteger(G,bot_bo3957492148770167129t_unit) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6494_sum_Oempty,axiom,
    ! [G: fun_o_nat] : groups8507830703676809646_o_nat(G,bot_bot_set_o) = zero_zero_nat ).

% sum.empty
tff(fact_6495_sum_Oempty,axiom,
    ! [G: fun_o_int] : groups8505340233167759370_o_int(G,bot_bot_set_o) = zero_zero_int ).

% sum.empty
tff(fact_6496_sum_Oempty,axiom,
    ! [G: fun_o_Code_integer] : groups4406642042086082107nteger(G,bot_bot_set_o) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6497_sum_Oempty,axiom,
    ! [G: fun_nat_int] : aa_set_nat_int(groups3539618377306564664at_int(G),bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_6498_sum_Oempty,axiom,
    ! [G: fun_nat_Code_integer] : groups7501900531339628137nteger(G,bot_bot_set_nat) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6499_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_6500_sum_Oempty,axiom,
    ! [G: fun_int_Code_integer] : groups7873554091576472773nteger(G,bot_bot_set_int) = zero_z3403309356797280102nteger ).

% sum.empty
tff(fact_6501_member__le__sum,axiom,
    ! [I2: code_integer,A: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),A)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,I2),bot_bo3990330152332043303nteger)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,X3)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F,I2)),groups879477027807139574nteger(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6502_member__le__sum,axiom,
    ! [I2: product_unit,A: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(I2),A)
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,I2),bot_bo3957492148770167129t_unit)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,X3)) )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro1705297278780387758nteger(F,I2)),groups5393439102872367408nteger(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6503_member__le__sum,axiom,
    ! [I2: $o,A: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(member_o((I2)),A)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(I2)),bot_bot_set_o)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(I2))),groups4406642042086082107nteger(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6504_member__le__sum,axiom,
    ! [I2: int,A: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(member_int(I2),A)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,I2),bot_bot_set_int)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X3)) )
       => ( aa_set_int_o(finite_finite_int,A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I2)),groups7873554091576472773nteger(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6505_member__le__sum,axiom,
    ! [I2: nat,A: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(member_nat(I2),A)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,I2),bot_bot_set_nat)))
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X3)) )
       => ( aa_set_nat_o(finite_finite_nat,A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I2)),groups7501900531339628137nteger(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6506_member__le__sum,axiom,
    ! [I2: code_integer,A: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I2),A)
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(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(F,X3)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,I2)),groups6602215022474089585er_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6507_member__le__sum,axiom,
    ! [I2: product_unit,A: set_Product_unit,F: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(member_Product_unit(I2),A)
     => ( ! [X3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(X3),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,I2),bot_bo3957492148770167129t_unit)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Product_unit_rat(F,X3)) )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Product_unit_rat(F,I2)),groups8131667117488461483it_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6508_member__le__sum,axiom,
    ! [I2: $o,A: set_o,F: fun_o_rat] :
      ( aa_set_o_o(member_o((I2)),A)
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(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(F,(X3))) )
       => ( aa_set_o_o(finite_finite_o,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I2))),groups7872700643590313910_o_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6509_member__le__sum,axiom,
    ! [I2: int,A: set_int,F: fun_int_rat] :
      ( aa_set_int_o(member_int(I2),A)
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(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(F,X3)) )
       => ( aa_set_int_o(finite_finite_int,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),groups3906332499630173760nt_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6510_member__le__sum,axiom,
    ! [I2: nat,A: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(I2),A)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(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(F,X3)) )
       => ( aa_set_nat_o(finite_finite_nat,A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),groups2906978787729119204at_rat(F,A)) ) ) ) ).

% member_le_sum
tff(fact_6511_sum_Ounion__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( groups879477027807139574nteger(G,sup_su848401254843788991nteger(A,B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups879477027807139574nteger(G,A)),groups879477027807139574nteger(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6512_sum_Ounion__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( groups6602215022474089585er_rat(G,sup_su848401254843788991nteger(A,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups6602215022474089585er_rat(G,A)),groups6602215022474089585er_rat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6513_sum_Ounion__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( groups7237345082560585321er_nat(G,sup_su848401254843788991nteger(A,B4)) = aa_nat_nat(plus_plus_nat(groups7237345082560585321er_nat(G,A)),groups7237345082560585321er_nat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6514_sum_Ounion__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( groups7234854612051535045er_int(G,sup_su848401254843788991nteger(A,B4)) = aa_int_int(plus_plus_int(groups7234854612051535045er_int(G,A)),groups7234854612051535045er_int(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6515_sum_Ounion__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,G: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( groups5393439102872367408nteger(G,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups5393439102872367408nteger(G,A)),groups5393439102872367408nteger(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6516_sum_Ounion__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( groups8131667117488461483it_rat(G,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups8131667117488461483it_rat(G,A)),groups8131667117488461483it_rat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6517_sum_Ounion__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,G: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( groups8766797177574957219it_nat(G,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_nat_nat(plus_plus_nat(groups8766797177574957219it_nat(G,A)),groups8766797177574957219it_nat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6518_sum_Ounion__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,G: fun_Product_unit_int] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( groups8764306707065906943it_int(G,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_int_int(plus_plus_int(groups8764306707065906943it_int(G,A)),groups8764306707065906943it_int(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6519_sum_Ounion__disjoint,axiom,
    ! [A: set_o,B4: set_o,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
         => ( groups4406642042086082107nteger(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,groups4406642042086082107nteger(G,A)),groups4406642042086082107nteger(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6520_sum_Ounion__disjoint,axiom,
    ! [A: set_o,B4: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
         => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups7872700643590313910_o_rat(G,A)),groups7872700643590313910_o_rat(G,B4)) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6521_sum__diff1,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_rat,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups6602215022474089585er_rat(F,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),A),aa_rat_rat(minus_minus_rat(groups6602215022474089585er_rat(F,A)),aa_Code_integer_rat(F,A2)),groups6602215022474089585er_rat(F,A)) ) ) ).

% sum_diff1
tff(fact_6522_sum__diff1,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_rat,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8131667117488461483it_rat(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(A2),A),aa_rat_rat(minus_minus_rat(groups8131667117488461483it_rat(F,A)),aa_Product_unit_rat(F,A2)),groups8131667117488461483it_rat(F,A)) ) ) ).

% sum_diff1
tff(fact_6523_sum__diff1,axiom,
    ! [A: set_o,F: fun_o_rat,A2: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups7872700643590313910_o_rat(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A2)),A),aa_rat_rat(minus_minus_rat(groups7872700643590313910_o_rat(F,A)),aa_o_rat(F,(A2))),groups7872700643590313910_o_rat(F,A)) ) ) ).

% sum_diff1
tff(fact_6524_sum__diff1,axiom,
    ! [A: set_int,F: fun_int_rat,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A2),A),aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A)),aa_int_rat(F,A2)),groups3906332499630173760nt_rat(F,A)) ) ) ).

% sum_diff1
tff(fact_6525_sum__diff1,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_int,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups7234854612051535045er_int(F,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),A),aa_int_int(minus_minus_int(groups7234854612051535045er_int(F,A)),aa_Code_integer_int(F,A2)),groups7234854612051535045er_int(F,A)) ) ) ).

% sum_diff1
tff(fact_6526_sum__diff1,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_int,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8764306707065906943it_int(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(A2),A),aa_int_int(minus_minus_int(groups8764306707065906943it_int(F,A)),aa_Product_unit_int(F,A2)),groups8764306707065906943it_int(F,A)) ) ) ).

% sum_diff1
tff(fact_6527_sum__diff1,axiom,
    ! [A: set_o,F: fun_o_int,A2: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups8505340233167759370_o_int(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A2)),A),aa_int_int(minus_minus_int(groups8505340233167759370_o_int(F,A)),aa_o_int(F,(A2))),groups8505340233167759370_o_int(F,A)) ) ) ).

% sum_diff1
tff(fact_6528_sum__diff1,axiom,
    ! [A: set_nat,F: fun_nat_rat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups2906978787729119204at_rat(F,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A2),A),aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A)),aa_nat_rat(F,A2)),groups2906978787729119204at_rat(F,A)) ) ) ).

% sum_diff1
tff(fact_6529_sum__diff1,axiom,
    ! [A: set_nat,F: fun_nat_int,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A2),A),aa_int_int(minus_minus_int(aa_set_nat_int(groups3539618377306564664at_int(F),A)),aa_nat_int(F,A2)),aa_set_nat_int(groups3539618377306564664at_int(F),A)) ) ) ).

% sum_diff1
tff(fact_6530_sum__diff1,axiom,
    ! [A: set_int,F: fun_int_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups4538972089207619220nt_int,F),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A2),A),aa_int_int(minus_minus_int(aa_set_int_int(aa_fun5337148683239322782nt_int(groups4538972089207619220nt_int,F),A)),aa_int_int(F,A2)),aa_set_int_int(aa_fun5337148683239322782nt_int(groups4538972089207619220nt_int,F),A)) ) ) ).

% sum_diff1
tff(fact_6531_sum_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups879477027807139574nteger(G,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(G,X)),groups879477027807139574nteger(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_6532_sum_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups6602215022474089585er_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(G,X)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_6533_sum_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups7237345082560585321er_nat(G,A) = aa_nat_nat(plus_plus_nat(aa_Code_integer_nat(G,X)),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_6534_sum_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups7234854612051535045er_int(G,A) = aa_int_int(plus_plus_int(aa_Code_integer_int(G,X)),groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_6535_sum_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups5393439102872367408nteger(G,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Pro1705297278780387758nteger(G,X)),groups5393439102872367408nteger(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% sum.remove
tff(fact_6536_sum_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups8131667117488461483it_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Product_unit_rat(G,X)),groups8131667117488461483it_rat(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% sum.remove
tff(fact_6537_sum_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups8766797177574957219it_nat(G,A) = aa_nat_nat(plus_plus_nat(aa_Product_unit_nat(G,X)),groups8766797177574957219it_nat(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% sum.remove
tff(fact_6538_sum_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Product_unit_int] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups8764306707065906943it_int(G,A) = aa_int_int(plus_plus_int(aa_Product_unit_int(G,X)),groups8764306707065906943it_int(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% sum.remove
tff(fact_6539_sum_Oremove,axiom,
    ! [A: set_o,X: $o,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( groups4406642042086082107nteger(G,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(G,(X))),groups4406642042086082107nteger(G,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_6540_sum_Oremove,axiom,
    ! [A: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( groups7872700643590313910_o_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ) ).

% sum.remove
tff(fact_6541_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Co8499238673658658103nteger,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups879477027807139574nteger(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(G,X)),groups879477027807139574nteger(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_6542_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups6602215022474089585er_rat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(G,X)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_6543_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_nat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups7237345082560585321er_nat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_nat_nat(plus_plus_nat(aa_Code_integer_nat(G,X)),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_6544_sum_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_int,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups7234854612051535045er_int(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_int_int(plus_plus_int(aa_Code_integer_int(G,X)),groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_6545_sum_Oinsert__remove,axiom,
    ! [A: set_Product_unit,G: fun_Pr1820225556744648189nteger,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups5393439102872367408nteger(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Pro1705297278780387758nteger(G,X)),groups5393439102872367408nteger(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% sum.insert_remove
tff(fact_6546_sum_Oinsert__remove,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_rat,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8131667117488461483it_rat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Product_unit_rat(G,X)),groups8131667117488461483it_rat(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% sum.insert_remove
tff(fact_6547_sum_Oinsert__remove,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_nat,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8766797177574957219it_nat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_nat_nat(plus_plus_nat(aa_Product_unit_nat(G,X)),groups8766797177574957219it_nat(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% sum.insert_remove
tff(fact_6548_sum_Oinsert__remove,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_int,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8764306707065906943it_int(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_int_int(plus_plus_int(aa_Product_unit_int(G,X)),groups8764306707065906943it_int(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% sum.insert_remove
tff(fact_6549_sum_Oinsert__remove,axiom,
    ! [A: set_o,G: fun_o_Code_integer,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups4406642042086082107nteger(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(G,(X))),groups4406642042086082107nteger(G,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_6550_sum_Oinsert__remove,axiom,
    ! [A: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X))),groups7872700643590313910_o_rat(G,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ).

% sum.insert_remove
tff(fact_6551_sum__strict__mono,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X3)),aa_Code_integer_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(F,A)),groups7237345082560585321er_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6552_sum__strict__mono,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,G: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Product_unit_nat(F,X3)),aa_Product_unit_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8766797177574957219it_nat(F,A)),groups8766797177574957219it_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6553_sum__strict__mono,axiom,
    ! [A: set_o,F: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(X3))),aa_o_nat(G,(X3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A)),groups8507830703676809646_o_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6554_sum__strict__mono,axiom,
    ! [A: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A)),groups4541462559716669496nt_nat(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6555_sum__strict__mono,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_int,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(F,X3)),aa_Code_integer_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups7234854612051535045er_int(F,A)),groups7234854612051535045er_int(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6556_sum__strict__mono,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_int,G: fun_Product_unit_int] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Product_unit_int(F,X3)),aa_Product_unit_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8764306707065906943it_int(F,A)),groups8764306707065906943it_int(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6557_sum__strict__mono,axiom,
    ! [A: set_o,F: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(X3))),aa_o_int(G,(X3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(F,A)),groups8505340233167759370_o_int(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6558_sum__strict__mono,axiom,
    ! [A: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A)),aa_set_nat_int(groups3539618377306564664at_int(G),A)) ) ) ) ).

% sum_strict_mono
tff(fact_6559_sum__strict__mono,axiom,
    ! [A: set_Code_integer,F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,X3)),aa_Cod1783974652533396728nteger(G,X3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups879477027807139574nteger(F,A)),groups879477027807139574nteger(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6560_sum__strict__mono,axiom,
    ! [A: set_Product_unit,F: fun_Pr1820225556744648189nteger,G: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Pro1705297278780387758nteger(F,X3)),aa_Pro1705297278780387758nteger(G,X3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups5393439102872367408nteger(F,A)),groups5393439102872367408nteger(G,A)) ) ) ) ).

% sum_strict_mono
tff(fact_6561_sum_Oinsert__if,axiom,
    ! [A: set_int,G: fun_int_Code_integer,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups7873554091576472773nteger(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(aa_set_int_o(member_int(X),A),groups7873554091576472773nteger(G,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_int_Code_integer(G,X)),groups7873554091576472773nteger(G,A))) ) ) ).

% sum.insert_if
tff(fact_6562_sum_Oinsert__if,axiom,
    ! [A: set_o,G: fun_o_Code_integer,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups4406642042086082107nteger(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = $ite(aa_set_o_o(member_o((X)),A),groups4406642042086082107nteger(G,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_o_Code_integer(G,(X))),groups4406642042086082107nteger(G,A))) ) ) ).

% sum.insert_if
tff(fact_6563_sum_Oinsert__if,axiom,
    ! [A: set_Product_unit,G: fun_Pr1820225556744648189nteger,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups5393439102872367408nteger(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),groups5393439102872367408nteger(G,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Pro1705297278780387758nteger(G,X)),groups5393439102872367408nteger(G,A))) ) ) ).

% sum.insert_if
tff(fact_6564_sum_Oinsert__if,axiom,
    ! [A: set_nat,G: fun_nat_Code_integer,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups7501900531339628137nteger(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_o(member_nat(X),A),groups7501900531339628137nteger(G,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(G,X)),groups7501900531339628137nteger(G,A))) ) ) ).

% sum.insert_if
tff(fact_6565_sum_Oinsert__if,axiom,
    ! [A: set_Code_integer,G: fun_Co8499238673658658103nteger,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups879477027807139574nteger(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A),groups879477027807139574nteger(G,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(G,X)),groups879477027807139574nteger(G,A))) ) ) ).

% sum.insert_if
tff(fact_6566_sum_Oinsert__if,axiom,
    ! [A: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups3906332499630173760nt_rat(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(aa_set_int_o(member_int(X),A),groups3906332499630173760nt_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(G,X)),groups3906332499630173760nt_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_6567_sum_Oinsert__if,axiom,
    ! [A: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = $ite(aa_set_o_o(member_o((X)),A),groups7872700643590313910_o_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_o_rat(G,(X))),groups7872700643590313910_o_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_6568_sum_Oinsert__if,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_rat,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8131667117488461483it_rat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),groups8131667117488461483it_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Product_unit_rat(G,X)),groups8131667117488461483it_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_6569_sum_Oinsert__if,axiom,
    ! [A: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_o(member_nat(X),A),groups2906978787729119204at_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,X)),groups2906978787729119204at_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_6570_sum_Oinsert__if,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups6602215022474089585er_rat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A),groups6602215022474089585er_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Code_integer_rat(G,X)),groups6602215022474089585er_rat(G,A))) ) ) ).

% sum.insert_if
tff(fact_6571_sum__pos,axiom,
    ! [I: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( ( I != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups7237345082560585321er_nat(F,I)) ) ) ) ).

% sum_pos
tff(fact_6572_sum__pos,axiom,
    ! [I: set_Product_unit,F: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( ( I != bot_bo3957492148770167129t_unit )
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Product_unit_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8766797177574957219it_nat(F,I)) ) ) ) ).

% sum_pos
tff(fact_6573_sum__pos,axiom,
    ! [I: set_o,F: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,I)
     => ( ( I != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(F,(I3))) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups8507830703676809646_o_nat(F,I)) ) ) ) ).

% sum_pos
tff(fact_6574_sum__pos,axiom,
    ! [I: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I)
     => ( ( I != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I)) ) ) ) ).

% sum_pos
tff(fact_6575_sum__pos,axiom,
    ! [I: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( ( I != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Code_integer_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups7234854612051535045er_int(F,I)) ) ) ) ).

% sum_pos
tff(fact_6576_sum__pos,axiom,
    ! [I: set_Product_unit,F: fun_Product_unit_int] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( ( I != bot_bo3957492148770167129t_unit )
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Product_unit_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups8764306707065906943it_int(F,I)) ) ) ) ).

% sum_pos
tff(fact_6577_sum__pos,axiom,
    ! [I: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I)
     => ( ( I != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(F,(I3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups8505340233167759370_o_int(F,I)) ) ) ) ).

% sum_pos
tff(fact_6578_sum__pos,axiom,
    ! [I: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I)
     => ( ( I != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_set_nat_int(groups3539618377306564664at_int(F),I)) ) ) ) ).

% sum_pos
tff(fact_6579_sum__pos,axiom,
    ! [I: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( ( I != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups879477027807139574nteger(F,I)) ) ) ) ).

% sum_pos
tff(fact_6580_sum__pos,axiom,
    ! [I: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( ( I != bot_bo3957492148770167129t_unit )
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),groups5393439102872367408nteger(F,I)) ) ) ) ).

% sum_pos
tff(fact_6581_sum_OUnion__disjoint,axiom,
    ! [C5: set_set_nat,G: fun_nat_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,C5)
         => aa_set_nat_o(finite_finite_nat,X3) )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,C5)
           => ! [Xa4: set_nat] :
                ( member_set_nat(Xa4,C5)
               => ( ( X3 != Xa4 )
                 => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Xa4) = bot_bot_set_nat ) ) ) )
       => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,G),comple7399068483239264473et_nat(C5)) = aa_set_set_nat_nat(comp_f5269661881483994078at_nat(groups8294997508430121362at_nat,groups3542108847815614940at_nat,G),C5) ) ) ) ).

% sum.Union_disjoint
tff(fact_6582_sum_OUnion__disjoint,axiom,
    ! [C5: set_set_int,G: fun_int_int] :
      ( ! [X3: set_int] :
          ( member_set_int(X3,C5)
         => aa_set_int_o(finite_finite_int,X3) )
     => ( ! [X3: set_int] :
            ( member_set_int(X3,C5)
           => ! [Xa4: set_int] :
                ( member_set_int(Xa4,C5)
               => ( ( X3 != Xa4 )
                 => ( inf_inf_set_int(X3,Xa4) = bot_bot_set_int ) ) ) )
       => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups4538972089207619220nt_int,G),comple3221217463730067765et_int(C5)) = aa_set_set_int_int(comp_f854714860224386566nt_int(groups1256056575759317066nt_int,groups4538972089207619220nt_int,G),C5) ) ) ) ).

% sum.Union_disjoint
tff(fact_6583_sum__diff1__nat,axiom,
    ! [F: fun_Product_unit_nat,A: set_Product_unit,A2: product_unit] :
      groups8766797177574957219it_nat(F,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),bot_bo3957492148770167129t_unit))) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(A2),A),aa_nat_nat(minus_minus_nat(groups8766797177574957219it_nat(F,A)),aa_Product_unit_nat(F,A2)),groups8766797177574957219it_nat(F,A)) ).

% sum_diff1_nat
tff(fact_6584_sum__diff1__nat,axiom,
    ! [F: fun_o_nat,A: set_o,A2: $o] :
      groups8507830703676809646_o_nat(F,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A2)),A),aa_nat_nat(minus_minus_nat(groups8507830703676809646_o_nat(F,A)),aa_o_nat(F,(A2))),groups8507830703676809646_o_nat(F,A)) ).

% sum_diff1_nat
tff(fact_6585_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A: set_int,A2: int] :
      groups4541462559716669496nt_nat(F,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A2),A),aa_nat_nat(minus_minus_nat(groups4541462559716669496nt_nat(F,A)),aa_int_nat(F,A2)),groups4541462559716669496nt_nat(F,A)) ).

% sum_diff1_nat
tff(fact_6586_sum__diff1__nat,axiom,
    ! [F: fun_fu3504661237974713170_o_nat,A: set_fu3555266933746557007_nat_o,A2: fun_Pr6460944699146269935_nat_o] :
      groups2076417071786835717_o_nat(F,minus_6456595905716665654_nat_o(A,aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(A2),bot_bo8632255151191677115_nat_o))) = $ite(aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(A2),A),aa_nat_nat(minus_minus_nat(groups2076417071786835717_o_nat(F,A)),aa_fun767828674014743683_o_nat(F,A2)),groups2076417071786835717_o_nat(F,A)) ).

% sum_diff1_nat
tff(fact_6587_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A: set_nat,A2: nat] :
      aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,F),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A2),A),aa_nat_nat(minus_minus_nat(aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,F),A)),aa_nat_nat(F,A2)),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,F),A)) ).

% sum_diff1_nat
tff(fact_6588_bezout__add__strong__nat,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D: nat,X3: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A2)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D) ) ) ) ).

% bezout_add_strong_nat
tff(fact_6589_bezout1__nat,axiom,
    ! [A2: nat,B: nat] :
    ? [D: nat,X3: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A2)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
      & ( ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = D )
        | ( aa_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,A2),Y2)) = D ) ) ) ).

% bezout1_nat
tff(fact_6590_bezout__add__nat,axiom,
    ! [A2: nat,B: nat] :
    ? [D: nat,X3: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A2)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y2)),D) ) ) ) ).

% bezout_add_nat
tff(fact_6591_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X) = aa_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(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y)),D2) ) )
         => ? [X3: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A2)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(plus_plus_nat(A2),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A2),B)),Y2)),D2) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(plus_plus_nat(A2),B)),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y2)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_6592_dvd__productE,axiom,
    ! [P2: nat,A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B))
     => ~ ! [X3: nat,Y2: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y2) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),A2)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_6593_dvd__productE,axiom,
    ! [P2: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B))
     => ~ ! [X3: int,Y2: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y2) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),A2)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y2),B) ) ) ) ).

% dvd_productE
tff(fact_6594_division__decomp,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B10: nat,C8: nat] :
          ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B10),C8) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B10),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C8),C) ) ) ).

% division_decomp
tff(fact_6595_division__decomp,axiom,
    ! [A2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B10: int,C8: int] :
          ( ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B10),C8) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B10),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C8),C) ) ) ).

% division_decomp
tff(fact_6596_card__Min__le__sum,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite4902975817058060853nteger(A)),aa_set_nat_nat(lattic8721135487736765967in_nat,image_951025933927791156er_nat(F,A)))),groups7237345082560585321er_nat(F,A)) ) ).

% card_Min_le_sum
tff(fact_6597_card__Min__le__sum,axiom,
    ! [A: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_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,aa_set_nat_nat(finite_card_nat,A)),aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(image_nat_nat2(F),A)))),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,F),A)) ) ).

% card_Min_le_sum
tff(fact_6598_Min__singleton,axiom,
    ! [X: product_unit] : lattic8554435388794137742t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) = X ).

% Min_singleton
tff(fact_6599_Min__singleton,axiom,
    ! [X: $o] :
      ( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o))
    <=> (X) ) ).

% Min_singleton
tff(fact_6600_Min__singleton,axiom,
    ! [X: nat] : aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) = X ).

% Min_singleton
tff(fact_6601_Min__singleton,axiom,
    ! [X: int] : lattic8718645017227715691in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) = X ).

% Min_singleton
tff(fact_6602_Min_Obounded__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),lattic1063845414844153500nteger(A))
        <=> ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),X2) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6603_Min_Obounded__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic8554435388794137742t_unit(A))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),X2) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6604_Min_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(X2)) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6605_Min_Obounded__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic8086005427650270231in_rat(A))
        <=> ! [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X2) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6606_Min_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic8721135487736765967in_nat,A))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X2) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6607_Min_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(A))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X2) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6608_Min__gr__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X),lattic8554435388794137742t_unit(A))
        <=> ! [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X),X2) ) ) ) ) ).

% Min_gr_iff
tff(fact_6609_Min__gr__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A))
        <=> ! [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(X2)) ) ) ) ) ).

% Min_gr_iff
tff(fact_6610_Min__gr__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_set_nat_nat(lattic8721135487736765967in_nat,A))
        <=> ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X2) ) ) ) ) ).

% Min_gr_iff
tff(fact_6611_Min__gr__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),lattic8718645017227715691in_int(A))
        <=> ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,X),X2) ) ) ) ) ).

% Min_gr_iff
tff(fact_6612_Min__gr__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),lattic1063845414844153500nteger(A))
        <=> ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),X2) ) ) ) ) ).

% Min_gr_iff
tff(fact_6613_Min__gr__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),lattic8086005427650270231in_rat(A))
        <=> ! [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X2) ) ) ) ) ).

% Min_gr_iff
tff(fact_6614_Min__in,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => aa_set5643387332196575111eger_o(member_Code_integer(lattic1063845414844153500nteger(A)),A) ) ) ).

% Min_in
tff(fact_6615_Min__in,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => aa_set8780210009671629607unit_o(member_Product_unit(lattic8554435388794137742t_unit(A)),A) ) ) ).

% Min_in
tff(fact_6616_Min__in,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => aa_set_o_o(member_o(aa_set_o_o(lattic1973801136483472281_Min_o,A)),A) ) ) ).

% Min_in
tff(fact_6617_Min__in,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic8721135487736765967in_nat,A)),A) ) ) ).

% Min_in
tff(fact_6618_Min__in,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => aa_set_int_o(member_int(lattic8718645017227715691in_int(A)),A) ) ) ).

% Min_in
tff(fact_6619_Min__eq__iff,axiom,
    ! [A: set_Code_integer,M4: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ( lattic1063845414844153500nteger(A) = M4 )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M4),A)
            & ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M4),X2) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6620_Min__eq__iff,axiom,
    ! [A: set_Product_unit,M4: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ( lattic8554435388794137742t_unit(A) = M4 )
        <=> ( aa_set8780210009671629607unit_o(member_Product_unit(M4),A)
            & ! [X2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,M4),X2) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6621_Min__eq__iff,axiom,
    ! [A: set_o,M4: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ( aa_set_o_o(lattic1973801136483472281_Min_o,A)
          <=> (M4) )
        <=> ( aa_set_o_o(member_o((M4)),A)
            & ! [X2: $o] :
                ( aa_set_o_o(member_o((X2)),A)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M4)),(X2)) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6622_Min__eq__iff,axiom,
    ! [A: set_rat,M4: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ( lattic8086005427650270231in_rat(A) = M4 )
        <=> ( aa_set_rat_o(member_rat(M4),A)
            & ! [X2: rat] :
                ( aa_set_rat_o(member_rat(X2),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M4),X2) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6623_Min__eq__iff,axiom,
    ! [A: set_nat,M4: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ( aa_set_nat_nat(lattic8721135487736765967in_nat,A) = M4 )
        <=> ( aa_set_nat_o(member_nat(M4),A)
            & ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),X2) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6624_Min__eq__iff,axiom,
    ! [A: set_int,M4: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ( lattic8718645017227715691in_int(A) = M4 )
        <=> ( aa_set_int_o(member_int(M4),A)
            & ! [X2: int] :
                ( aa_set_int_o(member_int(X2),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),X2) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6625_Min__le__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(A)),X)
        <=> ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),X) ) ) ) ) ).

% Min_le_iff
tff(fact_6626_Min__le__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic8554435388794137742t_unit(A)),X)
        <=> ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X2),X) ) ) ) ) ).

% Min_le_iff
tff(fact_6627_Min__le__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,A)),(X))
        <=> ? [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
              & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X2)),(X)) ) ) ) ) ).

% Min_le_iff
tff(fact_6628_Min__le__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(A)),X)
        <=> ? [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),X) ) ) ) ) ).

% Min_le_iff
tff(fact_6629_Min__le__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,A)),X)
        <=> ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),X) ) ) ) ) ).

% Min_le_iff
tff(fact_6630_Min__le__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(A)),X)
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),X) ) ) ) ) ).

% Min_le_iff
tff(fact_6631_eq__Min__iff,axiom,
    ! [A: set_Code_integer,M4: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ( M4 = lattic1063845414844153500nteger(A) )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M4),A)
            & ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M4),X2) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6632_eq__Min__iff,axiom,
    ! [A: set_Product_unit,M4: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ( M4 = lattic8554435388794137742t_unit(A) )
        <=> ( aa_set8780210009671629607unit_o(member_Product_unit(M4),A)
            & ! [X2: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
               => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,M4),X2) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6633_eq__Min__iff,axiom,
    ! [A: set_o,M4: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ( (M4)
          <=> aa_set_o_o(lattic1973801136483472281_Min_o,A) )
        <=> ( aa_set_o_o(member_o((M4)),A)
            & ! [X2: $o] :
                ( aa_set_o_o(member_o((X2)),A)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M4)),(X2)) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6634_eq__Min__iff,axiom,
    ! [A: set_rat,M4: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ( M4 = lattic8086005427650270231in_rat(A) )
        <=> ( aa_set_rat_o(member_rat(M4),A)
            & ! [X2: rat] :
                ( aa_set_rat_o(member_rat(X2),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M4),X2) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6635_eq__Min__iff,axiom,
    ! [A: set_nat,M4: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ( M4 = aa_set_nat_nat(lattic8721135487736765967in_nat,A) )
        <=> ( aa_set_nat_o(member_nat(M4),A)
            & ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),A)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),X2) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6636_eq__Min__iff,axiom,
    ! [A: set_int,M4: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ( M4 = lattic8718645017227715691in_int(A) )
        <=> ( aa_set_int_o(member_int(M4),A)
            & ! [X2: int] :
                ( aa_set_int_o(member_int(X2),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),X2) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6637_Min_OboundedE,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),lattic1063845414844153500nteger(A))
         => ! [A10: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A10),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A10) ) ) ) ) ).

% Min.boundedE
tff(fact_6638_Min_OboundedE,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic8554435388794137742t_unit(A))
         => ! [A10: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A10),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A10) ) ) ) ) ).

% Min.boundedE
tff(fact_6639_Min_OboundedE,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A))
         => ! [A10: $o] :
              ( aa_set_o_o(member_o((A10)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A10)) ) ) ) ) ).

% Min.boundedE
tff(fact_6640_Min_OboundedE,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic8086005427650270231in_rat(A))
         => ! [A10: rat] :
              ( aa_set_rat_o(member_rat(A10),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A10) ) ) ) ) ).

% Min.boundedE
tff(fact_6641_Min_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic8721135487736765967in_nat,A))
         => ! [A10: nat] :
              ( aa_set_nat_o(member_nat(A10),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A10) ) ) ) ) ).

% Min.boundedE
tff(fact_6642_Min_OboundedE,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(A))
         => ! [A10: int] :
              ( aa_set_int_o(member_int(A10),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A10) ) ) ) ) ).

% Min.boundedE
tff(fact_6643_Min_OboundedI,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A3) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),lattic1063845414844153500nteger(A)) ) ) ) ).

% Min.boundedI
tff(fact_6644_Min_OboundedI,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
             => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),A3) )
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),lattic8554435388794137742t_unit(A)) ) ) ) ).

% Min.boundedI
tff(fact_6645_Min_OboundedI,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(A3)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),aa_set_o_o(lattic1973801136483472281_Min_o,A)) ) ) ) ).

% Min.boundedI
tff(fact_6646_Min_OboundedI,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A3) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),lattic8086005427650270231in_rat(A)) ) ) ) ).

% Min.boundedI
tff(fact_6647_Min_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A3) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_set_nat_nat(lattic8721135487736765967in_nat,A)) ) ) ) ).

% Min.boundedI
tff(fact_6648_Min_OboundedI,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A3) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),lattic8718645017227715691in_int(A)) ) ) ) ).

% Min.boundedI
tff(fact_6649_Min__less__iff,axiom,
    ! [A: set_Product_unit,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,lattic8554435388794137742t_unit(A)),X)
        <=> ? [X2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X2),A)
              & aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le361264281704409273t_unit,X2),X) ) ) ) ) ).

% Min_less_iff
tff(fact_6650_Min__less__iff,axiom,
    ! [A: set_o,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_set_o_o(lattic1973801136483472281_Min_o,A)),(X))
        <=> ? [X2: $o] :
              ( aa_set_o_o(member_o((X2)),A)
              & aa_o_o(aa_o_fun_o_o(ord_less_o,(X2)),(X)) ) ) ) ) ).

% Min_less_iff
tff(fact_6651_Min__less__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,A)),X)
        <=> ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),X) ) ) ) ) ).

% Min_less_iff
tff(fact_6652_Min__less__iff,axiom,
    ! [A: set_int,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8718645017227715691in_int(A)),X)
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),A)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),X) ) ) ) ) ).

% Min_less_iff
tff(fact_6653_Min__less__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,lattic1063845414844153500nteger(A)),X)
        <=> ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),X) ) ) ) ) ).

% Min_less_iff
tff(fact_6654_Min__less__iff,axiom,
    ! [A: set_rat,X: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,lattic8086005427650270231in_rat(A)),X)
        <=> ? [X2: rat] :
              ( aa_set_rat_o(member_rat(X2),A)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),X) ) ) ) ) ).

% Min_less_iff
tff(fact_6655_Min__insert2,axiom,
    ! [A: set_o,A2: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ! [B2: $o] :
            ( aa_set_o_o(member_o((B2)),A)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A2)),(B2)) )
       => ( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(A2)),A))
        <=> (A2) ) ) ) ).

% Min_insert2
tff(fact_6656_Min__insert2,axiom,
    ! [A: set_Product_unit,A2: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ! [B2: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(B2),A)
           => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,A2),B2) )
       => ( lattic8554435388794137742t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,A2),A)) = A2 ) ) ) ).

% Min_insert2
tff(fact_6657_Min__insert2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [B2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(B2),A)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) )
       => ( lattic1063845414844153500nteger(aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,A2),A)) = A2 ) ) ) ).

% Min_insert2
tff(fact_6658_Min__insert2,axiom,
    ! [A: set_rat,A2: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ! [B2: rat] :
            ( aa_set_rat_o(member_rat(B2),A)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
       => ( lattic8086005427650270231in_rat(insert_rat(A2,A)) = A2 ) ) ) ).

% Min_insert2
tff(fact_6659_Min__insert2,axiom,
    ! [A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [B2: nat] :
            ( aa_set_nat_o(member_nat(B2),A)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,A2),A)) = A2 ) ) ) ).

% Min_insert2
tff(fact_6660_Min__insert2,axiom,
    ! [A: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [B2: int] :
            ( aa_set_int_o(member_int(B2),A)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) )
       => ( lattic8718645017227715691in_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,A2),A)) = A2 ) ) ) ).

% Min_insert2
tff(fact_6661_cInf__eq__Min,axiom,
    ! [X5: set_int] :
      ( aa_set_int_o(finite_finite_int,X5)
     => ( ( X5 != bot_bot_set_int )
       => ( complete_Inf_Inf_int(X5) = lattic8718645017227715691in_int(X5) ) ) ) ).

% cInf_eq_Min
tff(fact_6662_cInf__eq__Min,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,X5)
     => ( ( X5 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,X5) = aa_set_nat_nat(lattic8721135487736765967in_nat,X5) ) ) ) ).

% cInf_eq_Min
tff(fact_6663_cInf__eq__Min,axiom,
    ! [X5: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X5)
     => ( ( X5 != bot_bo3957492148770167129t_unit )
       => ( comple2584293577114468500t_unit(X5) = lattic8554435388794137742t_unit(X5) ) ) ) ).

% cInf_eq_Min
tff(fact_6664_Min__Inf,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( lattic8554435388794137742t_unit(A) = comple2584293577114468500t_unit(A) ) ) ) ).

% Min_Inf
tff(fact_6665_Min__antimono,axiom,
    ! [M: set_Code_integer,N: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(M),N)
     => ( ( M != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(N)),lattic1063845414844153500nteger(M)) ) ) ) ).

% Min_antimono
tff(fact_6666_Min__antimono,axiom,
    ! [M: set_Product_unit,N: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,M),N)
     => ( ( M != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,N)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic8554435388794137742t_unit(N)),lattic8554435388794137742t_unit(M)) ) ) ) ).

% Min_antimono
tff(fact_6667_Min__antimono,axiom,
    ! [M: set_o,N: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,M),N)
     => ( ( M != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,N)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,N)),aa_set_o_o(lattic1973801136483472281_Min_o,M)) ) ) ) ).

% Min_antimono
tff(fact_6668_Min__antimono,axiom,
    ! [M: set_rat,N: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(M),N)
     => ( ( M != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(N)),lattic8086005427650270231in_rat(M)) ) ) ) ).

% Min_antimono
tff(fact_6669_Min__antimono,axiom,
    ! [M: set_int,N: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,M),N)
     => ( ( M != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(N)),lattic8718645017227715691in_int(M)) ) ) ) ).

% Min_antimono
tff(fact_6670_Min__antimono,axiom,
    ! [M: set_nat,N: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M),N)
     => ( ( M != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,N)),aa_set_nat_nat(lattic8721135487736765967in_nat,M)) ) ) ) ).

% Min_antimono
tff(fact_6671_Min_Osubset__imp,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B4)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(B4)),lattic1063845414844153500nteger(A)) ) ) ) ).

% Min.subset_imp
tff(fact_6672_Min_Osubset__imp,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
     => ( ( A != bot_bo3957492148770167129t_unit )
       => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
         => aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,lattic8554435388794137742t_unit(B4)),lattic8554435388794137742t_unit(A)) ) ) ) ).

% Min.subset_imp
tff(fact_6673_Min_Osubset__imp,axiom,
    ! [A: set_o,B4: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
     => ( ( A != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,B4)),aa_set_o_o(lattic1973801136483472281_Min_o,A)) ) ) ) ).

% Min.subset_imp
tff(fact_6674_Min_Osubset__imp,axiom,
    ! [A: set_rat,B4: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A),B4)
     => ( ( A != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(B4)),lattic8086005427650270231in_rat(A)) ) ) ) ).

% Min.subset_imp
tff(fact_6675_Min_Osubset__imp,axiom,
    ! [A: set_int,B4: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
     => ( ( A != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(B4)),lattic8718645017227715691in_int(A)) ) ) ) ).

% Min.subset_imp
tff(fact_6676_Min_Osubset__imp,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => ( ( A != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,B4)),aa_set_nat_nat(lattic8721135487736765967in_nat,A)) ) ) ) ).

% Min.subset_imp
tff(fact_6677_minus__Min__eq__Max,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ( aa_int_int(uminus_uminus_int,lattic8718645017227715691in_int(S)) = lattic8263393255366662781ax_int(image_int_int2(uminus_uminus_int,S)) ) ) ) ).

% minus_Min_eq_Max
tff(fact_6678_minus__Min__eq__Max,axiom,
    ! [S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,lattic1063845414844153500nteger(S)) = lattic4901227151466704046nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(uminus1351360451143612070nteger),S)) ) ) ) ).

% minus_Min_eq_Max
tff(fact_6679_minus__Max__eq__Min,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ( aa_int_int(uminus_uminus_int,lattic8263393255366662781ax_int(S)) = lattic8718645017227715691in_int(image_int_int2(uminus_uminus_int,S)) ) ) ) ).

% minus_Max_eq_Min
tff(fact_6680_minus__Max__eq__Min,axiom,
    ! [S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,lattic4901227151466704046nteger(S)) = lattic1063845414844153500nteger(aa_set8622560600277679972nteger(image_4470545334726330049nteger(uminus1351360451143612070nteger),S)) ) ) ) ).

% minus_Max_eq_Min
tff(fact_6681_Gcd__0__iff,axiom,
    ! [A: set_int] :
      ( ( gcd_Gcd_int(A) = zero_zero_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),bot_bot_set_int)) ) ).

% Gcd_0_iff
tff(fact_6682_Gcd__0__iff,axiom,
    ! [A: set_nat] :
      ( ( aa_set_nat_nat(gcd_Gcd_nat,A) = zero_zero_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat)) ) ).

% Gcd_0_iff
tff(fact_6683_Gcd__fin__0__iff,axiom,
    ! [A: set_int] :
      ( ( semiri4256215615220890538in_int(A) = zero_zero_int )
    <=> ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),bot_bot_set_int))
        & aa_set_int_o(finite_finite_int,A) ) ) ).

% Gcd_fin_0_iff
tff(fact_6684_Gcd__fin__0__iff,axiom,
    ! [A: set_nat] :
      ( ( aa_set_nat_nat(semiri4258706085729940814in_nat,A) = zero_zero_nat )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat))
        & aa_set_nat_o(finite_finite_nat,A) ) ) ).

% Gcd_fin_0_iff
tff(fact_6685_sum__bounded__above__strict,axiom,
    ! [A: set_nat,F: fun_nat_rat,K4: rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6686_sum__bounded__above__strict,axiom,
    ! [A: set_int,F: fun_int_rat,K4: rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6687_sum__bounded__above__strict,axiom,
    ! [A: set_o,F: fun_o_rat,K4: rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(I3))),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups7872700643590313910_o_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_o(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6688_sum__bounded__above__strict,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_rat,K4: rat] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Product_unit_rat(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups8131667117488461483it_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite410649719033368117t_unit(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6689_sum__bounded__above__strict,axiom,
    ! [A: set_nat,F: fun_nat_int,K4: int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6690_sum__bounded__above__strict,axiom,
    ! [A: set_o,F: fun_o_int,K4: int] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(F,(I3))),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8505340233167759370_o_int(F,A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_o(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6691_sum__bounded__above__strict,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_int,K4: int] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Product_unit_int(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8764306707065906943it_int(F,A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite410649719033368117t_unit(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6692_sum__bounded__above__strict,axiom,
    ! [A: set_int,F: fun_int_nat,K4: nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6693_sum__bounded__above__strict,axiom,
    ! [A: set_o,F: fun_o_nat,K4: nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(F,(I3))),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8507830703676809646_o_nat(F,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_o(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6694_sum__bounded__above__strict,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_nat,K4: nat] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Product_unit_nat(F,I3)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite410649719033368117t_unit(A))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups8766797177574957219it_nat(F,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite410649719033368117t_unit(A))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_6695_prod__le__power,axiom,
    ! [A: set_nat,F: fun_nat_Code_natural,N2: code_natural,K3: nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,I3))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A)),K3)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N2)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2279045934846249631atural(F,A)),aa_nat_Code_natural(power_7079662738309270450atural(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6696_prod__le__power,axiom,
    ! [A: set_int,F: fun_int_Code_natural,N2: code_natural,K3: nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,I3))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A)),K3)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N2)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2650699495083094267atural(F,A)),aa_nat_Code_natural(power_7079662738309270450atural(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6697_prod__le__power,axiom,
    ! [A: set_o,F: fun_o_Code_natural,N2: code_natural,K3: nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_o_Code_natural(F,(I3)))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_o_Code_natural(F,(I3))),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A)),K3)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N2)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6518289543945209051atural(F,A)),aa_nat_Code_natural(power_7079662738309270450atural(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6698_prod__le__power,axiom,
    ! [A: set_Product_unit,F: fun_Pr6278675377255837256atural,N2: code_natural,K3: nat] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Pro528892430537105273atural(F,I3))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Pro528892430537105273atural(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A)),K3)
       => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),N2)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups7008954963700796496atural(F,A)),aa_nat_Code_natural(power_7079662738309270450atural(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6699_prod__le__power,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer,N2: code_integer,K3: nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A)),K3)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3455450783089532116nteger(F,A)),aa_nat_Code_integer(power_8256067586552552935nteger(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6700_prod__le__power,axiom,
    ! [A: set_int,F: fun_int_Code_integer,N2: code_integer,K3: nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A)),K3)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A)),aa_nat_Code_integer(power_8256067586552552935nteger(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6701_prod__le__power,axiom,
    ! [A: set_o,F: fun_o_Code_integer,N2: code_integer,K3: nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3)))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(I3))),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A)),K3)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(F,A)),aa_nat_Code_integer(power_8256067586552552935nteger(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6702_prod__le__power,axiom,
    ! [A: set_Product_unit,F: fun_Pr1820225556744648189nteger,N2: code_integer,K3: nat] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,I3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro1705297278780387758nteger(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite410649719033368117t_unit(A)),K3)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),N2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups8185359811944078981nteger(F,A)),aa_nat_Code_integer(power_8256067586552552935nteger(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6703_prod__le__power,axiom,
    ! [A: set_nat,F: fun_nat_rat,N2: rat,K3: nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A)),K3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),N2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A)),aa_nat_rat(power_power_rat(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6704_prod__le__power,axiom,
    ! [A: set_int,F: fun_int_rat,N2: rat,K3: nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),N2) ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A)),K3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),N2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A)),aa_nat_rat(power_power_rat(N2),K3)) ) ) ) ).

% prod_le_power
tff(fact_6705_prod__mono2,axiom,
    ! [B4: set_int,A: set_int,F: 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,A),B4)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,B2)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,A3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A)),groups3827104343326376752nteger(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6706_prod__mono2,axiom,
    ! [B4: set_o,A: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
       => ( ! [B2: $o] :
              ( aa_set_o_o(member_o((B2)),minus_minus_set_o(B4,A))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(F,(B2))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(A3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(F,A)),groups7694694392188491536nteger(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6707_prod__mono2,axiom,
    ! [B4: set_Product_unit,A: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
       => ( ! [B2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(B4,A))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro1705297278780387758nteger(F,B2)) )
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,A3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups8185359811944078981nteger(F,A)),groups8185359811944078981nteger(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6708_prod__mono2,axiom,
    ! [B4: set_Code_integer,A: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B4)
       => ( ! [B2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B4,A))
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,B2)) )
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,A3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3674199335183972705nteger(F,A)),groups3674199335183972705nteger(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6709_prod__mono2,axiom,
    ! [B4: set_int,A: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B2)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A)),groups1072433553688619179nt_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6710_prod__mono2,axiom,
    ! [B4: set_o,A: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
       => ( ! [B2: $o] :
              ( aa_set_o_o(member_o((B2)),minus_minus_set_o(B4,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(B2))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(A3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2869687844427037835_o_rat(F,A)),groups2869687844427037835_o_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6711_prod__mono2,axiom,
    ! [B4: set_Product_unit,A: set_Product_unit,F: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),B4)
       => ( ! [B2: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(B4,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Product_unit_rat(F,B2)) )
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Product_unit_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups8984663225794451456it_rat(F,A)),groups8984663225794451456it_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6712_prod__mono2,axiom,
    ! [B4: set_Code_integer,A: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B4)
       => ( ! [B2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B4,A))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(F,B2)) )
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2555765274223993564er_rat(F,A)),groups2555765274223993564er_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6713_prod__mono2,axiom,
    ! [B4: set_int,A: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),B4)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(F,B2)) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,F),A)),aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,F),B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6714_prod__mono2,axiom,
    ! [B4: set_o,A: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),B4)
       => ( ! [B2: $o] :
              ( aa_set_o_o(member_o((B2)),minus_minus_set_o(B4,A))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_o_int(F,(B2))) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),A)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(F,(A3))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3502327434004483295_o_int(F,A)),groups3502327434004483295_o_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_6715_neg__equal__iff__equal,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A2 = B ) ) ).

% neg_equal_iff_equal
tff(fact_6716_neg__equal__iff__equal,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A2 = B ) ) ).

% neg_equal_iff_equal
tff(fact_6717_add_Oinverse__inverse,axiom,
    ! [A2: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A2)) = A2 ).

% add.inverse_inverse
tff(fact_6718_add_Oinverse__inverse,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) = A2 ).

% add.inverse_inverse
tff(fact_6719_Compl__subset__Compl__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A) ) ).

% Compl_subset_Compl_iff
tff(fact_6720_Compl__anti__mono,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)) ) ).

% Compl_anti_mono
tff(fact_6721_neg__le__iff__le,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B) ) ).

% neg_le_iff_le
tff(fact_6722_neg__le__iff__le,axiom,
    ! [B: rat,A2: 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,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B) ) ).

% neg_le_iff_le
tff(fact_6723_neg__le__iff__le,axiom,
    ! [B: int,A2: 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,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B) ) ).

% neg_le_iff_le
tff(fact_6724_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = A2 )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_6725_neg__equal__zero,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = A2 )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_6726_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_6727_equal__neg__zero,axiom,
    ! [A2: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_6728_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_6729_neg__equal__0__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_6730_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A2) )
    <=> ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6731_neg__0__equal__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) )
    <=> ( zero_z3403309356797280102nteger = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6732_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_6733_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_6734_neg__less__iff__less,axiom,
    ! [B: rat,A2: 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,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% neg_less_iff_less
tff(fact_6735_neg__less__iff__less,axiom,
    ! [B: int,A2: 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,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% neg_less_iff_less
tff(fact_6736_neg__less__iff__less,axiom,
    ! [B: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B) ) ).

% neg_less_iff_less
tff(fact_6737_minus__add__distrib,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_6738_minus__add__distrib,axiom,
    ! [A2: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A2),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_6739_minus__add__distrib,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_6740_minus__add__cancel,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B)) = B ).

% minus_add_cancel
tff(fact_6741_minus__add__cancel,axiom,
    ! [A2: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(plus_plus_int(A2),B)) = B ).

% minus_add_cancel
tff(fact_6742_minus__add__cancel,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)) = B ).

% minus_add_cancel
tff(fact_6743_add__minus__cancel,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),B)) = B ).

% add_minus_cancel
tff(fact_6744_add__minus__cancel,axiom,
    ! [A2: int,B: int] : aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),B)) = B ).

% add_minus_cancel
tff(fact_6745_add__minus__cancel,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B)) = B ).

% add_minus_cancel
tff(fact_6746_mult__minus__left,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A2)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)) ).

% mult_minus_left
tff(fact_6747_mult__minus__left,axiom,
    ! [A2: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)) ).

% mult_minus_left
tff(fact_6748_mult__minus__left,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)) ).

% mult_minus_left
tff(fact_6749_minus__mult__minus,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B) ).

% minus_mult_minus
tff(fact_6750_minus__mult__minus,axiom,
    ! [A2: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) ).

% minus_mult_minus
tff(fact_6751_minus__mult__minus,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) ).

% minus_mult_minus
tff(fact_6752_mult__minus__right,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B)) ).

% mult_minus_right
tff(fact_6753_mult__minus__right,axiom,
    ! [A2: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)) ).

% mult_minus_right
tff(fact_6754_mult__minus__right,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B)) ).

% mult_minus_right
tff(fact_6755_minus__diff__eq,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A2),B)) = aa_rat_rat(minus_minus_rat(B),A2) ).

% minus_diff_eq
tff(fact_6756_minus__diff__eq,axiom,
    ! [A2: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A2),B)) = aa_int_int(minus_minus_int(B),A2) ).

% minus_diff_eq
tff(fact_6757_minus__diff__eq,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A2) ).

% minus_diff_eq
tff(fact_6758_Compl__disjoint,axiom,
    ! [A: set_Product_unit] : inf_in4660618365625256667t_unit(A,uminus5944136376168626660t_unit(A)) = bot_bo3957492148770167129t_unit ).

% Compl_disjoint
tff(fact_6759_Compl__disjoint,axiom,
    ! [A: set_o] : inf_inf_set_o(A,uminus_uminus_set_o(A)) = bot_bot_set_o ).

% Compl_disjoint
tff(fact_6760_Compl__disjoint,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_6761_Compl__disjoint,axiom,
    ! [A: set_int] : inf_inf_set_int(A,uminus1532241313380277803et_int(A)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_6762_Compl__disjoint,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(A,uminus2199602881456817606_nat_o(A)) = bot_bo8632255151191677115_nat_o ).

% Compl_disjoint
tff(fact_6763_Compl__disjoint2,axiom,
    ! [A: set_Product_unit] : inf_in4660618365625256667t_unit(uminus5944136376168626660t_unit(A),A) = bot_bo3957492148770167129t_unit ).

% Compl_disjoint2
tff(fact_6764_Compl__disjoint2,axiom,
    ! [A: set_o] : inf_inf_set_o(uminus_uminus_set_o(A),A) = bot_bot_set_o ).

% Compl_disjoint2
tff(fact_6765_Compl__disjoint2,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),A) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_6766_Compl__disjoint2,axiom,
    ! [A: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(A),A) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_6767_Compl__disjoint2,axiom,
    ! [A: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(uminus2199602881456817606_nat_o(A),A) = bot_bo8632255151191677115_nat_o ).

% Compl_disjoint2
tff(fact_6768_inter__compl__diff__conv,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)) = aa_set_nat_set_nat(minus_minus_set_nat(A),B4) ).

% inter_compl_diff_conv
tff(fact_6769_Diff__Compl,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) ).

% Diff_Compl
tff(fact_6770_Compl__Diff__eq,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(minus_minus_set_nat(A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),B4) ).

% Compl_Diff_eq
tff(fact_6771_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_6772_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_6773_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_6774_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_le_0_iff_le
tff(fact_6775_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% neg_le_0_iff_le
tff(fact_6776_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% neg_le_0_iff_le
tff(fact_6777_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_6778_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_6779_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_6780_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_less_eq_nonneg
tff(fact_6781_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% neg_less_eq_nonneg
tff(fact_6782_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% neg_less_eq_nonneg
tff(fact_6783_less__neg__neg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_6784_less__neg__neg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% less_neg_neg
tff(fact_6785_less__neg__neg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_6786_neg__less__pos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% neg_less_pos
tff(fact_6787_neg__less__pos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% neg_less_pos
tff(fact_6788_neg__less__pos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_less_pos
tff(fact_6789_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_6790_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_6791_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_6792_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% neg_less_0_iff_less
tff(fact_6793_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% neg_less_0_iff_less
tff(fact_6794_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_less_0_iff_less
tff(fact_6795_ab__left__minus,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2) = zero_zero_rat ).

% ab_left_minus
tff(fact_6796_ab__left__minus,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_left_minus
tff(fact_6797_ab__left__minus,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_6798_add_Oright__inverse,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(uminus_uminus_rat,A2)) = zero_zero_rat ).

% add.right_inverse
tff(fact_6799_add_Oright__inverse,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),aa_int_int(uminus_uminus_int,A2)) = zero_zero_int ).

% add.right_inverse
tff(fact_6800_add_Oright__inverse,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_6801_diff__0,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A2) = aa_rat_rat(uminus_uminus_rat,A2) ).

% diff_0
tff(fact_6802_diff__0,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(zero_zero_int),A2) = aa_int_int(uminus_uminus_int,A2) ).

% diff_0
tff(fact_6803_diff__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ).

% diff_0
tff(fact_6804_diff__minus__eq__add,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) ).

% diff_minus_eq_add
tff(fact_6805_diff__minus__eq__add,axiom,
    ! [A2: int,B: int] : aa_int_int(minus_minus_int(A2),aa_int_int(uminus_uminus_int,B)) = aa_int_int(plus_plus_int(A2),B) ).

% diff_minus_eq_add
tff(fact_6806_diff__minus__eq__add,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) ).

% diff_minus_eq_add
tff(fact_6807_uminus__add__conv__diff,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),B) = aa_rat_rat(minus_minus_rat(B),A2) ).

% uminus_add_conv_diff
tff(fact_6808_uminus__add__conv__diff,axiom,
    ! [A2: int,B: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),B) = aa_int_int(minus_minus_int(B),A2) ).

% uminus_add_conv_diff
tff(fact_6809_uminus__add__conv__diff,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),A2) ).

% uminus_add_conv_diff
tff(fact_6810_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_6811_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_6812_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_6813_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_6814_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_6815_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_6816_boolean__algebra_Ocompl__one,axiom,
    uminus2199602881456817606_nat_o(top_to9044565752620676511_nat_o) = bot_bo8632255151191677115_nat_o ).

% boolean_algebra.compl_one
tff(fact_6817_boolean__algebra_Ocompl__one,axiom,
    uminus5944136376168626660t_unit(top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).

% boolean_algebra.compl_one
tff(fact_6818_boolean__algebra_Ocompl__one,axiom,
    uminus_uminus_set_o(top_top_set_o) = bot_bot_set_o ).

% boolean_algebra.compl_one
tff(fact_6819_boolean__algebra_Ocompl__one,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,top_top_set_nat) = bot_bot_set_nat ).

% boolean_algebra.compl_one
tff(fact_6820_boolean__algebra_Ocompl__one,axiom,
    uminus1532241313380277803et_int(top_top_set_int) = bot_bot_set_int ).

% boolean_algebra.compl_one
tff(fact_6821_boolean__algebra_Ocompl__one,axiom,
    aa_assn_assn(uminus_uminus_assn,top_top_assn) = bot_bot_assn ).

% boolean_algebra.compl_one
tff(fact_6822_boolean__algebra_Ocompl__one,axiom,
    aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,top_top_Product_unit) = bot_bot_Product_unit ).

% boolean_algebra.compl_one
tff(fact_6823_boolean__algebra_Ocompl__zero,axiom,
    uminus2199602881456817606_nat_o(bot_bo8632255151191677115_nat_o) = top_to9044565752620676511_nat_o ).

% boolean_algebra.compl_zero
tff(fact_6824_boolean__algebra_Ocompl__zero,axiom,
    uminus5944136376168626660t_unit(bot_bo3957492148770167129t_unit) = top_to1996260823553986621t_unit ).

% boolean_algebra.compl_zero
tff(fact_6825_boolean__algebra_Ocompl__zero,axiom,
    uminus_uminus_set_o(bot_bot_set_o) = top_top_set_o ).

% boolean_algebra.compl_zero
tff(fact_6826_boolean__algebra_Ocompl__zero,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,bot_bot_set_nat) = top_top_set_nat ).

% boolean_algebra.compl_zero
tff(fact_6827_boolean__algebra_Ocompl__zero,axiom,
    uminus1532241313380277803et_int(bot_bot_set_int) = top_top_set_int ).

% boolean_algebra.compl_zero
tff(fact_6828_boolean__algebra_Ocompl__zero,axiom,
    aa_assn_assn(uminus_uminus_assn,bot_bot_assn) = top_top_assn ).

% boolean_algebra.compl_zero
tff(fact_6829_boolean__algebra_Ocompl__zero,axiom,
    aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,bot_bot_Product_unit) = top_top_Product_unit ).

% boolean_algebra.compl_zero
tff(fact_6830_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Product_unit] : inf_in4660618365625256667t_unit(X,uminus5944136376168626660t_unit(X)) = bot_bo3957492148770167129t_unit ).

% boolean_algebra.conj_cancel_right
tff(fact_6831_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_o] : inf_inf_set_o(X,uminus_uminus_set_o(X)) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_right
tff(fact_6832_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_6833_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] : inf_inf_set_int(X,uminus1532241313380277803et_int(X)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_6834_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(X,uminus2199602881456817606_nat_o(X)) = bot_bo8632255151191677115_nat_o ).

% boolean_algebra.conj_cancel_right
tff(fact_6835_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,X)) = bot_bot_assn ).

% boolean_algebra.conj_cancel_right
tff(fact_6836_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)) = bot_bot_Product_unit ).

% boolean_algebra.conj_cancel_right
tff(fact_6837_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Product_unit] : inf_in4660618365625256667t_unit(uminus5944136376168626660t_unit(X),X) = bot_bo3957492148770167129t_unit ).

% boolean_algebra.conj_cancel_left
tff(fact_6838_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_o] : inf_inf_set_o(uminus_uminus_set_o(X),X) = bot_bot_set_o ).

% boolean_algebra.conj_cancel_left
tff(fact_6839_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),X) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_6840_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(X),X) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_6841_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(uminus2199602881456817606_nat_o(X),X) = bot_bo8632255151191677115_nat_o ).

% boolean_algebra.conj_cancel_left
tff(fact_6842_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),X) = bot_bot_assn ).

% boolean_algebra.conj_cancel_left
tff(fact_6843_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),X) = bot_bot_Product_unit ).

% boolean_algebra.conj_cancel_left
tff(fact_6844_inf__compl__bot__right,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] : inf_in4660618365625256667t_unit(X,inf_in4660618365625256667t_unit(Y,uminus5944136376168626660t_unit(X))) = bot_bo3957492148770167129t_unit ).

% inf_compl_bot_right
tff(fact_6845_inf__compl__bot__right,axiom,
    ! [X: set_o,Y: set_o] : inf_inf_set_o(X,inf_inf_set_o(Y,uminus_uminus_set_o(X))) = bot_bot_set_o ).

% inf_compl_bot_right
tff(fact_6846_inf__compl__bot__right,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(inf_inf_set_nat,Y),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_6847_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] : inf_inf_set_int(X,inf_inf_set_int(Y,uminus1532241313380277803et_int(X))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_6848_inf__compl__bot__right,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(X,inf_in3376754634429691197_nat_o(Y,uminus2199602881456817606_nat_o(X))) = bot_bo8632255151191677115_nat_o ).

% inf_compl_bot_right
tff(fact_6849_inf__compl__bot__right,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(uminus_uminus_assn,X))) = bot_bot_assn ).

% inf_compl_bot_right
tff(fact_6850_inf__compl__bot__right,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Y),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X))) = bot_bot_Product_unit ).

% inf_compl_bot_right
tff(fact_6851_inf__compl__bot__left2,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] : inf_in4660618365625256667t_unit(X,inf_in4660618365625256667t_unit(uminus5944136376168626660t_unit(X),Y)) = bot_bo3957492148770167129t_unit ).

% inf_compl_bot_left2
tff(fact_6852_inf__compl__bot__left2,axiom,
    ! [X: set_o,Y: set_o] : inf_inf_set_o(X,inf_inf_set_o(uminus_uminus_set_o(X),Y)) = bot_bot_set_o ).

% inf_compl_bot_left2
tff(fact_6853_inf__compl__bot__left2,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(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_6854_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] : inf_inf_set_int(X,inf_inf_set_int(uminus1532241313380277803et_int(X),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_6855_inf__compl__bot__left2,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(X,inf_in3376754634429691197_nat_o(uminus2199602881456817606_nat_o(X),Y)) = bot_bo8632255151191677115_nat_o ).

% inf_compl_bot_left2
tff(fact_6856_inf__compl__bot__left2,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),Y)) = bot_bot_assn ).

% inf_compl_bot_left2
tff(fact_6857_inf__compl__bot__left2,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),Y)) = bot_bot_Product_unit ).

% inf_compl_bot_left2
tff(fact_6858_inf__compl__bot__left1,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] : inf_in4660618365625256667t_unit(uminus5944136376168626660t_unit(X),inf_in4660618365625256667t_unit(X,Y)) = bot_bo3957492148770167129t_unit ).

% inf_compl_bot_left1
tff(fact_6859_inf__compl__bot__left1,axiom,
    ! [X: set_o,Y: set_o] : inf_inf_set_o(uminus_uminus_set_o(X),inf_inf_set_o(X,Y)) = bot_bot_set_o ).

% inf_compl_bot_left1
tff(fact_6860_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_6861_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] : inf_inf_set_int(uminus1532241313380277803et_int(X),inf_inf_set_int(X,Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_6862_inf__compl__bot__left1,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(uminus2199602881456817606_nat_o(X),inf_in3376754634429691197_nat_o(X,Y)) = bot_bo8632255151191677115_nat_o ).

% inf_compl_bot_left1
tff(fact_6863_inf__compl__bot__left1,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = bot_bot_assn ).

% inf_compl_bot_left1
tff(fact_6864_inf__compl__bot__left1,axiom,
    ! [X: product_unit,Y: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y)) = bot_bot_Product_unit ).

% inf_compl_bot_left1
tff(fact_6865_subset__Compl__singleton,axiom,
    ! [A: set_Product_unit,B: product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),uminus5944136376168626660t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,B),bot_bo3957492148770167129t_unit)))
    <=> ~ aa_set8780210009671629607unit_o(member_Product_unit(B),A) ) ).

% subset_Compl_singleton
tff(fact_6866_subset__Compl__singleton,axiom,
    ! [A: set_o,B: $o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),uminus_uminus_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(B)),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B)),A) ) ).

% subset_Compl_singleton
tff(fact_6867_subset__Compl__singleton,axiom,
    ! [A: set_int,B: int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),uminus1532241313380277803et_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,B),bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A) ) ).

% subset_Compl_singleton
tff(fact_6868_subset__Compl__singleton,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),uminus2199602881456817606_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(B),bot_bo8632255151191677115_nat_o)))
    <=> ~ aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(B),A) ) ).

% subset_Compl_singleton
tff(fact_6869_subset__Compl__singleton,axiom,
    ! [A: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,B),bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A) ) ).

% subset_Compl_singleton
tff(fact_6870_prod_Oempty,axiom,
    ! [G: fun_Pr6626665806141890376t_assn] : aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),bot_bo3957492148770167129t_unit) = one_one_assn ).

% prod.empty
tff(fact_6871_prod_Oempty,axiom,
    ! [G: fun_Product_unit_nat] : aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),bot_bo3957492148770167129t_unit) = one_one_nat ).

% prod.empty
tff(fact_6872_prod_Oempty,axiom,
    ! [G: fun_Product_unit_int] : groups393930778517121108it_int(G,bot_bo3957492148770167129t_unit) = one_one_int ).

% prod.empty
tff(fact_6873_prod_Oempty,axiom,
    ! [G: fun_Pr1820225556744648189nteger] : groups8185359811944078981nteger(G,bot_bo3957492148770167129t_unit) = one_one_Code_integer ).

% prod.empty
tff(fact_6874_prod_Oempty,axiom,
    ! [G: fun_Pr6278675377255837256atural] : groups7008954963700796496atural(G,bot_bo3957492148770167129t_unit) = one_one_Code_natural ).

% prod.empty
tff(fact_6875_prod_Oempty,axiom,
    ! [G: fun_o_assn] : aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),bot_bot_set_o) = one_one_assn ).

% prod.empty
tff(fact_6876_prod_Oempty,axiom,
    ! [G: fun_o_nat] : aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),bot_bot_set_o) = one_one_nat ).

% prod.empty
tff(fact_6877_prod_Oempty,axiom,
    ! [G: fun_o_int] : groups3502327434004483295_o_int(G,bot_bot_set_o) = one_one_int ).

% prod.empty
tff(fact_6878_prod_Oempty,axiom,
    ! [G: fun_o_Code_integer] : groups7694694392188491536nteger(G,bot_bot_set_o) = one_one_Code_integer ).

% prod.empty
tff(fact_6879_prod_Oempty,axiom,
    ! [G: fun_o_Code_natural] : groups6518289543945209051atural(G,bot_bot_set_o) = one_one_Code_natural ).

% prod.empty
tff(fact_6880_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_assn] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A) = one_one_assn ) ) ).

% prod.infinite
tff(fact_6881_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A) = one_one_assn ) ) ).

% prod.infinite
tff(fact_6882_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_nat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6883_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A) = one_one_int ) ) ).

% prod.infinite
tff(fact_6884_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_int] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups3188404863801439024er_int(G,A) = one_one_int ) ) ).

% prod.infinite
tff(fact_6885_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_Code_integer] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( groups3455450783089532116nteger(G,A) = one_one_Code_integer ) ) ).

% prod.infinite
tff(fact_6886_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups3674199335183972705nteger(G,A) = one_one_Code_integer ) ) ).

% prod.infinite
tff(fact_6887_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_Code_natural] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( groups2279045934846249631atural(G,A) = one_one_Code_natural ) ) ).

% prod.infinite
tff(fact_6888_prod_Oinfinite,axiom,
    ! [A: set_Code_integer,G: fun_Co3734316457315071362atural] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups2497794486940690220atural(G,A) = one_one_Code_natural ) ) ).

% prod.infinite
tff(fact_6889_prod_Oinfinite,axiom,
    ! [A: set_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),A) = one_one_nat ) ) ).

% prod.infinite
tff(fact_6890_of__nat__mult,axiom,
    ! [M4: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,M4)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_mult
tff(fact_6891_of__nat__mult,axiom,
    ! [M4: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M4)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_mult
tff(fact_6892_of__nat__mult,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M4)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_mult
tff(fact_6893_of__nat__mult,axiom,
    ! [M4: nat,N2: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M4)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) ).

% of_nat_mult
tff(fact_6894_of__nat__1,axiom,
    aa_nat_Code_natural(semiri3763490453095760265atural,one_one_nat) = one_one_Code_natural ).

% of_nat_1
tff(fact_6895_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6896_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6897_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_6898_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_natural = aa_nat_Code_natural(semiri3763490453095760265atural,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6899_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_6900_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_6901_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_6902_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_Code_natural(semiri3763490453095760265atural,N2) = one_one_Code_natural )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6903_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_6904_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_6905_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_6906_Gcd__empty,axiom,
    aa_set_nat_nat(gcd_Gcd_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_empty
tff(fact_6907_Gcd__empty,axiom,
    gcd_Gcd_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_empty
tff(fact_6908_Gcd__UNIV,axiom,
    aa_set_nat_nat(gcd_Gcd_nat,top_top_set_nat) = one_one_nat ).

% Gcd_UNIV
tff(fact_6909_Gcd__UNIV,axiom,
    gcd_Gcd_int(top_top_set_int) = one_one_int ).

% Gcd_UNIV
tff(fact_6910_Gcd__fin_Oempty,axiom,
    aa_set_nat_nat(semiri4258706085729940814in_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_fin.empty
tff(fact_6911_Gcd__fin_Oempty,axiom,
    semiri4256215615220890538in_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_fin.empty
tff(fact_6912_Gcd__fin_Oinfinite,axiom,
    ! [A: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(semiri4258706085729940814in_nat,A) = one_one_nat ) ) ).

% Gcd_fin.infinite
tff(fact_6913_Gcd__fin_Oinfinite,axiom,
    ! [A: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A)
     => ( semiri4256215615220890538in_int(A) = one_one_int ) ) ).

% Gcd_fin.infinite
tff(fact_6914_is__unit__Gcd__fin__iff,axiom,
    ! [A: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(semiri4258706085729940814in_nat,A)),one_one_nat)
    <=> ( aa_set_nat_nat(semiri4258706085729940814in_nat,A) = one_one_nat ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_6915_is__unit__Gcd__fin__iff,axiom,
    ! [A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,semiri4256215615220890538in_int(A)),one_one_int)
    <=> ( semiri4256215615220890538in_int(A) = one_one_int ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_6916_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_6917_dbl__inc__simps_I4_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu5831290666863070958nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_6918_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_6919_add__neg__numeral__special_I7_J,axiom,
    aa_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_6920_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_6921_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_6922_add__neg__numeral__special_I8_J,axiom,
    aa_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_6923_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_6924_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_6925_diff__numeral__special_I12_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_6926_diff__numeral__special_I12_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_6927_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)) = one_one_rat ).

% minus_one_mult_self
tff(fact_6928_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2)) = one_one_int ).

% minus_one_mult_self
tff(fact_6929_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_6930_left__minus__one__mult__self,axiom,
    ! [N2: nat,A2: 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)),N2)),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)),N2)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6931_left__minus__one__mult__self,axiom,
    ! [N2: nat,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6932_left__minus__one__mult__self,axiom,
    ! [N2: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6933_of__nat__Suc,axiom,
    ! [M4: nat] : aa_nat_Code_natural(semiri3763490453095760265atural,aa_nat_nat(suc,M4)) = plus_p4538020629002901425atural(one_one_Code_natural,aa_nat_Code_natural(semiri3763490453095760265atural,M4)) ).

% of_nat_Suc
tff(fact_6934_of__nat__Suc,axiom,
    ! [M4: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M4)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M4)) ).

% of_nat_Suc
tff(fact_6935_of__nat__Suc,axiom,
    ! [M4: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M4)) = aa_int_int(plus_plus_int(one_one_int),aa_nat_int(semiri1314217659103216013at_int,M4)) ).

% of_nat_Suc
tff(fact_6936_of__nat__Suc,axiom,
    ! [M4: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M4)) = aa_nat_nat(plus_plus_nat(one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M4)) ).

% of_nat_Suc
tff(fact_6937_of__nat__Suc,axiom,
    ! [M4: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M4)) ).

% of_nat_Suc
tff(fact_6938_prod_Oinsert,axiom,
    ! [A: set_int,X: int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_assn_assn(times_times_assn(aa_int_assn(G,X)),aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A)) ) ) ) ).

% prod.insert
tff(fact_6939_prod_Oinsert,axiom,
    ! [A: set_o,X: $o,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_assn_assn(times_times_assn(aa_o_assn(G,(X))),aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A)) ) ) ) ).

% prod.insert
tff(fact_6940_prod_Oinsert,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_assn_assn(times_times_assn(aa_Product_unit_assn(G,X)),aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A)) ) ) ) ).

% prod.insert
tff(fact_6941_prod_Oinsert,axiom,
    ! [A: set_nat,X: nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_assn_assn(times_times_assn(aa_nat_assn(G,X)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A)) ) ) ) ).

% prod.insert
tff(fact_6942_prod_Oinsert,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,X)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A)) ) ) ) ).

% prod.insert
tff(fact_6943_prod_Oinsert,axiom,
    ! [A: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ~ aa_set_int_o(member_int(X),A)
       => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_6944_prod_Oinsert,axiom,
    ! [A: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ~ aa_set_o_o(member_o((X)),A)
       => ( groups2869687844427037835_o_rat(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_6945_prod_Oinsert,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ~ aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups8984663225794451456it_rat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Product_unit_rat(G,X)),groups8984663225794451456it_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_6946_prod_Oinsert,axiom,
    ! [A: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(X),A)
       => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_6947_prod_Oinsert,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups2555765274223993564er_rat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(G,X)),groups2555765274223993564er_rat(G,A)) ) ) ) ).

% prod.insert
tff(fact_6948_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_6949_dbl__dec__simps_I2_J,axiom,
    aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6950_minus__equation__iff,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A2 ) ) ).

% minus_equation_iff
tff(fact_6951_minus__equation__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A2 ) ) ).

% minus_equation_iff
tff(fact_6952_equation__minus__iff,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A2) ) ) ).

% equation_minus_iff
tff(fact_6953_equation__minus__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% equation_minus_iff
tff(fact_6954_prod_Oneutral,axiom,
    ! [A: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_nat_nat(G,X3) = one_one_nat ) )
     => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),A) = one_one_nat ) ) ).

% prod.neutral
tff(fact_6955_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_assn,A: set_nat] :
      ( ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A) != one_one_assn )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A)
           => ( aa_nat_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6956_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_assn,A: set_int] :
      ( ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A) != one_one_assn )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A)
           => ( aa_int_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6957_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_o_assn,A: set_o] :
      ( ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A) != one_one_assn )
     => ~ ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A)
           => ( aa_o_assn(G,(A3)) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6958_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Pr6626665806141890376t_assn,A: set_Product_unit] :
      ( ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A) != one_one_assn )
     => ~ ! [A3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
           => ( aa_Product_unit_assn(G,A3) = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6959_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A: set_int] :
      ( ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),A) != one_one_nat )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A)
           => ( aa_int_nat(G,A3) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6960_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_o_nat,A: set_o] :
      ( ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),A) != one_one_nat )
     => ~ ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A)
           => ( aa_o_nat(G,(A3)) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6961_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Product_unit_nat,A: set_Product_unit] :
      ( ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),A) != one_one_nat )
     => ~ ! [A3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(A3),A)
           => ( aa_Product_unit_nat(G,A3) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6962_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A: set_nat] :
      ( ( aa_set_nat_int(groups705719431365010083at_int(G),A) != one_one_int )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A)
           => ( aa_nat_int(G,A3) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6963_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_int,A: set_int] :
      ( ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),A) != one_one_int )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A)
           => ( aa_int_int(G,A3) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6964_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_o_int,A: set_o] :
      ( ( groups3502327434004483295_o_int(G,A) != one_one_int )
     => ~ ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A)
           => ( aa_o_int(G,(A3)) = one_one_int ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_6965_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_6966_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_6967_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_6968_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_6969_le__imp__neg__le,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ) ).

% le_imp_neg_le
tff(fact_6970_le__imp__neg__le,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2)) ) ).

% le_imp_neg_le
tff(fact_6971_le__imp__neg__le,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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,A2)) ) ).

% le_imp_neg_le
tff(fact_6972_minus__le__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A2) ) ).

% minus_le_iff
tff(fact_6973_minus__le__iff,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A2) ) ).

% minus_le_iff
tff(fact_6974_minus__le__iff,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A2) ) ).

% minus_le_iff
tff(fact_6975_le__minus__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ) ).

% le_minus_iff
tff(fact_6976_le__minus__iff,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),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,A2)) ) ).

% le_minus_iff
tff(fact_6977_le__minus__iff,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),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,A2)) ) ).

% le_minus_iff
tff(fact_6978_minus__less__iff,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A2) ) ).

% minus_less_iff
tff(fact_6979_minus__less__iff,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A2) ) ).

% minus_less_iff
tff(fact_6980_minus__less__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A2) ) ).

% minus_less_iff
tff(fact_6981_less__minus__iff,axiom,
    ! [A2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),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,A2)) ) ).

% less_minus_iff
tff(fact_6982_less__minus__iff,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),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,A2)) ) ).

% less_minus_iff
tff(fact_6983_less__minus__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ) ).

% less_minus_iff
tff(fact_6984_add_Oinverse__distrib__swap,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),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,A2)) ).

% add.inverse_distrib_swap
tff(fact_6985_add_Oinverse__distrib__swap,axiom,
    ! [A2: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A2),B)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A2)) ).

% add.inverse_distrib_swap
tff(fact_6986_add_Oinverse__distrib__swap,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ).

% add.inverse_distrib_swap
tff(fact_6987_group__cancel_Oneg1,axiom,
    ! [A: rat,K3: rat,A2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),A2) )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K3)),aa_rat_rat(uminus_uminus_rat,A2)) ) ) ).

% group_cancel.neg1
tff(fact_6988_group__cancel_Oneg1,axiom,
    ! [A: int,K3: int,A2: int] :
      ( ( A = aa_int_int(plus_plus_int(K3),A2) )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K3)),aa_int_int(uminus_uminus_int,A2)) ) ) ).

% group_cancel.neg1
tff(fact_6989_group__cancel_Oneg1,axiom,
    ! [A: code_integer,K3: code_integer,A2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),A2) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K3)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) ) ) ).

% group_cancel.neg1
tff(fact_6990_square__eq__iff,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A2 = B )
        | ( A2 = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_6991_square__eq__iff,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A2 = B )
        | ( A2 = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_6992_square__eq__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A2 = B )
        | ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_6993_minus__mult__commute,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A2)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_6994_minus__mult__commute,axiom,
    ! [A2: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_6995_minus__mult__commute,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_6996_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_6997_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_6998_minus__diff__commute,axiom,
    ! [B: rat,A2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A2)),B) ).

% minus_diff_commute
tff(fact_6999_minus__diff__commute,axiom,
    ! [B: int,A2: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,B)),A2) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A2)),B) ).

% minus_diff_commute
tff(fact_7000_minus__diff__commute,axiom,
    ! [B: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B) ).

% minus_diff_commute
tff(fact_7001_Gcd__1,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),A)
     => ( aa_set_nat_nat(gcd_Gcd_nat,A) = one_one_nat ) ) ).

% Gcd_1
tff(fact_7002_Gcd__1,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(member_int(one_one_int),A)
     => ( gcd_Gcd_int(A) = one_one_int ) ) ).

% Gcd_1
tff(fact_7003_minus__assn__def,axiom,
    ! [A2: assn,B: assn] : aa_assn_assn(minus_minus_assn(A2),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),aa_assn_assn(uminus_uminus_assn,B)) ).

% minus_assn_def
tff(fact_7004_prod__ge__1,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups3455450783089532116nteger(F,A)) ) ).

% prod_ge_1
tff(fact_7005_prod__ge__1,axiom,
    ! [A: set_int,F: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups3827104343326376752nteger(F,A)) ) ).

% prod_ge_1
tff(fact_7006_prod__ge__1,axiom,
    ! [A: set_o,F: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(F,(X3))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups7694694392188491536nteger(F,A)) ) ).

% prod_ge_1
tff(fact_7007_prod__ge__1,axiom,
    ! [A: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro1705297278780387758nteger(F,X3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),groups8185359811944078981nteger(F,A)) ) ).

% prod_ge_1
tff(fact_7008_prod__ge__1,axiom,
    ! [A: set_nat,F: fun_nat_Code_natural] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_nat_Code_natural(F,X3)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),groups2279045934846249631atural(F,A)) ) ).

% prod_ge_1
tff(fact_7009_prod__ge__1,axiom,
    ! [A: set_int,F: fun_int_Code_natural] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_int_Code_natural(F,X3)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),groups2650699495083094267atural(F,A)) ) ).

% prod_ge_1
tff(fact_7010_prod__ge__1,axiom,
    ! [A: set_o,F: fun_o_Code_natural] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_o_Code_natural(F,(X3))) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),groups6518289543945209051atural(F,A)) ) ).

% prod_ge_1
tff(fact_7011_prod__ge__1,axiom,
    ! [A: set_Product_unit,F: fun_Pr6278675377255837256atural] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),aa_Pro528892430537105273atural(F,X3)) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,one_one_Code_natural),groups7008954963700796496atural(F,A)) ) ).

% prod_ge_1
tff(fact_7012_prod__ge__1,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A)) ) ).

% prod_ge_1
tff(fact_7013_prod__ge__1,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A)) ) ).

% prod_ge_1
tff(fact_7014_prod_OUnion__comp,axiom,
    ! [B4: set_set_int,G: fun_int_assn] :
      ( ! [X3: set_int] :
          ( member_set_int(X3,B4)
         => aa_set_int_o(finite_finite_int,X3) )
     => ( ! [A12: set_int] :
            ( member_set_int(A12,B4)
           => ! [A23: set_int] :
                ( member_set_int(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: int] :
                      ( aa_set_int_o(member_int(X3),A12)
                     => ( aa_set_int_o(member_int(X3),A23)
                       => ( aa_int_assn(G,X3) = one_one_assn ) ) ) ) ) )
       => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),comple3221217463730067765et_int(B4)) = aa_set_set_int_assn(comp_f3572868634108735226t_assn(groups5950967880310095793t_assn,groups7882442080178216443t_assn,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7015_prod_OUnion__comp,axiom,
    ! [B4: set_set_o,G: fun_o_assn] :
      ( ! [X3: set_o] :
          ( member_set_o(X3,B4)
         => aa_set_o_o(finite_finite_o,X3) )
     => ( ! [A12: set_o] :
            ( member_set_o(A12,B4)
           => ! [A23: set_o] :
                ( member_set_o(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: $o] :
                      ( aa_set_o_o(member_o((X3)),A12)
                     => ( aa_set_o_o(member_o((X3)),A23)
                       => ( aa_o_assn(G,(X3)) = one_one_assn ) ) ) ) ) )
       => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),comple90263536869209701_set_o(B4)) = aa_set_set_o_assn(comp_f4038625060515458192o_assn(groups6963733262650904763o_assn,groups5301882518646026715o_assn,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7016_prod_OUnion__comp,axiom,
    ! [B4: set_set_Product_unit,G: fun_Pr6626665806141890376t_assn] :
      ( ! [X3: set_Product_unit] :
          ( member5877623283571906838t_unit(X3,B4)
         => aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X3) )
     => ( ! [A12: set_Product_unit] :
            ( member5877623283571906838t_unit(A12,B4)
           => ! [A23: set_Product_unit] :
                ( member5877623283571906838t_unit(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A12)
                     => ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A23)
                       => ( aa_Product_unit_assn(G,X3) = one_one_assn ) ) ) ) ) )
       => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),comple3349607820504768346t_unit(B4)) = aa_set7364560454212353593t_assn(comp_f2328362070581412293t_assn(groups2392624107559085104t_assn,groups5360233303959380816t_assn,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7017_prod_OUnion__comp,axiom,
    ! [B4: set_set_nat,G: fun_nat_assn] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,B4)
         => aa_set_nat_o(finite_finite_nat,X3) )
     => ( ! [A12: set_nat] :
            ( member_set_nat(A12,B4)
           => ! [A23: set_nat] :
                ( member_set_nat(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: nat] :
                      ( aa_set_nat_o(member_nat(X3),A12)
                     => ( aa_set_nat_o(member_nat(X3),A23)
                       => ( aa_nat_assn(G,X3) = one_one_assn ) ) ) ) ) )
       => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),comple7399068483239264473et_nat(B4)) = aa_set_set_nat_assn(comp_f4825699036011179494t_assn(groups1535173586962943829t_assn,groups6906906614972039071t_assn,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7018_prod_OUnion__comp,axiom,
    ! [B4: set_set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( ! [X3: set_Code_integer] :
          ( member1834298987299537450nteger(X3,B4)
         => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X3) )
     => ( ! [A12: set_Code_integer] :
            ( member1834298987299537450nteger(A12,B4)
           => ! [A23: set_Code_integer] :
                ( member1834298987299537450nteger(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: code_integer] :
                      ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A12)
                     => ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A23)
                       => ( aa_Code_integer_assn(G,X3) = one_one_assn ) ) ) ) ) )
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),comple739944243200306918nteger(B4)) = aa_set586451247737764655r_assn(comp_f96878100917031437r_assn(groups4245288774754739042r_assn,groups1304777262505850412r_assn,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7019_prod_OUnion__comp,axiom,
    ! [B4: set_set_int,G: fun_int_nat] :
      ( ! [X3: set_int] :
          ( member_set_int(X3,B4)
         => aa_set_int_o(finite_finite_int,X3) )
     => ( ! [A12: set_int] :
            ( member_set_int(A12,B4)
           => ! [A23: set_int] :
                ( member_set_int(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: int] :
                      ( aa_set_int_o(member_int(X3),A12)
                     => ( aa_set_int_o(member_int(X3),A23)
                       => ( aa_int_nat(G,X3) = one_one_nat ) ) ) ) ) )
       => ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),comple3221217463730067765et_int(B4)) = aa_set_set_int_nat(comp_f1986522502478886130nt_nat(groups6435469334873047129nt_nat,groups1707563613775114915nt_nat,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7020_prod_OUnion__comp,axiom,
    ! [B4: set_set_o,G: fun_o_nat] :
      ( ! [X3: set_o] :
          ( member_set_o(X3,B4)
         => aa_set_o_o(finite_finite_o,X3) )
     => ( ! [A12: set_o] :
            ( member_set_o(A12,B4)
           => ! [A23: set_o] :
                ( member_set_o(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: $o] :
                      ( aa_set_o_o(member_o((X3)),A12)
                     => ( aa_set_o_o(member_o((X3)),A23)
                       => ( aa_o_nat(G,(X3)) = one_one_nat ) ) ) ) ) )
       => ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),comple90263536869209701_set_o(B4)) = aa_set_set_o_nat(comp_f724240716414175032_o_nat(groups4868773677410035043_o_nat,groups3504817904513533571_o_nat,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7021_prod_OUnion__comp,axiom,
    ! [B4: set_set_Product_unit,G: fun_Product_unit_nat] :
      ( ! [X3: set_Product_unit] :
          ( member5877623283571906838t_unit(X3,B4)
         => aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,X3) )
     => ( ! [A12: set_Product_unit] :
            ( member5877623283571906838t_unit(A12,B4)
           => ! [A23: set_Product_unit] :
                ( member5877623283571906838t_unit(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A12)
                     => ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A23)
                       => ( aa_Product_unit_nat(G,X3) = one_one_nat ) ) ) ) ) )
       => ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),comple3349607820504768346t_unit(B4)) = aa_set1592508101343907553it_nat(comp_f5755674356557206637it_nat(groups8454329047868613336it_nat,groups396421249026171384it_nat,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7022_prod_OUnion__comp,axiom,
    ! [B4: set_set_Code_integer,G: fun_Code_integer_nat] :
      ( ! [X3: set_Code_integer] :
          ( member1834298987299537450nteger(X3,B4)
         => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X3) )
     => ( ! [A12: set_Code_integer] :
            ( member1834298987299537450nteger(A12,B4)
           => ! [A23: set_Code_integer] :
                ( member1834298987299537450nteger(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: code_integer] :
                      ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A12)
                     => ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A23)
                       => ( aa_Code_integer_nat(G,X3) = one_one_nat ) ) ) ) ) )
       => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),comple739944243200306918nteger(B4)) = aa_set5348425839952492503er_nat(comp_f7963283641980994565er_nat(groups6044384099095135754er_nat,groups3190895334310489300er_nat,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7023_prod_OUnion__comp,axiom,
    ! [B4: set_set_int,G: fun_int_int] :
      ( ! [X3: set_int] :
          ( member_set_int(X3,B4)
         => aa_set_int_o(finite_finite_int,X3) )
     => ( ! [A12: set_int] :
            ( member_set_int(A12,B4)
           => ! [A23: set_int] :
                ( member_set_int(A23,B4)
               => ( ( A12 != A23 )
                 => ! [X3: int] :
                      ( aa_set_int_o(member_int(X3),A12)
                     => ( aa_set_int_o(member_int(X3),A23)
                       => ( aa_int_int(G,X3) = one_one_int ) ) ) ) ) )
       => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),comple3221217463730067765et_int(B4)) = aa_set_set_int_int(comp_f854714860224386566nt_int(groups6432978864363996853nt_int,groups1705073143266064639nt_int,G),B4) ) ) ) ).

% prod.Union_comp
tff(fact_7024_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7025_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B )
    <=> ( aa_int_int(plus_plus_int(A2),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7026_neg__eq__iff__add__eq__0,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7027_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7028_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(plus_plus_int(A2),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7029_eq__neg__iff__add__eq__0,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7030_add_Oinverse__unique,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = B ) ) ).

% add.inverse_unique
tff(fact_7031_add_Oinverse__unique,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A2),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A2) = B ) ) ).

% add.inverse_unique
tff(fact_7032_add_Oinverse__unique,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = B ) ) ).

% add.inverse_unique
tff(fact_7033_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_7034_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_7035_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),A2) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_7036_add__eq__0__iff,axiom,
    ! [A2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% add_eq_0_iff
tff(fact_7037_add__eq__0__iff,axiom,
    ! [A2: int,B: int] :
      ( ( aa_int_int(plus_plus_int(A2),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A2) ) ) ).

% add_eq_0_iff
tff(fact_7038_add__eq__0__iff,axiom,
    ! [A2: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% add_eq_0_iff
tff(fact_7039_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_7040_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_7041_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_7042_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_7043_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_7044_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_7045_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_7046_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_7047_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_7048_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_7049_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_7050_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_7051_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_7052_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_7053_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7054_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: int,B: int] : aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(A2),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7055_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7056_diff__conv__add__uminus,axiom,
    ! [A2: rat,B: rat] : aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_7057_diff__conv__add__uminus,axiom,
    ! [A2: int,B: int] : aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(A2),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_7058_diff__conv__add__uminus,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_7059_group__cancel_Osub2,axiom,
    ! [B4: rat,K3: rat,B: rat,A2: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),B) )
     => ( aa_rat_rat(minus_minus_rat(A2),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K3)),aa_rat_rat(minus_minus_rat(A2),B)) ) ) ).

% group_cancel.sub2
tff(fact_7060_group__cancel_Osub2,axiom,
    ! [B4: int,K3: int,B: int,A2: int] :
      ( ( B4 = aa_int_int(plus_plus_int(K3),B) )
     => ( aa_int_int(minus_minus_int(A2),B4) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K3)),aa_int_int(minus_minus_int(A2),B)) ) ) ).

% group_cancel.sub2
tff(fact_7061_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
      ( ( B4 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K3),B) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A2),B)) ) ) ).

% group_cancel.sub2
tff(fact_7062_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_7063_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_7064_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_7065_inf__cancel__left2,axiom,
    ! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] : inf_in4660618365625256667t_unit(inf_in4660618365625256667t_unit(uminus5944136376168626660t_unit(X),A2),inf_in4660618365625256667t_unit(X,B)) = bot_bo3957492148770167129t_unit ).

% inf_cancel_left2
tff(fact_7066_inf__cancel__left2,axiom,
    ! [X: set_o,A2: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(uminus_uminus_set_o(X),A2),inf_inf_set_o(X,B)) = bot_bot_set_o ).

% inf_cancel_left2
tff(fact_7067_inf__cancel__left2,axiom,
    ! [X: set_nat,A2: set_nat,B: 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,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),B)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_7068_inf__cancel__left2,axiom,
    ! [X: set_int,A2: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(uminus1532241313380277803et_int(X),A2),inf_inf_set_int(X,B)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_7069_inf__cancel__left2,axiom,
    ! [X: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(inf_in3376754634429691197_nat_o(uminus2199602881456817606_nat_o(X),A2),inf_in3376754634429691197_nat_o(X,B)) = bot_bo8632255151191677115_nat_o ).

% inf_cancel_left2
tff(fact_7070_inf__cancel__left2,axiom,
    ! [X: assn,A2: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),A2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),B)) = bot_bot_assn ).

% inf_cancel_left2
tff(fact_7071_inf__cancel__left2,axiom,
    ! [X: product_unit,A2: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),A2)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),B)) = bot_bot_Product_unit ).

% inf_cancel_left2
tff(fact_7072_inf__cancel__left1,axiom,
    ! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] : inf_in4660618365625256667t_unit(inf_in4660618365625256667t_unit(X,A2),inf_in4660618365625256667t_unit(uminus5944136376168626660t_unit(X),B)) = bot_bo3957492148770167129t_unit ).

% inf_cancel_left1
tff(fact_7073_inf__cancel__left1,axiom,
    ! [X: set_o,A2: set_o,B: set_o] : inf_inf_set_o(inf_inf_set_o(X,A2),inf_inf_set_o(uminus_uminus_set_o(X),B)) = bot_bot_set_o ).

% inf_cancel_left1
tff(fact_7074_inf__cancel__left1,axiom,
    ! [X: set_nat,A2: set_nat,B: 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,X),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),B)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_7075_inf__cancel__left1,axiom,
    ! [X: set_int,A2: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(X,A2),inf_inf_set_int(uminus1532241313380277803et_int(X),B)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_7076_inf__cancel__left1,axiom,
    ! [X: set_fu3555266933746557007_nat_o,A2: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] : inf_in3376754634429691197_nat_o(inf_in3376754634429691197_nat_o(X,A2),inf_in3376754634429691197_nat_o(uminus2199602881456817606_nat_o(X),B)) = bot_bo8632255151191677115_nat_o ).

% inf_cancel_left1
tff(fact_7077_inf__cancel__left1,axiom,
    ! [X: assn,A2: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),A2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),B)) = bot_bot_assn ).

% inf_cancel_left1
tff(fact_7078_inf__cancel__left1,axiom,
    ! [X: product_unit,A2: product_unit,B: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),A2)),aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X)),B)) = bot_bot_Product_unit ).

% inf_cancel_left1
tff(fact_7079_subset__Compl__self__eq,axiom,
    ! [A: set_Product_unit] :
      ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),uminus5944136376168626660t_unit(A))
    <=> ( A = bot_bo3957492148770167129t_unit ) ) ).

% subset_Compl_self_eq
tff(fact_7080_subset__Compl__self__eq,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),uminus_uminus_set_o(A))
    <=> ( A = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_7081_subset__Compl__self__eq,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),uminus1532241313380277803et_int(A))
    <=> ( A = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_7082_subset__Compl__self__eq,axiom,
    ! [A: set_fu3555266933746557007_nat_o] :
      ( aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),uminus2199602881456817606_nat_o(A))
    <=> ( A = bot_bo8632255151191677115_nat_o ) ) ).

% subset_Compl_self_eq
tff(fact_7083_subset__Compl__self__eq,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A))
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_7084_Compl__UNIV__eq,axiom,
    uminus2199602881456817606_nat_o(top_to9044565752620676511_nat_o) = bot_bo8632255151191677115_nat_o ).

% Compl_UNIV_eq
tff(fact_7085_Compl__UNIV__eq,axiom,
    uminus5944136376168626660t_unit(top_to1996260823553986621t_unit) = bot_bo3957492148770167129t_unit ).

% Compl_UNIV_eq
tff(fact_7086_Compl__UNIV__eq,axiom,
    uminus_uminus_set_o(top_top_set_o) = bot_bot_set_o ).

% Compl_UNIV_eq
tff(fact_7087_Compl__UNIV__eq,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,top_top_set_nat) = bot_bot_set_nat ).

% Compl_UNIV_eq
tff(fact_7088_Compl__UNIV__eq,axiom,
    uminus1532241313380277803et_int(top_top_set_int) = bot_bot_set_int ).

% Compl_UNIV_eq
tff(fact_7089_Compl__empty__eq,axiom,
    uminus2199602881456817606_nat_o(bot_bo8632255151191677115_nat_o) = top_to9044565752620676511_nat_o ).

% Compl_empty_eq
tff(fact_7090_Compl__empty__eq,axiom,
    uminus5944136376168626660t_unit(bot_bo3957492148770167129t_unit) = top_to1996260823553986621t_unit ).

% Compl_empty_eq
tff(fact_7091_Compl__empty__eq,axiom,
    uminus_uminus_set_o(bot_bot_set_o) = top_top_set_o ).

% Compl_empty_eq
tff(fact_7092_Compl__empty__eq,axiom,
    aa_set_nat_set_nat(uminus5710092332889474511et_nat,bot_bot_set_nat) = top_top_set_nat ).

% Compl_empty_eq
tff(fact_7093_Compl__empty__eq,axiom,
    uminus1532241313380277803et_int(bot_bot_set_int) = top_top_set_int ).

% Compl_empty_eq
tff(fact_7094_Compl__partition2,axiom,
    ! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,uminus5944136376168626660t_unit(A)),A) = top_to1996260823553986621t_unit ).

% Compl_partition2
tff(fact_7095_Compl__partition2,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,uminus_uminus_set_o(A)),A) = top_top_set_o ).

% Compl_partition2
tff(fact_7096_Compl__partition2,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),A) = top_top_set_nat ).

% Compl_partition2
tff(fact_7097_Compl__partition2,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,uminus1532241313380277803et_int(A)),A) = top_top_set_int ).

% Compl_partition2
tff(fact_7098_Compl__partition,axiom,
    ! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),uminus5944136376168626660t_unit(A)) = top_to1996260823553986621t_unit ).

% Compl_partition
tff(fact_7099_Compl__partition,axiom,
    ! [A: set_o] : aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),uminus_uminus_set_o(A)) = top_top_set_o ).

% Compl_partition
tff(fact_7100_Compl__partition,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)) = top_top_set_nat ).

% Compl_partition
tff(fact_7101_Compl__partition,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),uminus1532241313380277803et_int(A)) = top_top_set_int ).

% Compl_partition
tff(fact_7102_Compl__eq__Diff__UNIV,axiom,
    ! [A: set_Product_unit] : uminus5944136376168626660t_unit(A) = minus_6452836326544984404t_unit(top_to1996260823553986621t_unit,A) ).

% Compl_eq_Diff_UNIV
tff(fact_7103_Compl__eq__Diff__UNIV,axiom,
    ! [A: set_o] : uminus_uminus_set_o(A) = minus_minus_set_o(top_top_set_o,A) ).

% Compl_eq_Diff_UNIV
tff(fact_7104_Compl__eq__Diff__UNIV,axiom,
    ! [A: set_int] : uminus1532241313380277803et_int(A) = minus_minus_set_int(top_top_set_int,A) ).

% Compl_eq_Diff_UNIV
tff(fact_7105_Compl__eq__Diff__UNIV,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,A) = aa_set_nat_set_nat(minus_minus_set_nat(top_top_set_nat),A) ).

% Compl_eq_Diff_UNIV
tff(fact_7106_Compl__Int,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)) ).

% Compl_Int
tff(fact_7107_Compl__Un,axiom,
    ! [A: set_nat,B4: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)) ).

% Compl_Un
tff(fact_7108_gbinomial__negated__upper,axiom,
    ! [A2: rat,K3: nat] : aa_nat_rat(gbinomial_rat(A2),K3) = 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)),K3)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K3)),A2)),one_one_rat)),K3)) ).

% gbinomial_negated_upper
tff(fact_7109_gbinomial__index__swap,axiom,
    ! [K3: nat,N2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K3)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),one_one_rat)),K3)) = 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)),N2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3))),one_one_rat)),N2)) ).

% gbinomial_index_swap
tff(fact_7110_prod_OUnion__disjoint,axiom,
    ! [C5: set_set_nat,G: fun_nat_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,C5)
         => aa_set_nat_o(finite_finite_nat,X3) )
     => ( ! [X3: set_nat] :
            ( member_set_nat(X3,C5)
           => ! [Xa4: set_nat] :
                ( member_set_nat(Xa4,C5)
               => ( ( X3 != Xa4 )
                 => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X3),Xa4) = bot_bot_set_nat ) ) ) )
       => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),comple7399068483239264473et_nat(C5)) = aa_set_set_nat_nat(comp_f5269661881483994078at_nat(groups4248547760180025341at_nat,groups708209901874060359at_nat,G),C5) ) ) ) ).

% prod.Union_disjoint
tff(fact_7111_Gcd__remove0__nat,axiom,
    ! [M: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M) = aa_set_nat_nat(gcd_Gcd_nat,aa_set_nat_set_nat(minus_minus_set_nat(M),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_7112_Gcd__eq__1__I,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_set_nat_o(member_nat(A2),A)
       => ( aa_set_nat_nat(gcd_Gcd_nat,A) = one_one_nat ) ) ) ).

% Gcd_eq_1_I
tff(fact_7113_Gcd__eq__1__I,axiom,
    ! [A2: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_set_int_o(member_int(A2),A)
       => ( gcd_Gcd_int(A) = one_one_int ) ) ) ).

% Gcd_eq_1_I
tff(fact_7114_prod__le__1,axiom,
    ! [A: set_nat,F: fun_nat_Code_natural] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_nat_Code_natural(F,X3))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_nat_Code_natural(F,X3)),one_one_Code_natural) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2279045934846249631atural(F,A)),one_one_Code_natural) ) ).

% prod_le_1
tff(fact_7115_prod__le__1,axiom,
    ! [A: set_int,F: fun_int_Code_natural] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_int_Code_natural(F,X3))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_int_Code_natural(F,X3)),one_one_Code_natural) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups2650699495083094267atural(F,A)),one_one_Code_natural) ) ).

% prod_le_1
tff(fact_7116_prod__le__1,axiom,
    ! [A: set_o,F: fun_o_Code_natural] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_o_Code_natural(F,(X3)))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_o_Code_natural(F,(X3))),one_one_Code_natural) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups6518289543945209051atural(F,A)),one_one_Code_natural) ) ).

% prod_le_1
tff(fact_7117_prod__le__1,axiom,
    ! [A: set_Product_unit,F: fun_Pr6278675377255837256atural] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,zero_z2226904508553997617atural),aa_Pro528892430537105273atural(F,X3))
            & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,aa_Pro528892430537105273atural(F,X3)),one_one_Code_natural) ) )
     => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(ord_le1926595141338095240atural,groups7008954963700796496atural(F,A)),one_one_Code_natural) ) ).

% prod_le_1
tff(fact_7118_prod__le__1,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3455450783089532116nteger(F,A)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7119_prod__le__1,axiom,
    ! [A: set_int,F: fun_int_Code_integer] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups3827104343326376752nteger(F,A)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7120_prod__le__1,axiom,
    ! [A: set_o,F: fun_o_Code_integer] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(X3)))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(X3))),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7694694392188491536nteger(F,A)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7121_prod__le__1,axiom,
    ! [A: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( ! [X3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(X3),A)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,X3))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro1705297278780387758nteger(F,X3)),one_one_Code_integer) ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups8185359811944078981nteger(F,A)),one_one_Code_integer) ) ).

% prod_le_1
tff(fact_7122_prod__le__1,axiom,
    ! [A: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A)),one_one_rat) ) ).

% prod_le_1
tff(fact_7123_prod__le__1,axiom,
    ! [A: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A)),one_one_rat) ) ).

% prod_le_1
tff(fact_7124_prod_Orelated,axiom,
    ! [R: fun_Co2081214437782444048eger_o,S: set_nat,Ha: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X12),X23)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,Y12),Y23) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23)) )
       => ( 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(R,aa_nat_Code_integer(Ha,X3)),aa_nat_Code_integer(G,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,groups3455450783089532116nteger(Ha,S)),groups3455450783089532116nteger(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7125_prod_Orelated,axiom,
    ! [R: fun_Co2081214437782444048eger_o,S: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,one_one_Code_integer),one_one_Code_integer)
     => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,X12),X23)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,Y12),Y23) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X12),Y12)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,aa_Cod1783974652533396728nteger(Ha,X3)),aa_Cod1783974652533396728nteger(G,X3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R,groups3674199335183972705nteger(Ha,S)),groups3674199335183972705nteger(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7126_prod_Orelated,axiom,
    ! [R: fun_Co7302518231560758906ural_o,S: set_nat,Ha: fun_nat_Code_natural,G: fun_nat_Code_natural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,one_one_Code_natural),one_one_Code_natural)
     => ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
            ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,X12),X23)
              & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,Y12),Y23) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X12),Y12)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,aa_nat_Code_natural(Ha,X3)),aa_nat_Code_natural(G,X3)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,groups2279045934846249631atural(Ha,S)),groups2279045934846249631atural(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7127_prod_Orelated,axiom,
    ! [R: fun_Co7302518231560758906ural_o,S: set_Code_integer,Ha: fun_Co3734316457315071362atural,G: fun_Co3734316457315071362atural] :
      ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,one_one_Code_natural),one_one_Code_natural)
     => ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
            ( ( aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,X12),X23)
              & aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,Y12),Y23) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X12),Y12)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,aa_Cod607569804290114243atural(Ha,X3)),aa_Cod607569804290114243atural(G,X3)) )
           => aa_Code_natural_o(aa_Cod1485054954640246627ural_o(R,groups2497794486940690220atural(Ha,S)),groups2497794486940690220atural(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7128_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S: set_nat,Ha: fun_nat_assn,G: fun_nat_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X12),X23)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y12),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(times_times_assn(X12),Y12)),aa_assn_assn(times_times_assn(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S)
         => ( ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),S)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_nat_assn(Ha,X3)),aa_nat_assn(G,X3)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,Ha),S)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),S)) ) ) ) ) ).

% prod.related
tff(fact_7129_prod_Orelated,axiom,
    ! [R: fun_assn_fun_assn_o,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(R,one_one_assn),one_one_assn)
     => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
            ( ( aa_assn_o(aa_assn_fun_assn_o(R,X12),X23)
              & aa_assn_o(aa_assn_fun_assn_o(R,Y12),Y23) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_assn_assn(times_times_assn(X12),Y12)),aa_assn_assn(times_times_assn(X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_assn_o(aa_assn_fun_assn_o(R,aa_Code_integer_assn(Ha,X3)),aa_Code_integer_assn(G,X3)) )
           => aa_assn_o(aa_assn_fun_assn_o(R,aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),S)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),S)) ) ) ) ) ).

% prod.related
tff(fact_7130_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S: set_nat,Ha: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X12),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y12),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23)) )
       => ( 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(R,aa_nat_rat(Ha,X3)),aa_nat_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(Ha,S)),groups73079841787564623at_rat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7131_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S: set_Code_integer,Ha: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X12),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y12),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X12),Y12)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_Code_integer_rat(Ha,X3)),aa_Code_integer_rat(G,X3)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2555765274223993564er_rat(Ha,S)),groups2555765274223993564er_rat(G,S)) ) ) ) ) ).

% prod.related
tff(fact_7132_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X12),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y12),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X12),Y12)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_Code_integer_nat(Ha,X3)),aa_Code_integer_nat(G,X3)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),S)),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),S)) ) ) ) ) ).

% prod.related
tff(fact_7133_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S: set_nat,Ha: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,one_one_int),one_one_int)
     => ( ! [X12: int,Y12: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X12),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y12),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(times_times_int,X12),Y12)),aa_int_int(aa_int_fun_int_int(times_times_int,X23),Y23)) )
       => ( 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(R,aa_nat_int(Ha,X3)),aa_nat_int(G,X3)) )
           => aa_int_o(aa_int_fun_int_o(R,aa_set_nat_int(groups705719431365010083at_int(Ha),S)),aa_set_nat_int(groups705719431365010083at_int(G),S)) ) ) ) ) ).

% prod.related
tff(fact_7134_prod_Oinsert__if,axiom,
    ! [A: set_int,G: fun_int_assn,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(aa_set_int_o(member_int(X),A),aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A),aa_assn_assn(times_times_assn(aa_int_assn(G,X)),aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A))) ) ) ).

% prod.insert_if
tff(fact_7135_prod_Oinsert__if,axiom,
    ! [A: set_o,G: fun_o_assn,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = $ite(aa_set_o_o(member_o((X)),A),aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A),aa_assn_assn(times_times_assn(aa_o_assn(G,(X))),aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A))) ) ) ).

% prod.insert_if
tff(fact_7136_prod_Oinsert__if,axiom,
    ! [A: set_Product_unit,G: fun_Pr6626665806141890376t_assn,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A),aa_assn_assn(times_times_assn(aa_Product_unit_assn(G,X)),aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A))) ) ) ).

% prod.insert_if
tff(fact_7137_prod_Oinsert__if,axiom,
    ! [A: set_nat,G: fun_nat_assn,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_o(member_nat(X),A),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A),aa_assn_assn(times_times_assn(aa_nat_assn(G,X)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A))) ) ) ).

% prod.insert_if
tff(fact_7138_prod_Oinsert__if,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A),aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,X)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A))) ) ) ).

% prod.insert_if
tff(fact_7139_prod_Oinsert__if,axiom,
    ! [A: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = $ite(aa_set_int_o(member_int(X),A),groups1072433553688619179nt_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,A))) ) ) ).

% prod.insert_if
tff(fact_7140_prod_Oinsert__if,axiom,
    ! [A: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups2869687844427037835_o_rat(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = $ite(aa_set_o_o(member_o((X)),A),groups2869687844427037835_o_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,A))) ) ) ).

% prod.insert_if
tff(fact_7141_prod_Oinsert__if,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_rat,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8984663225794451456it_rat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = $ite(aa_set8780210009671629607unit_o(member_Product_unit(X),A),groups8984663225794451456it_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Product_unit_rat(G,X)),groups8984663225794451456it_rat(G,A))) ) ) ).

% prod.insert_if
tff(fact_7142_prod_Oinsert__if,axiom,
    ! [A: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = $ite(aa_set_nat_o(member_nat(X),A),groups73079841787564623at_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A))) ) ) ).

% prod.insert_if
tff(fact_7143_prod_Oinsert__if,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups2555765274223993564er_rat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(X),A),groups2555765274223993564er_rat(G,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(G,X)),groups2555765274223993564er_rat(G,A))) ) ) ).

% prod.insert_if
tff(fact_7144_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_int,S: set_int,I2: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_assn,Ha: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
             => ( aa_int_int(I2,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
               => aa_set_int_o(member_int(aa_int_int(J,A3)),minus_minus_set_int(T2,T5)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T2,T5))
                 => ( aa_int_int(J,aa_int_int(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T2,T5))
                   => aa_set_int_o(member_int(aa_int_int(I2,B2)),minus_minus_set_int(S,S4)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S4)
                     => ( aa_int_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T5)
                       => ( aa_int_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S)
                         => ( aa_int_assn(Ha,aa_int_int(J,A3)) = aa_int_assn(G,A3) ) )
                     => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),S) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7145_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_o,S: set_int,I2: fun_o_int,J: fun_int_o,T2: set_o,G: fun_int_assn,Ha: fun_o_assn] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_o_o(finite_finite_o,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
             => ( aa_o_int(I2,aa_int_o(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
               => aa_set_o_o(member_o(aa_int_o(J,A3)),minus_minus_set_o(T2,T5)) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(T2,T5))
                 => ( aa_int_o(J,aa_o_int(I2,(B2)))
                  <=> (B2) ) )
             => ( ! [B2: $o] :
                    ( aa_set_o_o(member_o((B2)),minus_minus_set_o(T2,T5))
                   => aa_set_int_o(member_int(aa_o_int(I2,(B2))),minus_minus_set_int(S,S4)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S4)
                     => ( aa_int_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: $o] :
                        ( aa_set_o_o(member_o((B2)),T5)
                       => ( aa_o_assn(Ha,(B2)) = one_one_assn ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S)
                         => ( aa_o_assn(Ha,aa_int_o(J,A3)) = aa_int_assn(G,A3) ) )
                     => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),S) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7146_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_Product_unit,S: set_int,I2: fun_Product_unit_int,J: fun_int_Product_unit,T2: set_Product_unit,G: fun_int_assn,Ha: fun_Pr6626665806141890376t_assn] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
             => ( aa_Product_unit_int(I2,aa_int_Product_unit(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
               => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(J,A3)),minus_6452836326544984404t_unit(T2,T5)) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(T2,T5))
                 => ( aa_int_Product_unit(J,aa_Product_unit_int(I2,B2)) = B2 ) )
             => ( ! [B2: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(T2,T5))
                   => aa_set_int_o(member_int(aa_Product_unit_int(I2,B2)),minus_minus_set_int(S,S4)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S4)
                     => ( aa_int_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: product_unit] :
                        ( aa_set8780210009671629607unit_o(member_Product_unit(B2),T5)
                       => ( aa_Product_unit_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S)
                         => ( aa_Product_unit_assn(Ha,aa_int_Product_unit(J,A3)) = aa_int_assn(G,A3) ) )
                     => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),S) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7147_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_o,T5: set_int,S: set_o,I2: fun_int_o,J: fun_o_int,T2: set_int,G: fun_o_assn,Ha: fun_int_assn] :
      ( aa_set_o_o(finite_finite_o,S4)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),minus_minus_set_o(S,S4))
             => ( aa_int_o(I2,aa_o_int(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(S,S4))
               => aa_set_int_o(member_int(aa_o_int(J,(A3))),minus_minus_set_int(T2,T5)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T2,T5))
                 => ( aa_o_int(J,aa_int_o(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T2,T5))
                   => aa_set_o_o(member_o(aa_int_o(I2,B2)),minus_minus_set_o(S,S4)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o((A3)),S4)
                     => ( aa_o_assn(G,(A3)) = one_one_assn ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T5)
                       => ( aa_int_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o((A3)),S)
                         => ( aa_int_assn(Ha,aa_o_int(J,(A3))) = aa_o_assn(G,(A3)) ) )
                     => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),S) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7148_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_o,T5: set_o,S: set_o,I2: fun_o_o,J: fun_o_o,T2: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,S4)
     => ( aa_set_o_o(finite_finite_o,T5)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),minus_minus_set_o(S,S4))
             => ( aa_o_o(I2,aa_o_o(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(S,S4))
               => aa_set_o_o(member_o(aa_o_o(J,(A3))),minus_minus_set_o(T2,T5)) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(T2,T5))
                 => ( aa_o_o(J,aa_o_o(I2,(B2)))
                  <=> (B2) ) )
             => ( ! [B2: $o] :
                    ( aa_set_o_o(member_o((B2)),minus_minus_set_o(T2,T5))
                   => aa_set_o_o(member_o(aa_o_o(I2,(B2))),minus_minus_set_o(S,S4)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o((A3)),S4)
                     => ( aa_o_assn(G,(A3)) = one_one_assn ) )
                 => ( ! [B2: $o] :
                        ( aa_set_o_o(member_o((B2)),T5)
                       => ( aa_o_assn(Ha,(B2)) = one_one_assn ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o((A3)),S)
                         => ( aa_o_assn(Ha,aa_o_o(J,(A3))) = aa_o_assn(G,(A3)) ) )
                     => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),S) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7149_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_o,T5: set_Product_unit,S: set_o,I2: fun_Product_unit_o,J: fun_o_Product_unit,T2: set_Product_unit,G: fun_o_assn,Ha: fun_Pr6626665806141890376t_assn] :
      ( aa_set_o_o(finite_finite_o,S4)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T5)
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),minus_minus_set_o(S,S4))
             => ( aa_Product_unit_o(I2,aa_o_Product_unit(J,(A3)))
              <=> (A3) ) )
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(S,S4))
               => aa_set8780210009671629607unit_o(member_Product_unit(aa_o_Product_unit(J,(A3))),minus_6452836326544984404t_unit(T2,T5)) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(T2,T5))
                 => ( aa_o_Product_unit(J,aa_Product_unit_o(I2,B2)) = B2 ) )
             => ( ! [B2: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(T2,T5))
                   => aa_set_o_o(member_o(aa_Product_unit_o(I2,B2)),minus_minus_set_o(S,S4)) )
               => ( ! [A3: $o] :
                      ( aa_set_o_o(member_o((A3)),S4)
                     => ( aa_o_assn(G,(A3)) = one_one_assn ) )
                 => ( ! [B2: product_unit] :
                        ( aa_set8780210009671629607unit_o(member_Product_unit(B2),T5)
                       => ( aa_Product_unit_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: $o] :
                          ( aa_set_o_o(member_o((A3)),S)
                         => ( aa_Product_unit_assn(Ha,aa_o_Product_unit(J,(A3))) = aa_o_assn(G,(A3)) ) )
                     => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),S) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7150_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_Product_unit,T5: set_int,S: set_Product_unit,I2: fun_int_Product_unit,J: fun_Product_unit_int,T2: set_int,G: fun_Pr6626665806141890376t_assn,Ha: fun_int_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S4)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(S,S4))
             => ( aa_int_Product_unit(I2,aa_Product_unit_int(J,A3)) = A3 ) )
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(S,S4))
               => aa_set_int_o(member_int(aa_Product_unit_int(J,A3)),minus_minus_set_int(T2,T5)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T2,T5))
                 => ( aa_Product_unit_int(J,aa_int_Product_unit(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T2,T5))
                   => aa_set8780210009671629607unit_o(member_Product_unit(aa_int_Product_unit(I2,B2)),minus_6452836326544984404t_unit(S,S4)) )
               => ( ! [A3: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(A3),S4)
                     => ( aa_Product_unit_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T5)
                       => ( aa_int_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: product_unit] :
                          ( aa_set8780210009671629607unit_o(member_Product_unit(A3),S)
                         => ( aa_int_assn(Ha,aa_Product_unit_int(J,A3)) = aa_Product_unit_assn(G,A3) ) )
                     => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),S) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7151_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_Product_unit,T5: set_o,S: set_Product_unit,I2: fun_o_Product_unit,J: fun_Product_unit_o,T2: set_o,G: fun_Pr6626665806141890376t_assn,Ha: fun_o_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S4)
     => ( aa_set_o_o(finite_finite_o,T5)
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(S,S4))
             => ( aa_o_Product_unit(I2,aa_Product_unit_o(J,A3)) = A3 ) )
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(S,S4))
               => aa_set_o_o(member_o(aa_Product_unit_o(J,A3)),minus_minus_set_o(T2,T5)) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(T2,T5))
                 => ( aa_Product_unit_o(J,aa_o_Product_unit(I2,(B2)))
                  <=> (B2) ) )
             => ( ! [B2: $o] :
                    ( aa_set_o_o(member_o((B2)),minus_minus_set_o(T2,T5))
                   => aa_set8780210009671629607unit_o(member_Product_unit(aa_o_Product_unit(I2,(B2))),minus_6452836326544984404t_unit(S,S4)) )
               => ( ! [A3: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(A3),S4)
                     => ( aa_Product_unit_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: $o] :
                        ( aa_set_o_o(member_o((B2)),T5)
                       => ( aa_o_assn(Ha,(B2)) = one_one_assn ) )
                   => ( ! [A3: product_unit] :
                          ( aa_set8780210009671629607unit_o(member_Product_unit(A3),S)
                         => ( aa_o_assn(Ha,aa_Product_unit_o(J,A3)) = aa_Product_unit_assn(G,A3) ) )
                     => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),S) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7152_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_Product_unit,T5: set_Product_unit,S: set_Product_unit,I2: fun_Pr2191293291710260995t_unit,J: fun_Pr2191293291710260995t_unit,T2: set_Product_unit,G: fun_Pr6626665806141890376t_assn,Ha: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S4)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T5)
       => ( ! [A3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(S,S4))
             => ( aa_Pro8294289767544744956t_unit(I2,aa_Pro8294289767544744956t_unit(J,A3)) = A3 ) )
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(S,S4))
               => aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(J,A3)),minus_6452836326544984404t_unit(T2,T5)) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(T2,T5))
                 => ( aa_Pro8294289767544744956t_unit(J,aa_Pro8294289767544744956t_unit(I2,B2)) = B2 ) )
             => ( ! [B2: product_unit] :
                    ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(T2,T5))
                   => aa_set8780210009671629607unit_o(member_Product_unit(aa_Pro8294289767544744956t_unit(I2,B2)),minus_6452836326544984404t_unit(S,S4)) )
               => ( ! [A3: product_unit] :
                      ( aa_set8780210009671629607unit_o(member_Product_unit(A3),S4)
                     => ( aa_Product_unit_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: product_unit] :
                        ( aa_set8780210009671629607unit_o(member_Product_unit(B2),T5)
                       => ( aa_Product_unit_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: product_unit] :
                          ( aa_set8780210009671629607unit_o(member_Product_unit(A3),S)
                         => ( aa_Product_unit_assn(Ha,aa_Pro8294289767544744956t_unit(J,A3)) = aa_Product_unit_assn(G,A3) ) )
                     => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),S) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7153_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_Code_integer,S: set_int,I2: fun_Code_integer_int,J: fun_int_Code_integer,T2: set_Code_integer,G: fun_int_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
             => ( aa_Code_integer_int(I2,aa_int_Code_integer(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(S,S4))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_int_Code_integer(J,A3)),minus_2355218937544613996nteger(T2,T5)) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T2,T5))
                 => ( aa_int_Code_integer(J,aa_Code_integer_int(I2,B2)) = B2 ) )
             => ( ! [B2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(T2,T5))
                   => aa_set_int_o(member_int(aa_Code_integer_int(I2,B2)),minus_minus_set_int(S,S4)) )
               => ( ! [A3: int] :
                      ( aa_set_int_o(member_int(A3),S4)
                     => ( aa_int_assn(G,A3) = one_one_assn ) )
                 => ( ! [B2: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B2),T5)
                       => ( aa_Code_integer_assn(Ha,B2) = one_one_assn ) )
                   => ( ! [A3: int] :
                          ( aa_set_int_o(member_int(A3),S)
                         => ( aa_Code_integer_assn(Ha,aa_int_Code_integer(J,A3)) = aa_int_assn(G,A3) ) )
                     => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),S) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),T2) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7154_gbinomial__minus,axiom,
    ! [A2: rat,K3: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A2)),K3) = 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)),K3)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_nat_rat(semiri681578069525770553at_rat,K3))),one_one_rat)),K3)) ).

% gbinomial_minus
tff(fact_7155_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_7156_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_7157_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_7158_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_7159_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_7160_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_7161_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_7162_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_7163_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_7164_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_7165_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_7166_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_7167_inf__shunt,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(X,Y) = bot_bo3957492148770167129t_unit )
    <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,X),uminus5944136376168626660t_unit(Y)) ) ).

% inf_shunt
tff(fact_7168_inf__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o(X,Y) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,X),uminus_uminus_set_o(Y)) ) ).

% inf_shunt
tff(fact_7169_inf__shunt,axiom,
    ! [X: set_int,Y: set_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_7170_inf__shunt,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(X,Y) = bot_bo8632255151191677115_nat_o )
    <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(X),uminus2199602881456817606_nat_o(Y)) ) ).

% inf_shunt
tff(fact_7171_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_7172_inf__shunt,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = bot_bot_Product_unit )
    <=> aa_Product_unit_o(aa_Pro4895447120693452255unit_o(ord_le3221252021190050221t_unit,X),aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Y)) ) ).

% inf_shunt
tff(fact_7173_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),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% inf_shunt
tff(fact_7174_power__minus,axiom,
    ! [A2: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),N2) = 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)),N2)),aa_nat_rat(power_power_rat(A2),N2)) ).

% power_minus
tff(fact_7175_power__minus,axiom,
    ! [A2: int,N2: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2)),aa_nat_int(power_power_int(A2),N2)) ).

% power_minus
tff(fact_7176_power__minus,axiom,
    ! [A2: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ).

% power_minus
tff(fact_7177_disjoint__eq__subset__Compl,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
    <=> aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),uminus5944136376168626660t_unit(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7178_disjoint__eq__subset__Compl,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
    <=> aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),uminus_uminus_set_o(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7179_disjoint__eq__subset__Compl,axiom,
    ! [A: set_int,B4: set_int] :
      ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
    <=> aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),uminus1532241313380277803et_int(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7180_disjoint__eq__subset__Compl,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B4: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A,B4) = bot_bo8632255151191677115_nat_o )
    <=> aa_set5091322549519198405at_o_o(ord_le5394272759557949423_nat_o(A),uminus2199602881456817606_nat_o(B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7181_disjoint__eq__subset__Compl,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)) ) ).

% disjoint_eq_subset_Compl
tff(fact_7182_Compl__insert,axiom,
    ! [X: product_unit,A: set_Product_unit] : uminus5944136376168626660t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = minus_6452836326544984404t_unit(uminus5944136376168626660t_unit(A),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)) ).

% Compl_insert
tff(fact_7183_Compl__insert,axiom,
    ! [X: $o,A: set_o] : uminus_uminus_set_o(aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = minus_minus_set_o(uminus_uminus_set_o(A),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)) ).

% Compl_insert
tff(fact_7184_Compl__insert,axiom,
    ! [X: int,A: set_int] : uminus1532241313380277803et_int(aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = minus_minus_set_int(uminus1532241313380277803et_int(A),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)) ).

% Compl_insert
tff(fact_7185_Compl__insert,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,A: set_fu3555266933746557007_nat_o] : uminus2199602881456817606_nat_o(aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),A)) = minus_6456595905716665654_nat_o(uminus2199602881456817606_nat_o(A),aa_set1652341062091217276_nat_o(insert4923026906511862463_nat_o(X),bot_bo8632255151191677115_nat_o)) ).

% Compl_insert
tff(fact_7186_Compl__insert,axiom,
    ! [X: nat,A: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(uminus5710092332889474511et_nat,A)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)) ).

% Compl_insert
tff(fact_7187_Image__subset__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(R2),A)),B4)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(image_nat_nat(converse_nat_nat(R2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B4)))) ) ).

% Image_subset_eq
tff(fact_7188_less__1__prod2,axiom,
    ! [I: set_int,I2: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I)
     => ( aa_set_int_o(member_int(I2),I)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7189_less__1__prod2,axiom,
    ! [I: set_o,I2: $o,F: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I)
     => ( aa_set_o_o(member_o((I2)),I)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(F,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_o_Code_integer(F,(I3))) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7190_less__1__prod2,axiom,
    ! [I: set_Product_unit,I2: product_unit,F: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(I2),I)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Pro1705297278780387758nteger(F,I2))
         => ( ! [I3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Pro1705297278780387758nteger(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups8185359811944078981nteger(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7191_less__1__prod2,axiom,
    ! [I: set_nat,I2: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I)
     => ( aa_set_nat_o(member_nat(I2),I)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3455450783089532116nteger(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7192_less__1__prod2,axiom,
    ! [I: set_Code_integer,I2: code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,I3)) )
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3674199335183972705nteger(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7193_less__1__prod2,axiom,
    ! [I: set_int,I2: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I)
     => ( aa_set_int_o(member_int(I2),I)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I2))
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),I)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7194_less__1__prod2,axiom,
    ! [I: set_o,I2: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,I)
     => ( aa_set_o_o(member_o((I2)),I)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_o_rat(F,(I2)))
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),I)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_o_rat(F,(I3))) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2869687844427037835_o_rat(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7195_less__1__prod2,axiom,
    ! [I: set_Product_unit,I2: product_unit,F: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(I2),I)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Product_unit_rat(F,I2))
         => ( ! [I3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Product_unit_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups8984663225794451456it_rat(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7196_less__1__prod2,axiom,
    ! [I: set_nat,I2: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I)
     => ( aa_set_nat_o(member_nat(I2),I)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7197_less__1__prod2,axiom,
    ! [I: set_Code_integer,I2: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_Code_integer_rat(F,I2))
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Code_integer_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups2555765274223993564er_rat(F,I)) ) ) ) ) ).

% less_1_prod2
tff(fact_7198_less__1__prod,axiom,
    ! [I: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( ( I != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Code_integer_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3188404863801439024er_int(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7199_less__1__prod,axiom,
    ! [I: set_Product_unit,F: fun_Product_unit_int] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( ( I != bot_bo3957492148770167129t_unit )
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_Product_unit_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups393930778517121108it_int(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7200_less__1__prod,axiom,
    ! [I: set_o,F: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,I)
     => ( ( I != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_o_int(F,(I3))) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups3502327434004483295_o_int(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7201_less__1__prod,axiom,
    ! [I: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,I)
     => ( ( I != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_nat_int(groups705719431365010083at_int(F),I)) ) ) ) ).

% less_1_prod
tff(fact_7202_less__1__prod,axiom,
    ! [I: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,I)
     => ( ( I != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(F,I3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,F),I)) ) ) ) ).

% less_1_prod
tff(fact_7203_less__1__prod,axiom,
    ! [I: set_Code_integer,F: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I)
     => ( ( I != bot_bo3990330152332043303nteger )
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3674199335183972705nteger(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7204_less__1__prod,axiom,
    ! [I: set_Product_unit,F: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,I)
     => ( ( I != bot_bo3957492148770167129t_unit )
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Pro1705297278780387758nteger(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups8185359811944078981nteger(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7205_less__1__prod,axiom,
    ! [I: set_o,F: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,I)
     => ( ( I != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_o_Code_integer(F,(I3))) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups7694694392188491536nteger(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7206_less__1__prod,axiom,
    ! [I: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,I)
     => ( ( I != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3455450783089532116nteger(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7207_less__1__prod,axiom,
    ! [I: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,I)
     => ( ( I != bot_bot_set_int )
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),I)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(F,I3)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),groups3827104343326376752nteger(F,I)) ) ) ) ).

% less_1_prod
tff(fact_7208_prod_Oreindex__nontrivial,axiom,
    ! [A: set_nat,Ha: fun_nat_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_nat_nat(Ha,X3) = aa_nat_nat(Ha,Y2) )
                 => ( aa_nat_assn(G,aa_nat_nat(Ha,X3)) = one_one_assn ) ) ) ) )
       => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(image_nat_nat2(Ha),A)) = aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,comp_nat_assn_nat(G,Ha)),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7209_prod_Oreindex__nontrivial,axiom,
    ! [A: set_nat,Ha: fun_nat_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_nat_Code_integer(Ha,X3) = aa_nat_Code_integer(Ha,Y2) )
                 => ( aa_Code_integer_assn(G,aa_nat_Code_integer(Ha,X3)) = one_one_assn ) ) ) ) )
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),aa_set7438092518518805847nteger(image_1215581382706833972nteger(Ha),A)) = aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,comp_C8462447471145125178sn_nat(G,Ha)),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7210_prod_Oreindex__nontrivial,axiom,
    ! [A: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [X3: code_integer,Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
           => ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_Cod1783974652533396728nteger(Ha,X3) = aa_Cod1783974652533396728nteger(Ha,Y2) )
                 => ( aa_Code_integer_assn(G,aa_Cod1783974652533396728nteger(Ha,X3)) = one_one_assn ) ) ) ) )
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),aa_set8622560600277679972nteger(image_4470545334726330049nteger(Ha),A)) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,comp_C3026223900822616519nteger(G,Ha)),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7211_prod_Oreindex__nontrivial,axiom,
    ! [A: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [X3: code_integer,Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
           => ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_Cod1783974652533396728nteger(Ha,X3) = aa_Cod1783974652533396728nteger(Ha,Y2) )
                 => ( aa_Code_integer_nat(G,aa_Cod1783974652533396728nteger(Ha,X3)) = one_one_nat ) ) ) ) )
       => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),aa_set8622560600277679972nteger(image_4470545334726330049nteger(Ha),A)) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,comp_C7238605732685392495nteger(G,Ha)),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7212_prod_Oreindex__nontrivial,axiom,
    ! [A: set_nat,Ha: fun_nat_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_nat_nat(Ha,X3) = aa_nat_nat(Ha,Y2) )
                 => ( aa_nat_int(G,aa_nat_nat(Ha,X3)) = one_one_int ) ) ) ) )
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(image_nat_nat2(Ha),A)) = aa_set_nat_int(groups705719431365010083at_int(comp_nat_int_nat(G,Ha)),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7213_prod_Oreindex__nontrivial,axiom,
    ! [A: set_nat,Ha: fun_nat_Code_integer,G: fun_Code_integer_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_nat_Code_integer(Ha,X3) = aa_nat_Code_integer(Ha,Y2) )
                 => ( aa_Code_integer_int(G,aa_nat_Code_integer(Ha,X3)) = one_one_int ) ) ) ) )
       => ( groups3188404863801439024er_int(G,aa_set7438092518518805847nteger(image_1215581382706833972nteger(Ha),A)) = aa_set_nat_int(groups705719431365010083at_int(comp_C5048910307564104766nt_nat(G,Ha)),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7214_prod_Oreindex__nontrivial,axiom,
    ! [A: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [X3: code_integer,Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
           => ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_Cod1783974652533396728nteger(Ha,X3) = aa_Cod1783974652533396728nteger(Ha,Y2) )
                 => ( aa_Code_integer_int(G,aa_Cod1783974652533396728nteger(Ha,X3)) = one_one_int ) ) ) ) )
       => ( groups3188404863801439024er_int(G,aa_set8622560600277679972nteger(image_4470545334726330049nteger(Ha),A)) = groups3188404863801439024er_int(comp_C7610259292922237131nteger(G,Ha),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7215_prod_Oreindex__nontrivial,axiom,
    ! [A: set_nat,Ha: fun_nat_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_nat_nat(Ha,X3) = aa_nat_nat(Ha,Y2) )
                 => ( aa_nat_Code_integer(G,aa_nat_nat(Ha,X3)) = one_one_Code_integer ) ) ) ) )
       => ( groups3455450783089532116nteger(G,aa_set_nat_set_nat(image_nat_nat2(Ha),A)) = groups3455450783089532116nteger(comp_n3898172953802868194er_nat(G,Ha),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7216_prod_Oreindex__nontrivial,axiom,
    ! [A: set_nat,Ha: fun_nat_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [X3: nat,Y2: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
           => ( aa_set_nat_o(member_nat(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_nat_Code_integer(Ha,X3) = aa_nat_Code_integer(Ha,Y2) )
                 => ( aa_Cod1783974652533396728nteger(G,aa_nat_Code_integer(Ha,X3)) = one_one_Code_integer ) ) ) ) )
       => ( groups3674199335183972705nteger(G,aa_set7438092518518805847nteger(image_1215581382706833972nteger(Ha),A)) = groups3455450783089532116nteger(comp_C6974050283906349679er_nat(G,Ha),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7217_prod_Oreindex__nontrivial,axiom,
    ! [A: set_Code_integer,Ha: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [X3: code_integer,Y2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
           => ( aa_set5643387332196575111eger_o(member_Code_integer(Y2),A)
             => ( ( X3 != Y2 )
               => ( ( aa_Cod1783974652533396728nteger(Ha,X3) = aa_Cod1783974652533396728nteger(Ha,Y2) )
                 => ( aa_Cod1783974652533396728nteger(G,aa_Cod1783974652533396728nteger(Ha,X3)) = one_one_Code_integer ) ) ) ) )
       => ( groups3674199335183972705nteger(G,aa_set8622560600277679972nteger(image_4470545334726330049nteger(Ha),A)) = groups3674199335183972705nteger(comp_C7449957260575251196nteger(G,Ha),A) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7218_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A) = aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),minus_2355218937544613996nteger(A,B4))),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7219_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ( groups2555765274223993564er_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,B4))),groups2555765274223993564er_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7220_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),minus_2355218937544613996nteger(A,B4))),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7221_prod_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
       => ( groups3188404863801439024er_int(G,A) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,minus_2355218937544613996nteger(A,B4))),groups3188404863801439024er_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7222_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
     => ( aa_set_nat_o(finite_finite_nat,A)
       => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A) = aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7223_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
     => ( aa_set_nat_o(finite_finite_nat,A)
       => ( groups73079841787564623at_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7224_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
     => ( aa_set_nat_o(finite_finite_nat,A)
       => ( aa_set_nat_int(groups705719431365010083at_int(G),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_int(groups705719431365010083at_int(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7225_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A)
     => ( aa_set_nat_o(finite_finite_nat,A)
       => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7226_prod_Osame__carrier,axiom,
    ! [C5: set_int,A: set_int,B4: set_int,G: fun_int_assn,Ha: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C5)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C5,A))
               => ( aa_int_assn(G,A3) = one_one_assn ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B4))
                 => ( aa_int_assn(Ha,B2) = one_one_assn ) )
             => ( ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),B4) )
              <=> ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),C5) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7227_prod_Osame__carrier,axiom,
    ! [C5: set_o,A: set_o,B4: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,C5)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),C5)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C5,A))
               => ( aa_o_assn(G,(A3)) = one_one_assn ) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(C5,B4))
                 => ( aa_o_assn(Ha,(B2)) = one_one_assn ) )
             => ( ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),B4) )
              <=> ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),C5) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7228_prod_Osame__carrier,axiom,
    ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: fun_Pr6626665806141890376t_assn,Ha: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,C5)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),C5)
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),C5)
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(C5,A))
               => ( aa_Product_unit_assn(G,A3) = one_one_assn ) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(C5,B4))
                 => ( aa_Product_unit_assn(Ha,B2) = one_one_assn ) )
             => ( ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),B4) )
              <=> ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),C5) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7229_prod_Osame__carrier,axiom,
    ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C5)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C5,A))
               => ( aa_Code_integer_assn(G,A3) = one_one_assn ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B4))
                 => ( aa_Code_integer_assn(Ha,B2) = one_one_assn ) )
             => ( ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),B4) )
              <=> ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),C5) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7230_prod_Osame__carrier,axiom,
    ! [C5: set_int,A: set_int,B4: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C5)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C5,A))
               => ( aa_int_nat(G,A3) = one_one_nat ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B4))
                 => ( aa_int_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),A) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),B4) )
              <=> ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),C5) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7231_prod_Osame__carrier,axiom,
    ! [C5: set_o,A: set_o,B4: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C5)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),C5)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C5,A))
               => ( aa_o_nat(G,(A3)) = one_one_nat ) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(C5,B4))
                 => ( aa_o_nat(Ha,(B2)) = one_one_nat ) )
             => ( ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),A) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),B4) )
              <=> ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),C5) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7232_prod_Osame__carrier,axiom,
    ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: fun_Product_unit_nat,Ha: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,C5)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),C5)
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),C5)
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(C5,A))
               => ( aa_Product_unit_nat(G,A3) = one_one_nat ) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(C5,B4))
                 => ( aa_Product_unit_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),A) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),B4) )
              <=> ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),C5) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7233_prod_Osame__carrier,axiom,
    ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C5)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C5,A))
               => ( aa_Code_integer_nat(G,A3) = one_one_nat ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B4))
                 => ( aa_Code_integer_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),B4) )
              <=> ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),C5) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7234_prod_Osame__carrier,axiom,
    ! [C5: set_int,A: set_int,B4: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C5)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C5,A))
               => ( aa_int_int(G,A3) = one_one_int ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B4))
                 => ( aa_int_int(Ha,B2) = one_one_int ) )
             => ( ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),A) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),B4) )
              <=> ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),C5) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7235_prod_Osame__carrier,axiom,
    ! [C5: set_o,A: set_o,B4: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,C5)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),C5)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C5,A))
               => ( aa_o_int(G,(A3)) = one_one_int ) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(C5,B4))
                 => ( aa_o_int(Ha,(B2)) = one_one_int ) )
             => ( ( groups3502327434004483295_o_int(G,A) = groups3502327434004483295_o_int(Ha,B4) )
              <=> ( groups3502327434004483295_o_int(G,C5) = groups3502327434004483295_o_int(Ha,C5) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7236_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A: set_int,B4: set_int,G: fun_int_assn,Ha: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C5)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C5,A))
               => ( aa_int_assn(G,A3) = one_one_assn ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B4))
                 => ( aa_int_assn(Ha,B2) = one_one_assn ) )
             => ( ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),C5) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),C5) )
               => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7237_prod_Osame__carrierI,axiom,
    ! [C5: set_o,A: set_o,B4: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,C5)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),C5)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C5,A))
               => ( aa_o_assn(G,(A3)) = one_one_assn ) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(C5,B4))
                 => ( aa_o_assn(Ha,(B2)) = one_one_assn ) )
             => ( ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),C5) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),C5) )
               => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7238_prod_Osame__carrierI,axiom,
    ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: fun_Pr6626665806141890376t_assn,Ha: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,C5)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),C5)
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),C5)
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(C5,A))
               => ( aa_Product_unit_assn(G,A3) = one_one_assn ) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(C5,B4))
                 => ( aa_Product_unit_assn(Ha,B2) = one_one_assn ) )
             => ( ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),C5) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),C5) )
               => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7239_prod_Osame__carrierI,axiom,
    ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C5)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C5,A))
               => ( aa_Code_integer_assn(G,A3) = one_one_assn ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B4))
                 => ( aa_Code_integer_assn(Ha,B2) = one_one_assn ) )
             => ( ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),C5) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),C5) )
               => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7240_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A: set_int,B4: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C5)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C5,A))
               => ( aa_int_nat(G,A3) = one_one_nat ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B4))
                 => ( aa_int_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),C5) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),C5) )
               => ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),A) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7241_prod_Osame__carrierI,axiom,
    ! [C5: set_o,A: set_o,B4: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,C5)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),C5)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C5,A))
               => ( aa_o_nat(G,(A3)) = one_one_nat ) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(C5,B4))
                 => ( aa_o_nat(Ha,(B2)) = one_one_nat ) )
             => ( ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),C5) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),C5) )
               => ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),A) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7242_prod_Osame__carrierI,axiom,
    ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: fun_Product_unit_nat,Ha: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,C5)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,A),C5)
       => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,B4),C5)
         => ( ! [A3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(A3),minus_6452836326544984404t_unit(C5,A))
               => ( aa_Product_unit_nat(G,A3) = one_one_nat ) )
           => ( ! [B2: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(B2),minus_6452836326544984404t_unit(C5,B4))
                 => ( aa_Product_unit_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),C5) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),C5) )
               => ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),A) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7243_prod_Osame__carrierI,axiom,
    ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),C5)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C5)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C5,A))
               => ( aa_Code_integer_nat(G,A3) = one_one_nat ) )
           => ( ! [B2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(C5,B4))
                 => ( aa_Code_integer_nat(Ha,B2) = one_one_nat ) )
             => ( ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),C5) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),C5) )
               => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7244_prod_Osame__carrierI,axiom,
    ! [C5: set_int,A: set_int,B4: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,C5)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,A),C5)
       => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,B4),C5)
         => ( ! [A3: int] :
                ( aa_set_int_o(member_int(A3),minus_minus_set_int(C5,A))
               => ( aa_int_int(G,A3) = one_one_int ) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(C5,B4))
                 => ( aa_int_int(Ha,B2) = one_one_int ) )
             => ( ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),C5) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),C5) )
               => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),A) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7245_prod_Osame__carrierI,axiom,
    ! [C5: set_o,A: set_o,B4: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,C5)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,A),C5)
       => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,B4),C5)
         => ( ! [A3: $o] :
                ( aa_set_o_o(member_o((A3)),minus_minus_set_o(C5,A))
               => ( aa_o_int(G,(A3)) = one_one_int ) )
           => ( ! [B2: $o] :
                  ( aa_set_o_o(member_o((B2)),minus_minus_set_o(C5,B4))
                 => ( aa_o_int(Ha,(B2)) = one_one_int ) )
             => ( ( groups3502327434004483295_o_int(G,C5) = groups3502327434004483295_o_int(Ha,C5) )
               => ( groups3502327434004483295_o_int(G,A) = groups3502327434004483295_o_int(Ha,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7246_prod_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),S) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7247_prod_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),S) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7248_prod_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_int(G,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,S) = groups3188404863801439024er_int(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7249_prod_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(G,S) = groups3674199335183972705nteger(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7250_prod_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co3734316457315071362atural] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Cod607569804290114243atural(G,X3) = one_one_Code_natural ) )
         => ( groups2497794486940690220atural(G,S) = groups2497794486940690220atural(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7251_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_assn(G,X3) = one_one_assn ) )
         => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),S) = aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7252_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_int(G,X3) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),S) = aa_set_nat_int(groups705719431365010083at_int(G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7253_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( groups3455450783089532116nteger(G,S) = groups3455450783089532116nteger(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7254_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_Code_natural(G,X3) = one_one_Code_natural ) )
         => ( groups2279045934846249631atural(G,S) = groups2279045934846249631atural(G,T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7255_prod_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_nat(G,X3) = one_one_nat ) )
         => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),S) = aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),T2) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7256_prod_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),T2) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7257_prod_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),T2) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7258_prod_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_int(G,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,T2) = groups3188404863801439024er_int(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7259_prod_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(G,T2) = groups3674199335183972705nteger(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7260_prod_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co3734316457315071362atural] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Cod607569804290114243atural(G,X3) = one_one_Code_natural ) )
         => ( groups2497794486940690220atural(G,T2) = groups2497794486940690220atural(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7261_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_assn(G,X3) = one_one_assn ) )
         => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),T2) = aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7262_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_int(G,X3) = one_one_int ) )
         => ( aa_set_nat_int(groups705719431365010083at_int(G),T2) = aa_set_nat_int(groups705719431365010083at_int(G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7263_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( groups3455450783089532116nteger(G,T2) = groups3455450783089532116nteger(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7264_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_Code_natural(G,X3) = one_one_Code_natural ) )
         => ( groups2279045934846249631atural(G,T2) = groups2279045934846249631atural(G,S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7265_prod_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),T2)
       => ( ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(minus_minus_set_nat(T2),S))
             => ( aa_nat_nat(G,X3) = one_one_nat ) )
         => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),T2) = aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),S) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7266_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S: set_int,Ha: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S))
             => ( aa_int_assn(Ha,X3) = one_one_assn ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ( aa_int_assn(G,X3) = aa_int_assn(Ha,X3) ) )
           => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),S) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7267_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S: set_o,Ha: fun_o_assn,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),T2)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T2,S))
             => ( aa_o_assn(Ha,(X3)) = one_one_assn ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_assn(G,(X3)) = aa_o_assn(Ha,(X3)) ) )
           => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),S) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7268_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_Product_unit,S: set_Product_unit,Ha: fun_Pr6626665806141890376t_assn,G: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,S),T2)
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),minus_6452836326544984404t_unit(T2,S))
             => ( aa_Product_unit_assn(Ha,X3) = one_one_assn ) )
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),S)
               => ( aa_Product_unit_assn(G,X3) = aa_Product_unit_assn(Ha,X3) ) )
           => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),S) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7269_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,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(G,X3) = aa_Code_integer_assn(Ha,X3) ) )
           => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),S) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7270_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S))
             => ( aa_int_nat(Ha,X3) = one_one_nat ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ( aa_int_nat(G,X3) = aa_int_nat(Ha,X3) ) )
           => ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),S) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7271_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),T2)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T2,S))
             => ( aa_o_nat(Ha,(X3)) = one_one_nat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),S) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7272_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_Product_unit,S: set_Product_unit,Ha: fun_Product_unit_nat,G: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,S),T2)
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),minus_6452836326544984404t_unit(T2,S))
             => ( aa_Product_unit_nat(Ha,X3) = one_one_nat ) )
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),S)
               => ( aa_Product_unit_nat(G,X3) = aa_Product_unit_nat(Ha,X3) ) )
           => ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),S) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7273_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_nat(Ha,X3) = one_one_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),S) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7274_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_int,S: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S))
             => ( aa_int_int(Ha,X3) = one_one_int ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ( aa_int_int(G,X3) = aa_int_int(Ha,X3) ) )
           => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),S) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7275_prod_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),T2)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T2,S))
             => ( aa_o_int(Ha,(X3)) = one_one_int ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups3502327434004483295_o_int(G,S) = groups3502327434004483295_o_int(Ha,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7276_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S: set_int,G: fun_int_assn,Ha: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S))
             => ( aa_int_assn(G,X3) = one_one_assn ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ( aa_int_assn(G,X3) = aa_int_assn(Ha,X3) ) )
           => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),T2) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7277_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S: set_o,G: fun_o_assn,Ha: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),T2)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T2,S))
             => ( aa_o_assn(G,(X3)) = one_one_assn ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_assn(G,(X3)) = aa_o_assn(Ha,(X3)) ) )
           => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),T2) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7278_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_Product_unit,S: set_Product_unit,G: fun_Pr6626665806141890376t_assn,Ha: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,S),T2)
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),minus_6452836326544984404t_unit(T2,S))
             => ( aa_Product_unit_assn(G,X3) = one_one_assn ) )
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),S)
               => ( aa_Product_unit_assn(G,X3) = aa_Product_unit_assn(Ha,X3) ) )
           => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),T2) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7279_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Co4490034186337852034r_assn,Ha: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_assn(G,X3) = aa_Code_integer_assn(Ha,X3) ) )
           => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),T2) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7280_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S: set_int,G: fun_int_nat,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S))
             => ( aa_int_nat(G,X3) = one_one_nat ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ( aa_int_nat(G,X3) = aa_int_nat(Ha,X3) ) )
           => ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),T2) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7281_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S: set_o,G: fun_o_nat,Ha: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),T2)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T2,S))
             => ( aa_o_nat(G,(X3)) = one_one_nat ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
           => ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),T2) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7282_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_Product_unit,S: set_Product_unit,G: fun_Product_unit_nat,Ha: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T2)
     => ( aa_set8780210009671629607unit_o(aa_set4216242928598786527unit_o(ord_le3507040750410214029t_unit,S),T2)
       => ( ! [X3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(X3),minus_6452836326544984404t_unit(T2,S))
             => ( aa_Product_unit_nat(G,X3) = one_one_nat ) )
         => ( ! [X3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(X3),S)
               => ( aa_Product_unit_nat(G,X3) = aa_Product_unit_nat(Ha,X3) ) )
           => ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),T2) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7283_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,G: fun_Code_integer_nat,Ha: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T2)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( ! [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
               => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
           => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),T2) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7284_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_int,S: set_int,G: fun_int_int,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(aa_set5425576183772390127_int_o(ord_less_eq_set_int,S),T2)
       => ( ! [X3: int] :
              ( aa_set_int_o(member_int(X3),minus_minus_set_int(T2,S))
             => ( aa_int_int(G,X3) = one_one_int ) )
         => ( ! [X3: int] :
                ( aa_set_int_o(member_int(X3),S)
               => ( aa_int_int(G,X3) = aa_int_int(Ha,X3) ) )
           => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),T2) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7285_prod_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S: set_o,G: fun_o_int,Ha: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(aa_set_o_fun_set_o_o(ord_less_eq_set_o,S),T2)
       => ( ! [X3: $o] :
              ( aa_set_o_o(member_o((X3)),minus_minus_set_o(T2,S))
             => ( aa_o_int(G,(X3)) = one_one_int ) )
         => ( ! [X3: $o] :
                ( aa_set_o_o(member_o((X3)),S)
               => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
           => ( groups3502327434004483295_o_int(G,T2) = groups3502327434004483295_o_int(Ha,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7286_prod_Ounion__inter,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),sup_su848401254843788991nteger(A,B4))),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),inf_in1364745209274528805nteger(A,B4))) = aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),B4)) ) ) ) ).

% prod.union_inter
tff(fact_7287_prod_Ounion__inter,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) = aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),B4)) ) ) ) ).

% prod.union_inter
tff(fact_7288_prod_Ounion__inter,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A,B4))),groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A,B4))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,A)),groups2555765274223993564er_rat(G,B4)) ) ) ) ).

% prod.union_inter
tff(fact_7289_prod_Ounion__inter,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))),groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,A)),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.union_inter
tff(fact_7290_prod_Ounion__inter,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),sup_su848401254843788991nteger(A,B4))),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),inf_in1364745209274528805nteger(A,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A)),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),B4)) ) ) ) ).

% prod.union_inter
tff(fact_7291_prod_Ounion__inter,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,sup_su848401254843788991nteger(A,B4))),groups3188404863801439024er_int(G,inf_in1364745209274528805nteger(A,B4))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,A)),groups3188404863801439024er_int(G,B4)) ) ) ) ).

% prod.union_inter
tff(fact_7292_prod_Ounion__inter,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),A)),aa_set_nat_int(groups705719431365010083at_int(G),B4)) ) ) ) ).

% prod.union_inter
tff(fact_7293_prod_Ounion__inter,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4))),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),A)),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),B4)) ) ) ) ).

% prod.union_inter
tff(fact_7294_prod_OInt__Diff,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A) = aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),inf_in1364745209274528805nteger(A,B4))),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),minus_2355218937544613996nteger(A,B4))) ) ) ).

% prod.Int_Diff
tff(fact_7295_prod_OInt__Diff,axiom,
    ! [A: set_nat,G: fun_nat_assn,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A) = aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ) ) ).

% prod.Int_Diff
tff(fact_7296_prod_OInt__Diff,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups2555765274223993564er_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A,B4))),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,B4))) ) ) ).

% prod.Int_Diff
tff(fact_7297_prod_OInt__Diff,axiom,
    ! [A: set_nat,G: fun_nat_rat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups73079841787564623at_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),groups73079841787564623at_rat(G,aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ) ) ).

% prod.Int_Diff
tff(fact_7298_prod_OInt__Diff,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_nat,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),inf_in1364745209274528805nteger(A,B4))),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),minus_2355218937544613996nteger(A,B4))) ) ) ).

% prod.Int_Diff
tff(fact_7299_prod_OInt__Diff,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_int,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups3188404863801439024er_int(G,A) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,inf_in1364745209274528805nteger(A,B4))),groups3188404863801439024er_int(G,minus_2355218937544613996nteger(A,B4))) ) ) ).

% prod.Int_Diff
tff(fact_7300_prod_OInt__Diff,axiom,
    ! [A: set_nat,G: fun_nat_int,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_int(groups705719431365010083at_int(G),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ) ) ).

% prod.Int_Diff
tff(fact_7301_prod_OInt__Diff,axiom,
    ! [A: set_nat,G: fun_nat_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))) ) ) ).

% prod.Int_Diff
tff(fact_7302_prod_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,Ha: fun_int_assn,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),minus_minus_set_int(T2,S))
             => ( aa_int_assn(Ha,I3) = one_one_assn ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),minus_minus_set_int(S,T2))
               => ( aa_int_assn(G,I3) = one_one_assn ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S,T2))
                 => ( aa_int_assn(G,X3) = aa_int_assn(Ha,X3) ) )
             => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),S) = aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7303_prod_Omono__neutral__cong,axiom,
    ! [T2: set_o,S: set_o,Ha: fun_o_assn,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(finite_finite_o,S)
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T2,S))
             => ( aa_o_assn(Ha,(I3)) = one_one_assn ) )
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),minus_minus_set_o(S,T2))
               => ( aa_o_assn(G,(I3)) = one_one_assn ) )
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),inf_inf_set_o(S,T2))
                 => ( aa_o_assn(G,(X3)) = aa_o_assn(Ha,(X3)) ) )
             => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),S) = aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7304_prod_Omono__neutral__cong,axiom,
    ! [T2: set_Product_unit,S: set_Product_unit,Ha: fun_Pr6626665806141890376t_assn,G: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T2)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),minus_6452836326544984404t_unit(T2,S))
             => ( aa_Product_unit_assn(Ha,I3) = one_one_assn ) )
         => ( ! [I3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(I3),minus_6452836326544984404t_unit(S,T2))
               => ( aa_Product_unit_assn(G,I3) = one_one_assn ) )
           => ( ! [X3: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X3),inf_in4660618365625256667t_unit(S,T2))
                 => ( aa_Product_unit_assn(G,X3) = aa_Product_unit_assn(Ha,X3) ) )
             => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),S) = aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7305_prod_Omono__neutral__cong,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,Ha: fun_Co4490034186337852034r_assn,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_assn(Ha,I3) = one_one_assn ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T2))
               => ( aa_Code_integer_assn(G,I3) = one_one_assn ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T2))
                 => ( aa_Code_integer_assn(G,X3) = aa_Code_integer_assn(Ha,X3) ) )
             => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),S) = aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7306_prod_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,Ha: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),minus_minus_set_int(T2,S))
             => ( aa_int_nat(Ha,I3) = one_one_nat ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),minus_minus_set_int(S,T2))
               => ( aa_int_nat(G,I3) = one_one_nat ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S,T2))
                 => ( aa_int_nat(G,X3) = aa_int_nat(Ha,X3) ) )
             => ( aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,G),S) = aa_set_int_nat(aa_fun5122536703717624806nt_nat(groups1707563613775114915nt_nat,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7307_prod_Omono__neutral__cong,axiom,
    ! [T2: set_o,S: set_o,Ha: fun_o_nat,G: fun_o_nat] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(finite_finite_o,S)
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T2,S))
             => ( aa_o_nat(Ha,(I3)) = one_one_nat ) )
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),minus_minus_set_o(S,T2))
               => ( aa_o_nat(G,(I3)) = one_one_nat ) )
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),inf_inf_set_o(S,T2))
                 => ( aa_o_nat(G,(X3)) = aa_o_nat(Ha,(X3)) ) )
             => ( aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,G),S) = aa_set_o_nat(aa_fun8470299178418016604_o_nat(groups3504817904513533571_o_nat,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7308_prod_Omono__neutral__cong,axiom,
    ! [T2: set_Product_unit,S: set_Product_unit,Ha: fun_Product_unit_nat,G: fun_Product_unit_nat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,T2)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,S)
       => ( ! [I3: product_unit] :
              ( aa_set8780210009671629607unit_o(member_Product_unit(I3),minus_6452836326544984404t_unit(T2,S))
             => ( aa_Product_unit_nat(Ha,I3) = one_one_nat ) )
         => ( ! [I3: product_unit] :
                ( aa_set8780210009671629607unit_o(member_Product_unit(I3),minus_6452836326544984404t_unit(S,T2))
               => ( aa_Product_unit_nat(G,I3) = one_one_nat ) )
           => ( ! [X3: product_unit] :
                  ( aa_set8780210009671629607unit_o(member_Product_unit(X3),inf_in4660618365625256667t_unit(S,T2))
                 => ( aa_Product_unit_nat(G,X3) = aa_Product_unit_nat(Ha,X3) ) )
             => ( aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,G),S) = aa_set1974266514704922113it_nat(aa_fun5089188478758539420it_nat(groups396421249026171384it_nat,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7309_prod_Omono__neutral__cong,axiom,
    ! [T2: set_Code_integer,S: set_Code_integer,Ha: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ( ! [I3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(T2,S))
             => ( aa_Code_integer_nat(Ha,I3) = one_one_nat ) )
         => ( ! [I3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I3),minus_2355218937544613996nteger(S,T2))
               => ( aa_Code_integer_nat(G,I3) = one_one_nat ) )
           => ( ! [X3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(S,T2))
                 => ( aa_Code_integer_nat(G,X3) = aa_Code_integer_nat(Ha,X3) ) )
             => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),S) = aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7310_prod_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,Ha: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( aa_set_int_o(finite_finite_int,S)
       => ( ! [I3: int] :
              ( aa_set_int_o(member_int(I3),minus_minus_set_int(T2,S))
             => ( aa_int_int(Ha,I3) = one_one_int ) )
         => ( ! [I3: int] :
                ( aa_set_int_o(member_int(I3),minus_minus_set_int(S,T2))
               => ( aa_int_int(G,I3) = one_one_int ) )
           => ( ! [X3: int] :
                  ( aa_set_int_o(member_int(X3),inf_inf_set_int(S,T2))
                 => ( aa_int_int(G,X3) = aa_int_int(Ha,X3) ) )
             => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,G),S) = aa_set_int_int(aa_fun5337148683239322782nt_int(groups1705073143266064639nt_int,Ha),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7311_prod_Omono__neutral__cong,axiom,
    ! [T2: set_o,S: set_o,Ha: fun_o_int,G: fun_o_int] :
      ( aa_set_o_o(finite_finite_o,T2)
     => ( aa_set_o_o(finite_finite_o,S)
       => ( ! [I3: $o] :
              ( aa_set_o_o(member_o((I3)),minus_minus_set_o(T2,S))
             => ( aa_o_int(Ha,(I3)) = one_one_int ) )
         => ( ! [I3: $o] :
                ( aa_set_o_o(member_o((I3)),minus_minus_set_o(S,T2))
               => ( aa_o_int(G,(I3)) = one_one_int ) )
           => ( ! [X3: $o] :
                  ( aa_set_o_o(member_o((X3)),inf_inf_set_o(S,T2))
                 => ( aa_o_int(G,(X3)) = aa_o_int(Ha,(X3)) ) )
             => ( groups3502327434004483295_o_int(G,S) = groups3502327434004483295_o_int(Ha,T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_7312_prod_Oeq__fold,axiom,
    ! [G: fun_nat_nat,A: set_nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),A) = aa_set_nat_nat(finite_fold_nat_nat(comp_n7084669201029016610at_nat(times_times_nat,G),one_one_nat),A) ).

% prod.eq_fold
tff(fact_7313_sum__bounded__above,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer,K4: code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,I3)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A))),K4)) ) ).

% sum_bounded_above
tff(fact_7314_sum__bounded__above,axiom,
    ! [A: set_int,F: fun_int_Code_integer,K4: code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I3)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7315_sum__bounded__above,axiom,
    ! [A: set_o,F: fun_o_Code_integer,K4: code_integer] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(I3))),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups4406642042086082107nteger(F,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_o(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7316_sum__bounded__above,axiom,
    ! [A: set_Product_unit,F: fun_Pr1820225556744648189nteger,K4: code_integer] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Pro1705297278780387758nteger(F,I3)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups5393439102872367408nteger(F,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite410649719033368117t_unit(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7317_sum__bounded__above,axiom,
    ! [A: set_nat,F: fun_nat_rat,K4: rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),K4) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A))),K4)) ) ).

% sum_bounded_above
tff(fact_7318_sum__bounded__above,axiom,
    ! [A: set_int,F: fun_int_rat,K4: rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),K4) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7319_sum__bounded__above,axiom,
    ! [A: set_o,F: fun_o_rat,K4: rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(I3))),K4) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups7872700643590313910_o_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_o(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7320_sum__bounded__above,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_rat,K4: rat] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Product_unit_rat(F,I3)),K4) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups8131667117488461483it_rat(F,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite410649719033368117t_unit(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7321_sum__bounded__above,axiom,
    ! [A: set_int,F: fun_int_nat,K4: nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),K4) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_int(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7322_sum__bounded__above,axiom,
    ! [A: set_o,F: fun_o_nat,K4: nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(I3))),K4) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8507830703676809646_o_nat(F,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,finite_card_o(A))),K4)) ) ).

% sum_bounded_above
tff(fact_7323_sum__bounded__below,axiom,
    ! [A: set_nat,K4: code_integer,F: fun_nat_Code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_nat_Code_integer(F,I3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(finite_card_nat,A))),K4)),groups7501900531339628137nteger(F,A)) ) ).

% sum_bounded_below
tff(fact_7324_sum__bounded__below,axiom,
    ! [A: set_int,K4: code_integer,F: fun_int_Code_integer] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_int_Code_integer(F,I3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A))),K4)),groups7873554091576472773nteger(F,A)) ) ).

% sum_bounded_below
tff(fact_7325_sum__bounded__below,axiom,
    ! [A: set_o,K4: code_integer,F: fun_o_Code_integer] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_o_Code_integer(F,(I3))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_o(A))),K4)),groups4406642042086082107nteger(F,A)) ) ).

% sum_bounded_below
tff(fact_7326_sum__bounded__below,axiom,
    ! [A: set_Product_unit,K4: code_integer,F: fun_Pr1820225556744648189nteger] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_Pro1705297278780387758nteger(F,I3)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite410649719033368117t_unit(A))),K4)),groups5393439102872367408nteger(F,A)) ) ).

% sum_bounded_below
tff(fact_7327_sum__bounded__below,axiom,
    ! [A: set_nat,K4: rat,F: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K4),aa_nat_rat(F,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(finite_card_nat,A))),K4)),groups2906978787729119204at_rat(F,A)) ) ).

% sum_bounded_below
tff(fact_7328_sum__bounded__below,axiom,
    ! [A: set_int,K4: rat,F: fun_int_rat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K4),aa_int_rat(F,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_int(A))),K4)),groups3906332499630173760nt_rat(F,A)) ) ).

% sum_bounded_below
tff(fact_7329_sum__bounded__below,axiom,
    ! [A: set_o,K4: rat,F: fun_o_rat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K4),aa_o_rat(F,(I3))) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite_card_o(A))),K4)),groups7872700643590313910_o_rat(F,A)) ) ).

% sum_bounded_below
tff(fact_7330_sum__bounded__below,axiom,
    ! [A: set_Product_unit,K4: rat,F: fun_Product_unit_rat] :
      ( ! [I3: product_unit] :
          ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K4),aa_Product_unit_rat(F,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,finite410649719033368117t_unit(A))),K4)),groups8131667117488461483it_rat(F,A)) ) ).

% sum_bounded_below
tff(fact_7331_sum__bounded__below,axiom,
    ! [A: set_int,K4: nat,F: fun_int_nat] :
      ( ! [I3: int] :
          ( aa_set_int_o(member_int(I3),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),aa_int_nat(F,I3)) )
     => 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(semiri1316708129612266289at_nat,finite_card_int(A))),K4)),groups4541462559716669496nt_nat(F,A)) ) ).

% sum_bounded_below
tff(fact_7332_sum__bounded__below,axiom,
    ! [A: set_o,K4: nat,F: fun_o_nat] :
      ( ! [I3: $o] :
          ( aa_set_o_o(member_o((I3)),A)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),aa_o_nat(F,(I3))) )
     => 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(semiri1316708129612266289at_nat,finite_card_o(A))),K4)),groups8507830703676809646_o_nat(F,A)) ) ).

% sum_bounded_below
tff(fact_7333_gbinomial__absorb__comp,axiom,
    ! [A2: rat,K3: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K3))),aa_nat_rat(gbinomial_rat(A2),K3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K3)) ).

% gbinomial_absorb_comp
tff(fact_7334_gbinomial__mult__1,axiom,
    ! [A2: rat,K3: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(gbinomial_rat(A2),K3)) = 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,K3)),aa_nat_rat(gbinomial_rat(A2),K3))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K3))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K3)))) ).

% gbinomial_mult_1
tff(fact_7335_gbinomial__mult__1_H,axiom,
    ! [A2: rat,K3: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A2),K3)),A2) = 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,K3)),aa_nat_rat(gbinomial_rat(A2),K3))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K3))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K3)))) ).

% gbinomial_mult_1'
tff(fact_7336_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_fu3555266933746557007_nat_o,Y: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(X,Y) = bot_bo8632255151191677115_nat_o )
     => ( ( aa_set1652341062091217276_nat_o(aa_set7740321069979895062_nat_o(sup_su2896084399781877795_nat_o,X),Y) = top_to9044565752620676511_nat_o )
       => ( uminus2199602881456817606_nat_o(X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7337_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit(X,Y) = bot_bo3957492148770167129t_unit )
     => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y) = top_to1996260823553986621t_unit )
       => ( uminus5944136376168626660t_unit(X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7338_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o(X,Y) = bot_bot_set_o )
     => ( ( aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,X),Y) = top_top_set_o )
       => ( uminus_uminus_set_o(X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7339_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,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_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = top_top_set_nat )
       => ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7340_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int(X,Y) = bot_bot_set_int )
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = top_top_set_int )
       => ( uminus1532241313380277803et_int(X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7341_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
     => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = top_top_assn )
       => ( aa_assn_assn(uminus_uminus_assn,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7342_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,X),Y) = bot_bot_Product_unit )
     => ( ( aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,X),Y) = top_top_Product_unit )
       => ( aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_7343_prod__mono__strict,axiom,
    ! [A: set_Code_integer,F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F,I3)),aa_Cod1783974652533396728nteger(G,I3)) ) )
       => ( ( A != bot_bo3990330152332043303nteger )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3674199335183972705nteger(F,A)),groups3674199335183972705nteger(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7344_prod__mono__strict,axiom,
    ! [A: set_Product_unit,F: fun_Pr1820225556744648189nteger,G: fun_Pr1820225556744648189nteger] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Pro1705297278780387758nteger(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Pro1705297278780387758nteger(F,I3)),aa_Pro1705297278780387758nteger(G,I3)) ) )
       => ( ( A != bot_bo3957492148770167129t_unit )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups8185359811944078981nteger(F,A)),groups8185359811944078981nteger(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7345_prod__mono__strict,axiom,
    ! [A: set_o,F: fun_o_Code_integer,G: fun_o_Code_integer] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_o_Code_integer(F,(I3)))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_o_Code_integer(F,(I3))),aa_o_Code_integer(G,(I3))) ) )
       => ( ( A != bot_bot_set_o )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7694694392188491536nteger(F,A)),groups7694694392188491536nteger(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7346_prod__mono__strict,axiom,
    ! [A: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(F,I3)),aa_nat_Code_integer(G,I3)) ) )
       => ( ( A != bot_bot_set_nat )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3455450783089532116nteger(F,A)),groups3455450783089532116nteger(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7347_prod__mono__strict,axiom,
    ! [A: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A)
           => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(F,I3))
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,I3)),aa_int_Code_integer(G,I3)) ) )
       => ( ( A != bot_bot_set_int )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups3827104343326376752nteger(F,A)),groups3827104343326376752nteger(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7348_prod__mono__strict,axiom,
    ! [A: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [I3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I3),A)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,I3)),aa_Code_integer_rat(G,I3)) ) )
       => ( ( A != bot_bo3990330152332043303nteger )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2555765274223993564er_rat(F,A)),groups2555765274223993564er_rat(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7349_prod__mono__strict,axiom,
    ! [A: set_Product_unit,F: fun_Product_unit_rat,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( ! [I3: product_unit] :
            ( aa_set8780210009671629607unit_o(member_Product_unit(I3),A)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Product_unit_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Product_unit_rat(F,I3)),aa_Product_unit_rat(G,I3)) ) )
       => ( ( A != bot_bo3957492148770167129t_unit )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups8984663225794451456it_rat(F,A)),groups8984663225794451456it_rat(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7350_prod__mono__strict,axiom,
    ! [A: set_o,F: fun_o_rat,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ! [I3: $o] :
            ( aa_set_o_o(member_o((I3)),A)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_o_rat(F,(I3)))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(I3))),aa_o_rat(G,(I3))) ) )
       => ( ( A != bot_bot_set_o )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2869687844427037835_o_rat(F,A)),groups2869687844427037835_o_rat(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7351_prod__mono__strict,axiom,
    ! [A: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),A)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
       => ( ( A != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups73079841787564623at_rat(F,A)),groups73079841787564623at_rat(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7352_prod__mono__strict,axiom,
    ! [A: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [I3: int] :
            ( aa_set_int_o(member_int(I3),A)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
       => ( ( A != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A)),groups1072433553688619179nt_rat(G,A)) ) ) ) ).

% prod_mono_strict
tff(fact_7353_prod_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Co4490034186337852034r_assn,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,X)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% prod.insert_remove
tff(fact_7354_prod_Oinsert__remove,axiom,
    ! [A: set_Product_unit,G: fun_Pr6626665806141890376t_assn,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_assn_assn(times_times_assn(aa_Product_unit_assn(G,X)),aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% prod.insert_remove
tff(fact_7355_prod_Oinsert__remove,axiom,
    ! [A: set_o,G: fun_o_assn,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_assn_assn(times_times_assn(aa_o_assn(G,(X))),aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_7356_prod_Oinsert__remove,axiom,
    ! [A: set_int,G: fun_int_assn,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_assn_assn(times_times_assn(aa_int_assn(G,X)),aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_7357_prod_Oinsert__remove,axiom,
    ! [A: set_nat,G: fun_nat_assn,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_assn_assn(times_times_assn(aa_nat_assn(G,X)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_7358_prod_Oinsert__remove,axiom,
    ! [A: set_Code_integer,G: fun_Code_integer_rat,X: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( groups2555765274223993564er_rat(G,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(G,X)),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ).

% prod.insert_remove
tff(fact_7359_prod_Oinsert__remove,axiom,
    ! [A: set_Product_unit,G: fun_Product_unit_rat,X: product_unit] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( groups8984663225794451456it_rat(G,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Product_unit_rat(G,X)),groups8984663225794451456it_rat(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ).

% prod.insert_remove
tff(fact_7360_prod_Oinsert__remove,axiom,
    ! [A: set_o,G: fun_o_rat,X: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( groups2869687844427037835_o_rat(G,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ).

% prod.insert_remove
tff(fact_7361_prod_Oinsert__remove,axiom,
    ! [A: set_int,G: fun_int_rat,X: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) ) ) ).

% prod.insert_remove
tff(fact_7362_prod_Oinsert__remove,axiom,
    ! [A: set_nat,G: fun_nat_rat,X: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),groups73079841787564623at_rat(G,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) ) ) ).

% prod.insert_remove
tff(fact_7363_prod_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A) = aa_assn_assn(times_times_assn(aa_Code_integer_assn(G,X)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% prod.remove
tff(fact_7364_prod_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A) = aa_assn_assn(times_times_assn(aa_Product_unit_assn(G,X)),aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% prod.remove
tff(fact_7365_prod_Oremove,axiom,
    ! [A: set_o,X: $o,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A) = aa_assn_assn(times_times_assn(aa_o_assn(G,(X))),aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_7366_prod_Oremove,axiom,
    ! [A: set_int,X: int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A) = aa_assn_assn(times_times_assn(aa_int_assn(G,X)),aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_7367_prod_Oremove,axiom,
    ! [A: set_nat,X: nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A) = aa_assn_assn(times_times_assn(aa_nat_assn(G,X)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_7368_prod_Oremove,axiom,
    ! [A: set_Code_integer,X: code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(X),A)
       => ( groups2555765274223993564er_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(G,X)),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(A,aa_set8622560600277679972nteger(aa_Cod6004039685867158334nteger(insert_Code_integer,X),bot_bo3990330152332043303nteger)))) ) ) ) ).

% prod.remove
tff(fact_7369_prod_Oremove,axiom,
    ! [A: set_Product_unit,X: product_unit,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(member_Product_unit(X),A)
       => ( groups8984663225794451456it_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Product_unit_rat(G,X)),groups8984663225794451456it_rat(G,minus_6452836326544984404t_unit(A,aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,X),bot_bo3957492148770167129t_unit)))) ) ) ) ).

% prod.remove
tff(fact_7370_prod_Oremove,axiom,
    ! [A: set_o,X: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(member_o((X)),A)
       => ( groups2869687844427037835_o_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_o_rat(G,(X))),groups2869687844427037835_o_rat(G,minus_minus_set_o(A,aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,(X)),bot_bot_set_o)))) ) ) ) ).

% prod.remove
tff(fact_7371_prod_Oremove,axiom,
    ! [A: set_int,X: int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(X),A)
       => ( groups1072433553688619179nt_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,minus_minus_set_int(A,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,X),bot_bot_set_int)))) ) ) ) ).

% prod.remove
tff(fact_7372_prod_Oremove,axiom,
    ! [A: set_nat,X: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(X),A)
       => ( groups73079841787564623at_rat(G,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,X)),groups73079841787564623at_rat(G,aa_set_nat_set_nat(minus_minus_set_nat(A),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,X),bot_bot_set_nat)))) ) ) ) ).

% prod.remove
tff(fact_7373_prod_Ounion__inter__neutral,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co8499238673658658103nteger] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A,B4))
             => ( aa_Cod1783974652533396728nteger(G,X3) = one_one_Code_integer ) )
         => ( groups3674199335183972705nteger(G,sup_su848401254843788991nteger(A,B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3674199335183972705nteger(G,A)),groups3674199335183972705nteger(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7374_prod_Ounion__inter__neutral,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co3734316457315071362atural] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A,B4))
             => ( aa_Cod607569804290114243atural(G,X3) = one_one_Code_natural ) )
         => ( groups2497794486940690220atural(G,sup_su848401254843788991nteger(A,B4)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,groups2497794486940690220atural(G,A)),groups2497794486940690220atural(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7375_prod_Ounion__inter__neutral,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( 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,A),B4))
             => ( aa_nat_Code_integer(G,X3) = one_one_Code_integer ) )
         => ( groups3455450783089532116nteger(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,groups3455450783089532116nteger(G,A)),groups3455450783089532116nteger(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7376_prod_Ounion__inter__neutral,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_Code_natural] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( 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,A),B4))
             => ( aa_nat_Code_natural(G,X3) = one_one_Code_natural ) )
         => ( groups2279045934846249631atural(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,groups2279045934846249631atural(G,A)),groups2279045934846249631atural(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7377_prod_Ounion__inter__neutral,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A,B4))
             => ( aa_Code_integer_assn(G,X3) = one_one_assn ) )
         => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),sup_su848401254843788991nteger(A,B4)) = aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7378_prod_Ounion__inter__neutral,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( 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,A),B4))
             => ( aa_nat_assn(G,X3) = one_one_assn ) )
         => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7379_prod_Ounion__inter__neutral,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A,B4))
             => ( aa_Code_integer_rat(G,X3) = one_one_rat ) )
         => ( groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,A)),groups2555765274223993564er_rat(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7380_prod_Ounion__inter__neutral,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( 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,A),B4))
             => ( aa_nat_rat(G,X3) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,A)),groups73079841787564623at_rat(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7381_prod_Ounion__inter__neutral,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A,B4))
             => ( aa_Code_integer_nat(G,X3) = one_one_nat ) )
         => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),sup_su848401254843788991nteger(A,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),A)),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7382_prod_Ounion__inter__neutral,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),inf_in1364745209274528805nteger(A,B4))
             => ( aa_Code_integer_int(G,X3) = one_one_int ) )
         => ( groups3188404863801439024er_int(G,sup_su848401254843788991nteger(A,B4)) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3188404863801439024er_int(G,A)),groups3188404863801439024er_int(G,B4)) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7383_prod_Ounion__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),sup_su848401254843788991nteger(A,B4)) = aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),A)),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7384_prod_Ounion__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,G: fun_Pr6626665806141890376t_assn] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_assn_assn(times_times_assn(aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),A)),aa_set6681470406643656025t_assn(aa_fun5848166259052469404t_assn(groups5360233303959380816t_assn,G),B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7385_prod_Ounion__disjoint,axiom,
    ! [A: set_o,B4: set_o,G: fun_o_assn] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
         => ( aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) = aa_assn_assn(times_times_assn(aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),A)),aa_set_o_assn(aa_fun1914167119812169564o_assn(groups5301882518646026715o_assn,G),B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7386_prod_Ounion__disjoint,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
         => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),A)),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7387_prod_Ounion__disjoint,axiom,
    ! [A: set_int,B4: set_int,G: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
         => ( aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_assn_assn(times_times_assn(aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),A)),aa_set_int_assn(aa_fun6757465843467192982t_assn(groups7882442080178216443t_assn,G),B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7388_prod_Ounion__disjoint,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ( inf_in1364745209274528805nteger(A,B4) = bot_bo3990330152332043303nteger )
         => ( groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A,B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2555765274223993564er_rat(G,A)),groups2555765274223993564er_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7389_prod_Ounion__disjoint,axiom,
    ! [A: set_Product_unit,B4: set_Product_unit,G: fun_Product_unit_rat] :
      ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,A)
     => ( aa_set8780210009671629607unit_o(finite4290736615968046902t_unit,B4)
       => ( ( inf_in4660618365625256667t_unit(A,B4) = bot_bo3957492148770167129t_unit )
         => ( groups8984663225794451456it_rat(G,aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups8984663225794451456it_rat(G,A)),groups8984663225794451456it_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7390_prod_Ounion__disjoint,axiom,
    ! [A: set_o,B4: set_o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(finite_finite_o,B4)
       => ( ( inf_inf_set_o(A,B4) = bot_bot_set_o )
         => ( groups2869687844427037835_o_rat(G,aa_set_o_set_o(aa_set2146880807996572927_set_o(sup_sup_set_o,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2869687844427037835_o_rat(G,A)),groups2869687844427037835_o_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7391_prod_Ounion__disjoint,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4) = bot_bot_set_nat )
         => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,A)),groups73079841787564623at_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7392_prod_Ounion__disjoint,axiom,
    ! [A: set_int,B4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ( inf_inf_set_int(A,B4) = bot_bot_set_int )
         => ( groups1072433553688619179nt_rat(G,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups1072433553688619179nt_rat(G,A)),groups1072433553688619179nt_rat(G,B4)) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7393_prod_Ounion__diff2,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Co4490034186337852034r_assn] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),sup_su848401254843788991nteger(A,B4)) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),minus_2355218937544613996nteger(A,B4))),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),minus_2355218937544613996nteger(B4,A)))),aa_set7873363181767135481r_assn(aa_fun4520394192108194424r_assn(groups1304777262505850412r_assn,G),inf_in1364745209274528805nteger(A,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7394_prod_Ounion__diff2,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)))),aa_set_nat_assn(aa_fun7221275712416644062t_assn(groups6906906614972039071t_assn,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7395_prod_Ounion__diff2,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups2555765274223993564er_rat(G,sup_su848401254843788991nteger(A,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(G,minus_2355218937544613996nteger(A,B4))),groups2555765274223993564er_rat(G,minus_2355218937544613996nteger(B4,A)))),groups2555765274223993564er_rat(G,inf_in1364745209274528805nteger(A,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7396_prod_Ounion__diff2,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),groups73079841787564623at_rat(G,aa_set_nat_set_nat(minus_minus_set_nat(B4),A)))),groups73079841787564623at_rat(G,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7397_prod_Ounion__diff2,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),sup_su848401254843788991nteger(A,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),minus_2355218937544613996nteger(A,B4))),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),minus_2355218937544613996nteger(B4,A)))),aa_set4154159416444559777er_nat(aa_fun597722859780706760er_nat(groups3190895334310489300er_nat,G),inf_in1364745209274528805nteger(A,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7398_prod_Ounion__diff2,axiom,
    ! [A: set_Code_integer,B4: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( groups3188404863801439024er_int(G,sup_su848401254843788991nteger(A,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(G,minus_2355218937544613996nteger(A,B4))),groups3188404863801439024er_int(G,minus_2355218937544613996nteger(B4,A)))),groups3188404863801439024er_int(G,inf_in1364745209274528805nteger(A,B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7399_prod_Ounion__diff2,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B4)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)))),aa_set_nat_int(groups705719431365010083at_int(G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7400_prod_Ounion__diff2,axiom,
    ! [A: set_nat,B4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),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(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(minus_minus_set_nat(A),B4))),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(minus_minus_set_nat(B4),A)))),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,G),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B4))) ) ) ) ).

% prod.union_diff2
tff(fact_7401_Suc__times__gbinomial,axiom,
    ! [K3: nat,A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K3))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat)),aa_nat_nat(suc,K3))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat)),aa_nat_rat(gbinomial_rat(A2),K3)) ).

% Suc_times_gbinomial
tff(fact_7402_gbinomial__absorption,axiom,
    ! [K3: nat,A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K3))),aa_nat_rat(gbinomial_rat(A2),aa_nat_nat(suc,K3))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),K3)) ).

% gbinomial_absorption
tff(fact_7403_gbinomial__trinomial__revision,axiom,
    ! [K3: nat,M4: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M4)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A2),M4)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M4)),K3)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A2),K3)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K3))),aa_nat_nat(minus_minus_nat(M4),K3))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7404_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N2)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(plus_plus_nat(N2),K3)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(minus_minus_nat(N2),K3)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7405_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N2)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(plus_plus_nat(N2),K3)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(minus_minus_nat(N2),K3)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7406_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)
     => ? [N3: 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,N3)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_7407_gbinomial__absorption_H,axiom,
    ! [K3: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_rat(gbinomial_rat(A2),K3) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(A2),aa_nat_rat(semiri681578069525770553at_rat,K3))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A2),one_one_rat)),aa_nat_nat(minus_minus_nat(K3),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7408_pochhammer__minus,axiom,
    ! [B: rat,K3: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K3) = 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)),K3)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K3))),one_one_rat)),K3)) ).

% pochhammer_minus
tff(fact_7409_pochhammer__minus,axiom,
    ! [B: int,K3: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K3)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(B),aa_nat_int(semiri1314217659103216013at_int,K3))),one_one_int)),K3)) ).

% pochhammer_minus
tff(fact_7410_pochhammer__minus,axiom,
    ! [B: code_integer,K3: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K3)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,B),aa_nat_Code_integer(semiri4939895301339042750nteger,K3))),one_one_Code_integer)),K3)) ).

% pochhammer_minus
tff(fact_7411_pochhammer__minus_H,axiom,
    ! [B: rat,K3: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K3))),one_one_rat)),K3) = 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)),K3)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K3)) ).

% pochhammer_minus'
tff(fact_7412_pochhammer__minus_H,axiom,
    ! [B: int,K3: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(B),aa_nat_int(semiri1314217659103216013at_int,K3))),one_one_int)),K3) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K3)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K3)) ).

% pochhammer_minus'
tff(fact_7413_pochhammer__minus_H,axiom,
    ! [B: code_integer,K3: 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,K3))),one_one_Code_integer)),K3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K3)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K3)) ).

% pochhammer_minus'
tff(fact_7414_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K3: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
       => 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,K3)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K3)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7415_nat__mult__div__cancel__disj,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)) = $ite(K3 = zero_zero_nat,zero_zero_nat,aa_nat_nat(divide_divide_nat(M4),N2)) ).

% nat_mult_div_cancel_disj
tff(fact_7416_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M4: int,N2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M4),N2) = one_one_int )
     => ( ( M4 = one_one_int )
        | ( M4 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_7417_zmult__eq__1__iff,axiom,
    ! [M4: int,N2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M4),N2) = one_one_int )
    <=> ( ( ( M4 = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M4 = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_7418_zdvd__mono,axiom,
    ! [K3: int,M4: int,T4: int] :
      ( ( K3 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M4),T4)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K3),M4)),aa_int_int(aa_int_fun_int_int(times_times_int,K3),T4)) ) ) ).

% zdvd_mono
tff(fact_7419_zdvd__mult__cancel,axiom,
    ! [K3: int,M4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K3),M4)),aa_int_int(aa_int_fun_int_int(times_times_int,K3),N2))
     => ( ( K3 != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M4),N2) ) ) ).

% zdvd_mult_cancel
tff(fact_7420_zdvd__reduce,axiom,
    ! [K3: int,N2: int,M4: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K3),aa_int_int(plus_plus_int(N2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),M4)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K3),N2) ) ).

% zdvd_reduce
tff(fact_7421_zdvd__period,axiom,
    ! [A2: int,D2: int,X: int,T4: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),D2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(X),T4))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2))),T4)) ) ) ).

% zdvd_period
tff(fact_7422_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_7423_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_7424_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_7425_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_7426_int__div__pos__eq,axiom,
    ! [A2: int,B: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(divide_divide_int(A2),B) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_7427_int__div__neg__eq,axiom,
    ! [A2: int,B: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(divide_divide_int(A2),B) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_7428_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_7429_split__zdiv,axiom,
    ! [P: fun_int_o,N2: int,K3: int] :
      ( aa_int_o(P,aa_int_int(divide_divide_int(N2),K3))
    <=> ( ( ( K3 = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K3)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K3)
                & ( N2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),I4)),J3) ) )
             => aa_int_o(P,I4) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K3),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K3),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( N2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),I4)),J3) ) )
             => aa_int_o(P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_7430_incr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_o,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(plus_plus_int(X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K3)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(plus_plus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_7431_pos__zmult__eq__1__iff,axiom,
    ! [M4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M4)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M4),N2) = one_one_int )
      <=> ( ( M4 = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_7432_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K3),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K3),J)) ) ) ).

% zmult_zless_mono2
tff(fact_7433_minusinfinity,axiom,
    ! [D2: int,P1: fun_int_o,P: 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(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(P,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_7434_plusinfinity,axiom,
    ! [D2: int,P5: fun_int_o,P: 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(P5,X3)
          <=> aa_int_o(P5,aa_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(P,X3)
              <=> aa_int_o(P5,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P5,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_7435_decr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_o,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(minus_minus_int(X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K3)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_7436_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_7437_times__int__code_I1_J,axiom,
    ! [K3: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K3),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_7438_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(Z1),Z22)),W) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_7439_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(plus_plus_int(Z1),Z22)) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_7440_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(minus_minus_int(Z1),Z22)) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_7441_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(minus_minus_int(Z1),Z22)),W) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_7442_nat__mult__div__cancel1,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)) = aa_nat_nat(divide_divide_nat(M4),N2) ) ) ).

% nat_mult_div_cancel1
tff(fact_7443_div__mult__self__is__m,axiom,
    ! [N2: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)),N2) = M4 ) ) ).

% div_mult_self_is_m
tff(fact_7444_div__mult__self1__is__m,axiom,
    ! [N2: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M4)),N2) = M4 ) ) ).

% div_mult_self1_is_m
tff(fact_7445_div__mult2__eq,axiom,
    ! [M4: nat,N2: nat,Q3: nat] : aa_nat_nat(divide_divide_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(divide_divide_nat(M4),N2)),Q3) ).

% div_mult2_eq
tff(fact_7446_zdiv__zmult2__eq,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A2),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_7447_less__mult__imp__div__less,axiom,
    ! [M4: nat,I2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),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(divide_divide_nat(M4),N2)),I2) ) ).

% less_mult_imp_div_less
tff(fact_7448_div__times__less__eq__dividend,axiom,
    ! [M4: 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(divide_divide_nat(M4),N2)),N2)),M4) ).

% div_times_less_eq_dividend
tff(fact_7449_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M4: 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(divide_divide_nat(M4),N2))),M4) ).

% times_div_less_eq_dividend
tff(fact_7450_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(divide_divide_nat(M4),Q3)),N2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_7451_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),aa_nat_nat(divide_divide_nat(N2),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),Q3)),N2) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_7452_div__nat__eqI,axiom,
    ! [N2: nat,Q3: nat,M4: 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),Q3)),M4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(divide_divide_nat(M4),N2) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_7453_dividend__less__times__div,axiom,
    ! [N2: nat,M4: 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,M4),aa_nat_nat(plus_plus_nat(N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(divide_divide_nat(M4),N2)))) ) ).

% dividend_less_times_div
tff(fact_7454_dividend__less__div__times,axiom,
    ! [N2: nat,M4: 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,M4),aa_nat_nat(plus_plus_nat(N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M4),N2)),N2))) ) ).

% dividend_less_div_times
tff(fact_7455_split__div,axiom,
    ! [P: fun_nat_o,M4: nat,N2: nat] :
      ( aa_nat_o(P,aa_nat_nat(divide_divide_nat(M4),N2))
    <=> ( ( ( N2 = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N2)
             => ( ( M4 = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I4)),J3) )
               => aa_nat_o(P,I4) ) ) ) ) ) ).

% split_div
tff(fact_7456_split__div_H,axiom,
    ! [P: fun_nat_o,M4: nat,N2: nat] :
      ( aa_nat_o(P,aa_nat_nat(divide_divide_nat(M4),N2))
    <=> ( ( ( N2 = zero_zero_nat )
          & aa_nat_o(P,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)),M4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q6)))
            & aa_nat_o(P,Q6) ) ) ) ).

% split_div'
tff(fact_7457_int__ops_I7_J,axiom,
    ! [A2: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_7458_eucl__rel__int__iff,axiom,
    ! [K3: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K3,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2))
    <=> ( ( K3 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_7459_eucl__rel__int__dividesI,axiom,
    ! [L: int,K3: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K3,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_7460_split__pos__lemma,axiom,
    ! [K3: int,P: fun_int_fun_int_o,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K3)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(N2),K3)),aa_int_int(modulo_modulo_int(N2),K3))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K3)
              & ( N2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_7461_split__neg__lemma,axiom,
    ! [K3: int,P: fun_int_fun_int_o,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K3),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(N2),K3)),aa_int_int(modulo_modulo_int(N2),K3))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K3),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( N2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_7462_decr__lemma,axiom,
    ! [D2: int,X: int,Z: 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(minus_minus_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),Z))),one_one_int)),D2))),Z) ) ).

% decr_lemma
tff(fact_7463_incr__lemma,axiom,
    ! [D2: int,Z: 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,Z),aa_int_int(plus_plus_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(X),Z))),one_one_int)),D2))) ) ).

% incr_lemma
tff(fact_7464_zmod__eq__0__iff,axiom,
    ! [M4: int,D2: int] :
      ( ( aa_int_int(modulo_modulo_int(M4),D2) = zero_zero_int )
    <=> ? [Q6: int] : M4 = aa_int_int(aa_int_fun_int_int(times_times_int,D2),Q6) ) ).

% zmod_eq_0_iff
tff(fact_7465_zmod__eq__0D,axiom,
    ! [M4: int,D2: int] :
      ( ( aa_int_int(modulo_modulo_int(M4),D2) = zero_zero_int )
     => ? [Q7: int] : M4 = aa_int_int(aa_int_fun_int_int(times_times_int,D2),Q7) ) ).

% zmod_eq_0D
tff(fact_7466_abs__zmult__eq__1,axiom,
    ! [M4: int,N2: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M4),N2)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M4) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7467_div__mod__decomp__int,axiom,
    ! [A: int,N2: int] : A = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),N2)),N2)),aa_int_int(modulo_modulo_int(A),N2)) ).

% div_mod_decomp_int
tff(fact_7468_zdvd__mult__cancel1,axiom,
    ! [M4: int,N2: int] :
      ( ( M4 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M4),N2)),M4)
      <=> ( aa_int_int(abs_abs_int,N2) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7469_split__zmod,axiom,
    ! [P: fun_int_o,N2: int,K3: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(N2),K3))
    <=> ( ( ( K3 = zero_zero_int )
         => aa_int_o(P,N2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K3)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K3)
                & ( N2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),I4)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K3),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K3),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( N2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),I4)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_7470_int__mod__neg__eq,axiom,
    ! [A2: int,B: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(modulo_modulo_int(A2),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_7471_int__mod__pos__eq,axiom,
    ! [A2: int,B: int,Q3: int,R2: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(modulo_modulo_int(A2),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_7472_zmod__zmult2__eq,axiom,
    ! [C: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A2),B)),C))),aa_int_int(modulo_modulo_int(A2),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_7473_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K5: int] :
            ( ( A1 = K5 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K5) ) )
        | ? [L2: int,K5: int,Q6: int] :
            ( ( A1 = K5 )
            & ( A22 = L2 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),zero_zero_int) )
            & ( L2 != zero_zero_int )
            & ( K5 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L2) ) )
        | ? [R5: int,L2: int,K5: int,Q6: int] :
            ( ( A1 = K5 )
            & ( A22 = L2 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L2) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L2))
            & ( K5 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L2)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_7474_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K3: nat,M4: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3)),M4))),N2) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M4)),N2) ).

% Suc_mod_mult_self4
tff(fact_7475_Suc__mod__mult__self3,axiom,
    ! [K3: nat,N2: nat,M4: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)),M4))),N2) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M4)),N2) ).

% Suc_mod_mult_self3
tff(fact_7476_Suc__mod__mult__self2,axiom,
    ! [M4: nat,N2: nat,K3: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K3)))),N2) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M4)),N2) ).

% Suc_mod_mult_self2
tff(fact_7477_Suc__mod__mult__self1,axiom,
    ! [M4: nat,K3: nat,N2: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)))),N2) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M4)),N2) ).

% Suc_mod_mult_self1
tff(fact_7478_Suc__times__numeral__mod__eq,axiom,
    ! [K3: num,N2: nat] :
      ( ( numeral_numeral_nat(K3) != one_one_nat )
     => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K3)),N2))),numeral_numeral_nat(K3)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_7479_dvd__mult__sgn__iff,axiom,
    ! [L: int,K3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K3)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_7480_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K3))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K3)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_7481_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K3)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K3)
        & ( ( R2 = zero_zero_int )
         => ( K3 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_7482_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K3)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K3)
        & ( ( R2 = zero_zero_int )
         => ( K3 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_7483_int__sgnE,axiom,
    ! [K3: int] :
      ~ ! [N3: nat,L3: int] : K3 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N3)) ).

% int_sgnE
tff(fact_7484_mod__eq__0D,axiom,
    ! [M4: nat,D2: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M4),D2) = zero_zero_nat )
     => ? [Q7: nat] : M4 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D2),Q7) ) ).

% mod_eq_0D
tff(fact_7485_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(X),N2) = aa_nat_nat(modulo_modulo_nat(Y),N2) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(plus_plus_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q1)) = aa_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_7486_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(X),N2) = aa_nat_nat(modulo_modulo_nat(Y),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ? [Q7: nat] : X = aa_nat_nat(plus_plus_nat(Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q7)) ) ) ).

% nat_mod_eq_lemma
tff(fact_7487_mod__eq__nat2E,axiom,
    ! [M4: nat,Q3: nat,N2: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M4),Q3) = aa_nat_nat(modulo_modulo_nat(N2),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
       => ~ ! [S6: nat] : N2 != aa_nat_nat(plus_plus_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S6)) ) ) ).

% mod_eq_nat2E
tff(fact_7488_mod__eq__nat1E,axiom,
    ! [M4: nat,Q3: nat,N2: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M4),Q3) = aa_nat_nat(modulo_modulo_nat(N2),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4)
       => ~ ! [S6: nat] : M4 != aa_nat_nat(plus_plus_nat(N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S6)) ) ) ).

% mod_eq_nat1E
tff(fact_7489_div__mod__decomp,axiom,
    ! [A: nat,N2: nat] : A = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(A),N2)),N2)),aa_nat_nat(modulo_modulo_nat(A),N2)) ).

% div_mod_decomp
tff(fact_7490_mod__mult2__eq,axiom,
    ! [M4: nat,N2: nat,Q3: nat] : aa_nat_nat(modulo_modulo_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(modulo_modulo_nat(aa_nat_nat(divide_divide_nat(M4),N2)),Q3))),aa_nat_nat(modulo_modulo_nat(M4),N2)) ).

% mod_mult2_eq
tff(fact_7491_modulo__nat__def,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(modulo_modulo_nat(M4),N2) = aa_nat_nat(minus_minus_nat(M4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M4),N2)),N2)) ).

% modulo_nat_def
tff(fact_7492_div__sgn__abs__cancel,axiom,
    ! [V: int,K3: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K3)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_7493_div__dvd__sgn__abs,axiom,
    ! [L: int,K3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K3)
     => ( aa_int_int(divide_divide_int(K3),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K3)),aa_int_int(sgn_sgn_int,L))),aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K3)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_7494_split__mod,axiom,
    ! [P: fun_nat_o,M4: nat,N2: nat] :
      ( aa_nat_o(P,aa_nat_nat(modulo_modulo_nat(M4),N2))
    <=> ( ( ( N2 = zero_zero_nat )
         => aa_nat_o(P,M4) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N2)
             => ( ( M4 = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I4)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_7495_Suc__times__mod__eq,axiom,
    ! [M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M4)
     => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2))),M4) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_7496_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K3: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K3 = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K3,L),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_7497_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(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q7: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q7),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q7),A22) ) ) )
         => ~ ! [R6: int,Q7: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q7),R6) )
               => ( ( aa_int_int(sgn_sgn_int,R6) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R6)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,Q7),A22)),R6) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_7498_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( Y = $ite(Xa = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(modulo_modulo_nat(X),Xa)))),aa_int_int(minus_minus_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(modulo_modulo_nat(X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(modulo_modulo_nat(X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Xa)))))) ) ) ).

% bezw.elims
tff(fact_7499_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y)))),aa_int_int(minus_minus_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_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(modulo_modulo_nat(X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Y)))))) ).

% bezw.simps
tff(fact_7500_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(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y)))),aa_int_int(minus_minus_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_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(modulo_modulo_nat(X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Y))))) ) ) ).

% bezw_non_0
tff(fact_7501_divide__int__unfold,axiom,
    ! [K3: int,M4: nat,L: int,N2: nat] :
      aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K3)),aa_nat_int(semiri1314217659103216013at_int,M4))),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,K3) = zero_zero_int )
        | ( N2 = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K3) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(M4),N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(plus_plus_nat(aa_nat_nat(divide_divide_nat(M4),N2)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M4)))))) ) ).

% divide_int_unfold
tff(fact_7502_modulo__int__unfold,axiom,
    ! [K3: int,M4: nat,L: int,N2: nat] :
      aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K3)),aa_nat_int(semiri1314217659103216013at_int,M4))),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,K3) = zero_zero_int )
        | ( N2 = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K3)),aa_nat_int(semiri1314217659103216013at_int,M4)),
        $ite(aa_int_int(sgn_sgn_int,K3) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(M4),N2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N2),M4))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(M4),N2))))) ) ).

% modulo_int_unfold
tff(fact_7503_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(modulo_modulo_nat(X),Xa)))),aa_int_int(minus_minus_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(modulo_modulo_nat(X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(modulo_modulo_nat(X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Xa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_7504_semiring__norm_I12_J,axiom,
    ! [N2: num] : times_times_num(one,N2) = N2 ).

% semiring_norm(12)
tff(fact_7505_semiring__norm_I11_J,axiom,
    ! [M4: num] : times_times_num(M4,one) = M4 ).

% semiring_norm(11)
tff(fact_7506_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A2: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(plus_plus_int(A2),one_one_int),B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_7507_semiring__norm_I13_J,axiom,
    ! [M4: num,N2: num] : times_times_num(bit0(M4),bit0(N2)) = bit0(bit0(times_times_num(M4,N2))) ).

% semiring_norm(13)
tff(fact_7508_num__double,axiom,
    ! [N2: num] : times_times_num(bit0(one),N2) = bit0(N2) ).

% num_double
tff(fact_7509_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(numeral_numeral_int(bit0(V))),numeral_numeral_int(bit0(W))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(modulo_modulo_int(numeral_numeral_int(V)),numeral_numeral_int(W))) ).

% zmod_numeral_Bit0
tff(fact_7510_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one)))) = aa_nat_nat(minus_minus_nat(N2),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_7511_Suc__double__not__eq__double,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M4)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N2) ).

% Suc_double_not_eq_double
tff(fact_7512_double__not__eq__Suc__double,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M4) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N2)) ).

% double_not_eq_Suc_double
tff(fact_7513_nat__bit__induct,axiom,
    ! [P: fun_nat_o,N2: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( aa_nat_o(P,N3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N3)) ) )
       => ( ! [N3: nat] :
              ( aa_nat_o(P,N3)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N3))) )
         => aa_nat_o(P,N2) ) ) ) ).

% nat_bit_induct
tff(fact_7514_int__bit__induct,axiom,
    ! [P: fun_int_o,K3: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K: int] :
              ( aa_int_o(P,K)
             => ( ( K != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K),numeral_numeral_int(bit0(one)))) ) )
         => ( ! [K: int] :
                ( aa_int_o(P,K)
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K),numeral_numeral_int(bit0(one))))) ) )
           => aa_int_o(P,K3) ) ) ) ) ).

% int_bit_induct
tff(fact_7515_pos__zdiv__mult__2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A2)) = aa_int_int(divide_divide_int(B),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_7516_neg__zdiv__mult__2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A2)) = aa_int_int(divide_divide_int(aa_int_int(plus_plus_int(B),one_one_int)),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_7517_pos__zmod__mult__2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A2)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(modulo_modulo_int(B),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_7518_neg__zmod__mult__2,axiom,
    ! [A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A2)) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B),one_one_int)),A2))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_7519_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A2: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_7520_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K3: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),numeral_numeral_int(bit0(K3))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),numeral_numeral_int(K3))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_Suc_bit0
tff(fact_7521_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K3: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K3)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),aa_int_int(uminus_uminus_int,numeral_numeral_int(K3)))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_7522_signed__take__bit__mult,axiom,
    ! [N2: nat,K3: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),K3)),aa_int_int(bit_ri631733984087533419it_int(N2),L))) = aa_int_int(bit_ri631733984087533419it_int(N2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),L)) ).

% signed_take_bit_mult
tff(fact_7523_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xa),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_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,numeral_numeral_int(bit0(one))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(Xa),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.elims
tff(fact_7524_and__int_Osimps,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K3),L) = $ite(
        ( aa_set_int_o(member_int(K3),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_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,numeral_numeral_int(bit0(one))),K3)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),
        aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K3)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ) ).

% and_int.simps
tff(fact_7525_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K3: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit1(K3)))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(K3))),one_one_int))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_7526_concat__bit__Suc,axiom,
    ! [N2: nat,K3: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N2),K3),L) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_concat_bit(N2,aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),L))) ).

% concat_bit_Suc
tff(fact_7527_semiring__norm_I14_J,axiom,
    ! [M4: num,N2: num] : times_times_num(bit0(M4),bit1(N2)) = bit0(times_times_num(M4,bit1(N2))) ).

% semiring_norm(14)
tff(fact_7528_semiring__norm_I15_J,axiom,
    ! [M4: num,N2: num] : times_times_num(bit1(M4),bit0(N2)) = bit0(times_times_num(bit1(M4),N2)) ).

% semiring_norm(15)
tff(fact_7529_semiring__norm_I16_J,axiom,
    ! [M4: num,N2: num] : times_times_num(bit1(M4),bit1(N2)) = bit1(plus_plus_num(plus_plus_num(M4,N2),bit0(times_times_num(M4,N2)))) ).

% semiring_norm(16)
tff(fact_7530_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(numeral_numeral_int(bit1(V))),numeral_numeral_int(bit0(W))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(modulo_modulo_int(numeral_numeral_int(V)),numeral_numeral_int(W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_7531_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K3: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),numeral_numeral_int(bit1(K3))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),numeral_numeral_int(K3))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_7532_take__bit__mult,axiom,
    ! [N2: nat,K3: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N2),K3)),aa_int_int(bit_se2923211474154528505it_int(N2),L))) = aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(aa_int_fun_int_int(times_times_int,K3),L)) ).

% take_bit_mult
tff(fact_7533_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K3: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K3)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(uminus_uminus_int,numeral_numeral_int(K3)))),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_minus_bit0
tff(fact_7534_and__int__rec,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K3),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K3)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ).

% and_int_rec
tff(fact_7535_and__int__unfold,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K3),L) = $ite(
        ( ( K3 = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K3 = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K3,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(L),numeral_numeral_int(bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one))))))) ) ) ).

% and_int_unfold
tff(fact_7536_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K3: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit1(K3)))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(K3))),one_one_int))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_7537_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K3: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),numeral_numeral_int(bit1(K3))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),numeral_numeral_int(K3))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_7538_and__int_Opsimps,axiom,
    ! [K3: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K3),L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K3),L) = $ite(
            ( aa_set_int_o(member_int(K3),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_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,numeral_numeral_int(bit0(one))),K3)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K3)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.psimps
tff(fact_7539_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(X),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xa),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_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,numeral_numeral_int(bit0(one))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),
                  aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(Xa),numeral_numeral_int(bit0(one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_7540_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K3: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),numeral_numeral_int(bit0(K3))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),numeral_numeral_int(K3))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_numeral_bit0
tff(fact_7541_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K3: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K3)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(K3)))),numeral_numeral_int(bit0(one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_7542_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K3: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(K3)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(K3)))),numeral_numeral_int(bit0(one))) ).

% take_bit_numeral_minus_bit0
tff(fact_7543_and__nat__unfold,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(bit_se727722235901077358nd_nat(M4),N2) = $ite(
        ( ( M4 = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(modulo_modulo_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(modulo_modulo_nat(N2),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(divide_divide_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one)))))) ) ).

% and_nat_unfold
tff(fact_7544_and__nat__rec,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(bit_se727722235901077358nd_nat(M4),N2) = aa_nat_nat(plus_plus_nat(aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M4)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(divide_divide_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one)))))) ).

% and_nat_rec
tff(fact_7545_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [K: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(product_Pair_int_int(K),L3))
           => ( ( ~ ( aa_set_int_o(member_int(K),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L3),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,zero_zero_int),aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,aa_int_int(uminus_uminus_int,one_one_int)),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(K),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L3),numeral_numeral_int(bit0(one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_7546_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K3: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit1(K3)))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K3))))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_7547_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K3: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(bit1(K3)))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K3))))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_7548_binomial__code,axiom,
    ! [N2: nat,K3: nat] :
      aa_nat_nat(binomial(N2),K3) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),K3),
        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,numeral_numeral_nat(bit0(one))),K3)),aa_nat_nat(binomial(N2),aa_nat_nat(minus_minus_nat(N2),K3)),aa_nat_nat(divide_divide_nat(aa_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(plus_plus_nat(aa_nat_nat(minus_minus_nat(N2),K3)),one_one_nat),N2),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K3))) ) ).

% binomial_code
tff(fact_7549_modulo__int__def,axiom,
    ! [K3: int,L: int] :
      aa_int_int(modulo_modulo_int(K3),L) = $ite(
        L = zero_zero_int,
        K3,
        $ite(aa_int_int(sgn_sgn_int,K3) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K3))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K3)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K3))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_7550_or__int__unfold,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K3),L) = $ite(
        ( ( K3 = 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(
          K3 = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K3,aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(modulo_modulo_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(L),numeral_numeral_int(bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one))))))) ) ) ).

% or_int_unfold
tff(fact_7551_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(binomial(N2),K3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(suc,N2)),aa_nat_nat(suc,K3))),aa_nat_nat(suc,K3)) ).

% Suc_times_binomial_eq
tff(fact_7552_Suc__times__binomial,axiom,
    ! [K3: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),aa_nat_nat(binomial(aa_nat_nat(suc,N2)),aa_nat_nat(suc,K3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(binomial(N2),K3)) ).

% Suc_times_binomial
tff(fact_7553_choose__mult__lemma,axiom,
    ! [M4: nat,R2: nat,K3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(M4),R2)),K3)),aa_nat_nat(plus_plus_nat(M4),K3))),aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(M4),K3)),K3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(plus_plus_nat(M4),R2)),K3)),K3)),aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(M4),R2)),M4)) ).

% choose_mult_lemma
tff(fact_7554_Suc__times__binomial__add,axiom,
    ! [A2: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A2)),aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(A2),B))),aa_nat_nat(suc,A2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(plus_plus_nat(A2),B))),A2)) ).

% Suc_times_binomial_add
tff(fact_7555_binomial__fact__lemma,axiom,
    ! [K3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),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,K3)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(N2),K3)))),aa_nat_nat(binomial(N2),K3)) = aa_nat_nat(semiri1408675320244567234ct_nat,N2) ) ) ).

% binomial_fact_lemma
tff(fact_7556_choose__mult,axiom,
    ! [K3: nat,M4: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(N2),M4)),aa_nat_nat(binomial(M4),K3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(N2),K3)),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N2),K3)),aa_nat_nat(minus_minus_nat(M4),K3))) ) ) ) ).

% choose_mult
tff(fact_7557_binomial__absorb__comp,axiom,
    ! [N2: nat,K3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(N2),K3)),aa_nat_nat(binomial(N2),K3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N2),one_one_nat)),K3)) ).

% binomial_absorb_comp
tff(fact_7558_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K3: nat] : aa_nat_nat(binomial(aa_nat_nat(suc,N2)),aa_nat_nat(suc,K3)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(binomial(N2),K3))),aa_nat_nat(suc,K3)) ).

% binomial_Suc_Suc_eq_times
tff(fact_7559_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_7560_mult__inc,axiom,
    ! [X: num,Y: num] : times_times_num(X,inc(Y)) = plus_plus_num(times_times_num(X,Y),X) ).

% mult_inc
tff(fact_7561_binomial__absorption,axiom,
    ! [K3: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),aa_nat_nat(binomial(N2),aa_nat_nat(suc,K3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N2),one_one_nat)),K3)) ).

% binomial_absorption
tff(fact_7562_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_7563_binomial__altdef__nat,axiom,
    ! [K3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N2)
     => ( aa_nat_nat(binomial(N2),K3) = aa_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,K3)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(N2),K3)))) ) ) ).

% binomial_altdef_nat
tff(fact_7564_times__binomial__minus1__eq,axiom,
    ! [K3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K3)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),aa_nat_nat(binomial(N2),K3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(N2),one_one_nat)),aa_nat_nat(minus_minus_nat(K3),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7565_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7566_choose__two,axiom,
    ! [N2: nat] : aa_nat_nat(binomial(N2),numeral_numeral_nat(bit0(one))) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(minus_minus_nat(N2),one_one_nat))),numeral_numeral_nat(bit0(one))) ).

% choose_two
tff(fact_7567_or__int__rec,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K3),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K3)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ).

% or_int_rec
tff(fact_7568_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N2: nat,K3: int] : aa_int_int(bit_ri631733984087533419it_int(N2),K3) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),K3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,N2))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K3),N2)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_7569_or__nat__unfold,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(bit_se1412395901928357646or_nat(M4),N2) = $ite(
        M4 = zero_zero_nat,
        N2,
        $ite(N2 = zero_zero_nat,M4,aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(modulo_modulo_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(modulo_modulo_nat(N2),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(divide_divide_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one))))))) ) ).

% or_nat_unfold
tff(fact_7570_push__bit__int__def,axiom,
    ! [N2: nat,K3: int] : aa_int_int(bit_se545348938243370406it_int(N2),K3) = aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N2)) ).

% push_bit_int_def
tff(fact_7571_push__bit__nat__def,axiom,
    ! [N2: nat,M4: nat] : aa_nat_nat(bit_se547839408752420682it_nat(N2),M4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),N2)) ).

% push_bit_nat_def
tff(fact_7572_set__bit__eq,axiom,
    ! [N2: nat,K3: int] : aa_int_int(bit_se7879613467334960850it_int(N2),K3) = aa_int_int(plus_plus_int(K3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K3),N2))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N2))) ).

% set_bit_eq
tff(fact_7573_unset__bit__eq,axiom,
    ! [N2: nat,K3: int] : aa_int_int(bit_se4203085406695923979it_int(N2),K3) = aa_int_int(minus_minus_int(K3),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K3),N2))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N2))) ).

% unset_bit_eq
tff(fact_7574_or__nat__rec,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(bit_se1412395901928357646or_nat(M4),N2) = aa_nat_nat(plus_plus_nat(aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M4)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(divide_divide_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one)))))) ).

% or_nat_rec
tff(fact_7575_xor__nat__unfold,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(bit_se6528837805403552850or_nat(M4),N2) = $ite(
        M4 = zero_zero_nat,
        N2,
        $ite(N2 = zero_zero_nat,M4,aa_nat_nat(plus_plus_nat(aa_nat_nat(modulo_modulo_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(modulo_modulo_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(modulo_modulo_nat(N2),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(divide_divide_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one))))))) ) ).

% xor_nat_unfold
tff(fact_7576_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K3: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),K3) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N2)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K3),N2)))),aa_int_int(bit_se2923211474154528505it_int(N2),K3)) ).

% take_bit_Suc_from_most
tff(fact_7577_xor__int__rec,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K3),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K3) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one)))))) ).

% xor_int_rec
tff(fact_7578_xor__nat__rec,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(bit_se6528837805403552850or_nat(M4),N2) = aa_nat_nat(plus_plus_nat(aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M4) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(divide_divide_nat(M4),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(N2),numeral_numeral_nat(bit0(one)))))) ).

% xor_nat_rec
tff(fact_7579_xor__int__unfold,axiom,
    ! [K3: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K3),L) = $ite(
        K3 = 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,K3),
          $ite(
            K3 = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K3,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(modulo_modulo_int(L),numeral_numeral_int(bit0(one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))),aa_int_int(divide_divide_int(L),numeral_numeral_int(bit0(one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_7580_not__int__rec,axiom,
    ! [K3: int] : aa_int_int(bit_ri7919022796975470100ot_int,K3) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K3))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one)))))) ).

% not_int_rec
tff(fact_7581_or__not__numerals_I8_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(bit1(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit0(N2)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2))))) ).

% or_not_numerals(8)
tff(fact_7582_or__not__numerals_I9_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(bit1(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit1(N2)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2))))) ).

% or_not_numerals(9)
tff(fact_7583_and__not__numerals_I8_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(bit1(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit0(N2)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2))))) ).

% and_not_numerals(8)
tff(fact_7584_bool__assn__proper_I4_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_ae(P),Q)) ) ) ).

% bool_assn_proper(4)
tff(fact_7585_bool__assn__proper_I3_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => ( aa_fun1675921005893230629at_o_o(proper,Q)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_af(P),Q)) ) ) ).

% bool_assn_proper(3)
tff(fact_7586_bool__assn__proper_I2_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_ag) ).

% bool_assn_proper(2)
tff(fact_7587_bool__assn__proper_I5_J,axiom,
    ! [P: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,P)
     => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_ah,P)) ) ).

% bool_assn_proper(5)
tff(fact_7588_bot__assn__def,axiom,
    bot_bot_assn = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_ag) ).

% bot_assn_def
tff(fact_7589_sup__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_ai(P),Q)) ).

% sup_assn_def
tff(fact_7590_inf__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_aj(P),Q)) ).

% inf_assn_def
tff(fact_7591_nat__times__as__int,axiom,
    ! [X4: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_times_as_int
tff(fact_7592_divide__nat__def,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_nat(divide_divide_nat(M4),N2) = $ite(N2 = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_ak(M4),N2)))) ).

% divide_nat_def
tff(fact_7593_uminus__assn__def,axiom,
    ! [P: assn] : aa_assn_assn(uminus_uminus_assn,P) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_al,P)) ).

% uminus_assn_def
tff(fact_7594_Gcd__eq__Max,axiom,
    ! [M: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M)
     => ( ( M != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_an),M))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_7595_and__not__numerals_I5_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(bit0(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit0(N2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2)))) ).

% and_not_numerals(5)
tff(fact_7596_and__not__numerals_I6_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(bit0(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit1(N2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2)))) ).

% and_not_numerals(6)
tff(fact_7597_and__not__numerals_I9_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(bit1(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit1(N2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se725231765392027082nd_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2)))) ).

% and_not_numerals(9)
tff(fact_7598_or__not__numerals_I6_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(bit0(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit1(N2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2)))) ).

% or_not_numerals(6)
tff(fact_7599_or__not__numerals_I5_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(bit0(M4))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit0(N2)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se1409905431419307370or_int(numeral_numeral_int(M4)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N2))))) ).

% or_not_numerals(5)
tff(fact_7600_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat) ).

% atMost_0
tff(fact_7601_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2) = aa_set_nat_set_nat(minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N2)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_7602_vandermonde,axiom,
    ! [M4: nat,N2: nat,R2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ao(M4),N2),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(binomial(aa_nat_nat(plus_plus_nat(M4),N2)),R2) ).

% vandermonde
tff(fact_7603_fact__eq__fact__times,axiom,
    ! [N2: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups708209901874060359at_nat,aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N2)),M4))) ) ) ).

% fact_eq_fact_times
tff(fact_7604_binomial,axiom,
    ! [A2: nat,B: nat,N2: nat] : aa_nat_nat(power_power_nat(aa_nat_nat(plus_plus_nat(A2),B)),N2) = aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aq(A2),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial
tff(fact_7605_choose__square__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aTP_Lamm_ar(N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N2)),N2) ).

% choose_square_sum
tff(fact_7606_gauss__sum__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,N2))),numeral_numeral_nat(bit0(one))) ).

% gauss_sum_nat
tff(fact_7607_binomial__r__part__sum,axiom,
    ! [M4: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,binomial(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M4)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M4)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M4)) ).

% binomial_r_part_sum
tff(fact_7608_choose__linear__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aTP_Lamm_as(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(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(N2),one_one_nat))) ).

% choose_linear_sum
tff(fact_7609_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aa_nat_fun_nat_nat(aTP_Lamm_at(A2),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),D2)))),numeral_numeral_nat(bit0(one))) ).

% arith_series_nat
tff(fact_7610_Sum__Icc__nat,axiom,
    ! [M4: nat,N2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aTP_Lamm_ap),aa_nat_set_nat(set_or1269000886237332187st_nat(M4),N2)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(plus_plus_nat(N2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Icc_nat
tff(fact_7611_periodic__finite__ex,axiom,
    ! [D2: int,P: 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(P,X3)
          <=> aa_int_o(P,aa_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(P,X_12)
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D2))
              & aa_int_o(P,X2) ) ) ) ) ).

% periodic_finite_ex
tff(fact_7612_simp__from__to,axiom,
    ! [I2: int,J: int] :
      aa_int_set_int(set_or1266510415728281911st_int(I2),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),bot_bot_set_int,aa_set_int_set_int(aa_int7955932875426912299et_int(insert_int,I2),aa_int_set_int(set_or1266510415728281911st_int(aa_int_int(plus_plus_int(I2),one_one_int)),J))) ).

% simp_from_to
tff(fact_7613_cpmi,axiom,
    ! [D3: int,P: fun_int_o,P5: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z5)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P5,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D3))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),B4)
                     => ( X3 != aa_int_int(plus_plus_int(Xb2),Xa3) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(minus_minus_int(X3),D3)) ) )
         => ( ! [X3: int,K: int] :
                ( aa_int_o(P5,X3)
              <=> aa_int_o(P5,aa_int_int(minus_minus_int(X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D3))
                    & aa_int_o(P5,X2) )
                | ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D3))
                    & ? [Xa2: int] :
                        ( aa_set_int_o(member_int(Xa2),B4)
                        & aa_int_o(P,aa_int_int(plus_plus_int(Xa2),X2)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_7614_cppi,axiom,
    ! [D3: int,P: fun_int_o,P5: fun_int_o,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),X3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P5,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( aa_set_int_o(member_int(Xa3),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D3))
                 => ! [Xb2: int] :
                      ( aa_set_int_o(member_int(Xb2),A)
                     => ( X3 != aa_int_int(minus_minus_int(Xb2),Xa3) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(plus_plus_int(X3),D3)) ) )
         => ( ! [X3: int,K: int] :
                ( aa_int_o(P5,X3)
              <=> aa_int_o(P5,aa_int_int(minus_minus_int(X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D3))
                    & aa_int_o(P5,X2) )
                | ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D3))
                    & ? [Xa2: int] :
                        ( aa_set_int_o(member_int(Xa2),A)
                        & aa_int_o(P,aa_int_int(minus_minus_int(Xa2),X2)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_7615_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_au(L)),Qr) ).

% divmod_step_nat_def
tff(fact_7616_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_av(L),Qr) ).

% divmod_step_int_def
tff(fact_7617_Sum__Icc__int,axiom,
    ! [M4: int,N2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M4),N2)
     => ( aa_set_int_int(aa_fun5337148683239322782nt_int(groups4538972089207619220nt_int,aTP_Lamm_aw),aa_int_set_int(set_or1266510415728281911st_int(M4),N2)) = aa_int_int(divide_divide_int(aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,N2),aa_int_int(plus_plus_int(N2),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M4),aa_int_int(minus_minus_int(M4),one_one_int)))),numeral_numeral_int(bit0(one))) ) ) ).

% Sum_Icc_int
tff(fact_7618_Sum__Ico__nat,axiom,
    ! [M4: nat,N2: nat] : aa_set_nat_nat(aa_fun3777887704360129838at_nat(groups3542108847815614940at_nat,aTP_Lamm_ap),aa_nat_set_nat(set_or4665077453230672383an_nat(M4),N2)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(minus_minus_nat(N2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(minus_minus_nat(M4),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Ico_nat
tff(fact_7619_atLeastLessThan__singleton,axiom,
    ! [M4: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M4),aa_nat_nat(suc,M4)) = aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,M4),bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_7620_atLeastLessThan0,axiom,
    ! [M4: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M4),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_7621_atLeastLessThanSuc,axiom,
    ! [M4: nat,N2: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M4),aa_nat_nat(suc,N2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N2),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,N2),aa_nat_set_nat(set_or4665077453230672383an_nat(M4),N2)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_7622_atLeastLessThan__nat__numeral,axiom,
    ! [M4: nat,K3: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M4),numeral_numeral_nat(K3)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),pred_numeral(K3)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,pred_numeral(K3)),aa_nat_set_nat(set_or4665077453230672383an_nat(M4),pred_numeral(K3))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_7623_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X: nat,Y: nat] :
      aa_set_nat_set_nat(image_nat_nat2(aTP_Lamm_ax(C)),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(minus_minus_nat(X),C)),aa_nat_nat(minus_minus_nat(Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_7624_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ay(L),Qr) ).

% divmod_step_integer_def
tff(fact_7625_triangle__def,axiom,
    ! [N2: nat] : aa_nat_nat(nat_triangle,N2) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,N2))),numeral_numeral_nat(bit0(one))) ).

% triangle_def
tff(fact_7626_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa_set8622560600277679972nteger(image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_az,L)),aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,U),L))) = aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(L),U) ).

% image_add_integer_atLeastLessThan
tff(fact_7627_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),one_one_Code_integer)) = aa_Cod2813182003028390446nteger(set_or189985376899183464nteger(L),U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_7628_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_7629_times__integer__code_I1_J,axiom,
    ! [K3: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_7630_integer__of__int__code,axiom,
    ! [K3: int] :
      aa_int_Code_integer(code_integer_of_int,K3) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K3),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K3))),
        $ite(
          K3 = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K3),numeral_numeral_int(bit0(one))))),
            $ite(aa_int_int(modulo_modulo_int(K3),numeral_numeral_int(bit0(one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_7631_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(L),U)) ).

% finite_atLeastLessThan_integer
tff(fact_7632_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_Cod2813182003028390446nteger(set_or189985376899183464nteger(L),U)) ).

% finite_atLeastAtMost_integer
tff(fact_7633_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(zero_z3403309356797280102nteger),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_7634_times__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).

% times_integer.rsp
tff(fact_7635_times__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).

% times_natural.rsp
tff(fact_7636_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,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(times_times_int,Xa),X)) ).

% times_integer.abs_eq
tff(fact_7637_divmod__integer__eq__cases,axiom,
    ! [K3: code_integer,L: code_integer] :
      aa_Cod6977894654539884864nteger(code_divmod_integer(K3),L) = $ite(
        K3 = zero_z3403309356797280102nteger,
        aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(zero_z3403309356797280102nteger),K3),
          aa_Pro4693618923521437820nteger(aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer),L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K3) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),aa_Cod6977894654539884864nteger(code_divmod_abs(K3),L),produc6916734918728496179nteger(aTP_Lamm_ba(L),aa_Cod6977894654539884864nteger(code_divmod_abs(K3),L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_7638_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_bb,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_7639_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_bc,times_times_rat) ).

% times_rat.transfer
tff(fact_7640_int__of__integer__code,axiom,
    ! [K3: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K3) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K3),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K3))),
        $ite(K3 = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_bd,aa_Cod6977894654539884864nteger(code_divmod_integer(K3),numera6620942414471956472nteger(bit0(one))))) ) ).

% int_of_integer_code
tff(fact_7641_positive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_be,positive) ).

% positive.transfer
tff(fact_7642_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_bg),times_times_int) ).

% times_int.transfer
tff(fact_7643_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(produc4528391901023682385at_nat(aTP_Lamm_bg),Xa),X)) ).

% times_int.abs_eq
tff(fact_7644_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa)) ).

% times_integer.rep_eq
tff(fact_7645_positive__mult,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(times_times_rat,X),Y)) ) ) ).

% positive_mult
tff(fact_7646_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_7647_type__definition__integer,axiom,
    aa_set_int_o(type_d8366093980585677751er_int(code_int_of_integer,code_integer_of_int),top_top_set_int) ).

% type_definition_integer
tff(fact_7648_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_7649_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_7650_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_bg),produc4528391901023682385at_nat(aTP_Lamm_bg)) ).

% times_int.rsp
tff(fact_7651_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N2) = aa_set_nat_set_nat(minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N2)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,zero_zero_nat),bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_7652_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_nat(ord_min_nat(aa_nat_nat(suc,A2)),B) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_7653_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B)
     => ( aa_nat_nat(ord_min_nat(B),aa_nat_nat(suc,A2)) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_7654_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7655_nat__mult__min__right,axiom,
    ! [M4: nat,N2: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),aa_nat_nat(ord_min_nat(N2),Q3)) = aa_nat_nat(ord_min_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),Q3)) ).

% nat_mult_min_right
tff(fact_7656_nat__mult__min__left,axiom,
    ! [M4: nat,N2: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(ord_min_nat(M4),N2)),Q3) = aa_nat_nat(ord_min_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q3)) ).

% nat_mult_min_left
tff(fact_7657_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N2) = bot_bot_set_nat )
    <=> ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_7658_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
     => ( aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(zero_z3403309356797280102nteger),U) = aa_set7438092518518805847nteger(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_7659_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(gcd_gcd_nat(X),Y)) = aa_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_7660_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(set_or1210151606488870762an_nat),top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_7661_gcd__mult__distrib__nat,axiom,
    ! [K3: nat,M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),aa_nat_nat(gcd_gcd_nat(M4),N2)) = aa_nat_nat(gcd_gcd_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N2)) ).

% gcd_mult_distrib_nat
tff(fact_7662_Gcd__in,axiom,
    ! [A: set_nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_set_nat_o(member_nat(A3),A)
         => ( aa_set_nat_o(member_nat(B2),A)
           => aa_set_nat_o(member_nat(aa_nat_nat(gcd_gcd_nat(A3),B2)),A) ) )
     => ( ( A != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A)),A) ) ) ).

% Gcd_in
tff(fact_7663_bezout__nat,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [X3: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(gcd_gcd_nat(A2),B)) ) ).

% bezout_nat
tff(fact_7664_bezout__gcd__nat_H,axiom,
    ! [B: nat,A2: nat] :
    ? [X3: nat,Y2: 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),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3))
        & ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = aa_nat_nat(gcd_gcd_nat(A2),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3))
        & ( aa_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,A2),Y2)) = aa_nat_nat(gcd_gcd_nat(A2),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_7665_greaterThan__Suc,axiom,
    ! [K3: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K3)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K3)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,aa_nat_nat(suc,K3)),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_7666_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_bb,aTP_Lamm_bb) ).

% plus_rat.rsp
tff(fact_7667_positive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_be,aTP_Lamm_be) ).

% positive.rsp
tff(fact_7668_ratrel__iff,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),Y)
    <=> ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X) != zero_zero_int )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Y) != 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,Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X)) ) ) ) ).

% ratrel_iff
tff(fact_7669_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V3: int] : aa_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(gcd_gcd_int(X),Y) ).

% bezout_int
tff(fact_7670_gcd__mult__distrib__int,axiom,
    ! [K3: int,M4: int,N2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K3)),aa_int_int(gcd_gcd_int(M4),N2)) = aa_int_int(gcd_gcd_int(aa_int_int(aa_int_fun_int_int(times_times_int,K3),M4)),aa_int_int(aa_int_fun_int_int(times_times_int,K3),N2)) ).

% gcd_mult_distrib_int
tff(fact_7671_ratrel__def,axiom,
    ! [X4: product_prod_int_int,Xa3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X4),Xa3)
    <=> ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,X4) != zero_zero_int )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa3)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X4)) ) ) ) ).

% ratrel_def
tff(fact_7672_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_bc,aTP_Lamm_bc) ).

% times_rat.rsp
tff(fact_7673_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(product_Pair_int_int(aa_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_7674_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_7675_times__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(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xa)),aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_fst_int_int,X))),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)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_7676_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_Cod2813182003028390446nteger(set_or4266950643985792945nteger(L),U)) ).

% finite_greaterThanLessThan_integer
tff(fact_7677_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_Cod2813182003028390446nteger(set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer)),U) = aa_Cod2813182003028390446nteger(set_or4266950643985792945nteger(L),U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_7678_positive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_be) ).

% positive_def
tff(fact_7679_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,aa_Cod2813182003028390446nteger(set_or2715278749043346189nteger(L),U)) ).

% finite_greaterThanAtMost_integer
tff(fact_7680_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_Cod2813182003028390446nteger(set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer)),U) = aa_Cod2813182003028390446nteger(set_or2715278749043346189nteger(L),U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_7681_atLeast__Suc,axiom,
    ! [K3: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K3)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K3)),aa_set_nat_set_nat(aa_nat5279478944698590999et_nat(insert_nat,K3),bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_7682_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_bh(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_7683_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K3: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_bi(P),K3),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_7684_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_7685_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X3)
         => aa_nat_o(P,X3) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_7686_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N7: nat] :
        ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N4)
         => aa_nat_o(P,N4) ) ) ).

% eventually_sequentially
tff(fact_7687_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( aa_filter_nat_o(aa_fil403049915117619187_nat_o(ord_le2510731241096832064er_nat,F2),at_top_nat)
    <=> ! [N7: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N7),F2) ) ).

% le_sequentially
tff(fact_7688_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_7689_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_7690_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_7691_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_bj,at_top_nat) ).

% eventually_False_sequentially
tff(fact_7692_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_bb) ).

% plus_rat_def
tff(fact_7693_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N7: nat] :
        ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M3)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N4)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M3)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_7694_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_bc) ).

% times_rat_def
tff(fact_7695_add__rat,axiom,
    ! [B: int,D2: int,A2: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(fract(A2),B)),aa_int_rat(fract(C),D2)) = aa_int_rat(fract(aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).

% add_rat
tff(fact_7696_divide__rat,axiom,
    ! [A2: int,B: int,C: int,D2: int] : aa_rat_rat(divide_divide_rat(aa_int_rat(fract(A2),B)),aa_int_rat(fract(C),D2)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A2),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% divide_rat
tff(fact_7697_sgn__rat,axiom,
    ! [A2: int,B: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(fract(A2),B)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A2)),aa_int_int(sgn_sgn_int,B))) ).

% sgn_rat
tff(fact_7698_mult__rat,axiom,
    ! [A2: int,B: int,C: int,D2: int] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(fract(A2),B)),aa_int_rat(fract(C),D2)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ).

% mult_rat
tff(fact_7699_le__rat,axiom,
    ! [B: int,D2: int,A2: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A2),B)),aa_int_rat(fract(C),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,A2),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,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))) ) ) ) ).

% le_rat
tff(fact_7700_less__rat,axiom,
    ! [B: int,D2: int,A2: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A2),B)),aa_int_rat(fract(C),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,A2),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,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2))) ) ) ) ).

% less_rat
tff(fact_7701_diff__rat,axiom,
    ! [B: int,D2: int,A2: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(fract(A2),B)),aa_int_rat(fract(C),D2)) = aa_int_rat(fract(aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).

% diff_rat
tff(fact_7702_eq__rat_I1_J,axiom,
    ! [B: int,D2: int,A2: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( aa_int_rat(fract(A2),B) = aa_int_rat(fract(C),D2) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),D2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_7703_mult__rat__cancel,axiom,
    ! [C: int,A2: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_rat(fract(A2),B) ) ) ).

% mult_rat_cancel
tff(fact_7704_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_7705_positive__rat,axiom,
    ! [A2: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(A2),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,A2),B)) ) ).

% positive_rat
tff(fact_7706_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_7707_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P2),Q3)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_7708_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_bl(Q3),quotient_of(P2)) ).

% rat_minus_code
tff(fact_7709_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_bn(Q3),quotient_of(P2)) ).

% rat_plus_code
tff(fact_7710_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_bp(Q3),quotient_of(P2)) ).

% rat_times_code
tff(fact_7711_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_br(Q3)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_7712_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_bt(Q3)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_7713_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_bv(Q3),quotient_of(P2)) ).

% rat_divide_code
tff(fact_7714_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = produc4245557441103728435nt_int(aTP_Lamm_bw,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_7715_divide__rat__def,axiom,
    ! [Q3: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q3),R2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Q3),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_7716_sub__BitM__One__eq,axiom,
    ! [N2: num] : neg_numeral_sub_int(bitM(N2),one) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),neg_numeral_sub_int(N2,one)) ).

% sub_BitM_One_eq
tff(fact_7717_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_7718_cofinite__eq__sequentially,axiom,
    cofinite_nat = at_top_nat ).

% cofinite_eq_sequentially
tff(fact_7719_INFM__nat__inductI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [I3: nat] :
            ( aa_nat_o(P,I3)
           => ? [J4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J4)
                & aa_nat_o(P,J4)
                & aa_nat_o(Q,J4) ) )
       => frequently_nat(Q,cofinite_nat) ) ) ).

% INFM_nat_inductI
tff(fact_7720_lcm__altdef__int,axiom,
    ! [A2: int,B: int] : aa_int_int(gcd_lcm_int(A2),B) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B))),aa_int_int(gcd_gcd_int(A2),B)) ).

% lcm_altdef_int
tff(fact_7721_times__num__def,axiom,
    ! [M4: num,N2: num] : times_times_num(M4,N2) = num_of_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,nat_of_num(M4)),nat_of_num(N2))) ).

% times_num_def
tff(fact_7722_nat__of__num__mult,axiom,
    ! [X: num,Y: num] : nat_of_num(times_times_num(X,Y)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,nat_of_num(X)),nat_of_num(Y)) ).

% nat_of_num_mult
tff(fact_7723_prod__gcd__lcm__int,axiom,
    ! [M4: int,N2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,M4)),aa_int_int(abs_abs_int,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(gcd_gcd_int(M4),N2)),aa_int_int(gcd_lcm_int(M4),N2)) ).

% prod_gcd_lcm_int
tff(fact_7724_prod__gcd__lcm__nat,axiom,
    ! [M4: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M4),N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(gcd_gcd_nat(M4),N2)),aa_nat_nat(gcd_lcm_nat(M4),N2)) ).

% prod_gcd_lcm_nat
tff(fact_7725_lcm__code__integer,axiom,
    ! [A2: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_lcm_Code_integer,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(gcd_gcd_Code_integer,A2),B)) ).

% lcm_code_integer
tff(fact_7726_lcm__nat__def,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(gcd_lcm_nat(X),Y) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),aa_nat_nat(gcd_gcd_nat(X),Y)) ).

% lcm_nat_def
tff(fact_7727_Lcm__eq__Max__nat,axiom,
    ! [M: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M)
     => ( ( M != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M)
         => ( ! [M2: nat,N3: nat] :
                ( aa_set_nat_o(member_nat(M2),M)
               => ( aa_set_nat_o(member_nat(N3),M)
                 => aa_set_nat_o(member_nat(aa_nat_nat(gcd_lcm_nat(M2),N3)),M) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_7728_nat__of__num__sqr,axiom,
    ! [X: num] : nat_of_num(sqr(X)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,nat_of_num(X)),nat_of_num(X)) ).

% nat_of_num_sqr
tff(fact_7729_Lcm__nat__empty,axiom,
    aa_set_nat_nat(gcd_Lcm_nat,bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_7730_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = times_times_num(X,X) ).

% sqr_conv_mult
tff(fact_7731_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M)
     => ( ( M != bot_bot_set_nat )
       => ( ! [M2: nat,N3: nat] :
              ( aa_set_nat_o(member_nat(M2),M)
             => ( aa_set_nat_o(member_nat(N3),M)
               => aa_set_nat_o(member_nat(aa_nat_nat(gcd_lcm_nat(M2),N3)),M) ) )
         => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Lcm_nat,M)),M) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_7732_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : pow(X,bit1(Y)) = times_times_num(sqr(pow(X,Y)),X) ).

% pow.simps(3)
tff(fact_7733_coprime__crossproduct__nat,axiom,
    ! [A2: nat,D2: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A2),D2)
     => ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,B),C)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2) )
        <=> ( ( A2 = B )
            & ( C = D2 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_7734_coprime__crossproduct__int,axiom,
    ! [A2: int,D2: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A2),D2)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,B),C)
       => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,B)),aa_int_int(abs_abs_int,D2)) )
        <=> ( ( aa_int_int(abs_abs_int,A2) = aa_int_int(abs_abs_int,B) )
            & ( aa_int_int(abs_abs_int,C) = aa_int_int(abs_abs_int,D2) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_7735_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ! [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),As2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_hea5848002036693495265it_nat(lim_Product_unit,H)) ) ) ) ).

% in_range.elims(3)
tff(fact_7736_in__range_Osimps,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,As: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(Ha),As))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),As)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),aa_hea5848002036693495265it_nat(lim_Product_unit,Ha)) ) ) ).

% in_range.simps
tff(fact_7737_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ( (Y)
            <=> ~ ! [X2: nat] :
                    ( aa_set_nat_o(member_nat(X2),As2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),aa_hea5848002036693495265it_nat(lim_Product_unit,H)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_7738_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
           => ~ ! [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),As2)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_hea5848002036693495265it_nat(lim_Product_unit,H)) ) ) ) ).

% in_range.elims(2)
tff(fact_7739_UNIV__bool,axiom,
    top_top_set_o = aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$false),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o)) ).

% UNIV_bool
tff(fact_7740_Rep__unit__induct,axiom,
    ! [Y: $o,P: fun_o_o] :
      ( aa_set_o_o(member_o((Y)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o))
     => ( ! [X3: product_unit] : aa_o_o(P,aa_Product_unit_o(product_Rep_unit,X3))
       => aa_o_o(P,(Y)) ) ) ).

% Rep_unit_induct
tff(fact_7741_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( aa_set_o_o(member_o((X)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o))
     => ( aa_set_o_o(member_o((Y)),aa_set_o_set_o(aa_o_fun_set_o_set_o(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_7742_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_7743_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_7744_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$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_7745_type__definition__unit,axiom,
    aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o)) ).

% type_definition_unit
tff(fact_7746_Rep__unit,axiom,
    ! [X: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(product_Rep_unit,X)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o)) ).

% Rep_unit
tff(fact_7747_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y2: $o] :
          ( ( X = aa_o_Product_unit(product_Abs_unit,(Y2)) )
         => ~ aa_set_o_o(member_o((Y2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_7748_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( aa_set_o_o(member_o((Y)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o))
     => ~ ! [X3: product_unit] :
            ( (Y)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X3) ) ) ).

% Rep_unit_cases
tff(fact_7749_Abs__unit__induct,axiom,
    ! [P: fun_Product_unit_o,X: product_unit] :
      ( ! [Y2: $o] :
          ( aa_set_o_o(member_o((Y2)),aa_set_o_set_o(aa_o_fun_set_o_set_o(insert_o,$true),bot_bot_set_o))
         => aa_Product_unit_o(P,aa_o_Product_unit(product_Abs_unit,(Y2))) )
     => aa_Product_unit_o(P,X) ) ).

% Abs_unit_induct
tff(fact_7750_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))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B62)))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B52)))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B42)))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B32)))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B22)))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B1)))),numera6620942414471956472nteger(bit0(one)))),aa_o_Code_integer(zero_n356916108424825756nteger,(B0))) ).

% integer_of_char_code
tff(fact_7751_sum__encode__def,axiom,
    ! [X: sum_sum_nat_nat] : nat_sum_encode(X) = sum_ca6763686470577984908at_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aTP_Lamm_bx,X) ).

% sum_encode_def
tff(fact_7752_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero_nat,J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_7753_upt__merge,axiom,
    ! [I2: nat,J: nat,K3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K3) )
     => ( append_nat(upt(I2,J),upt(J,K3)) = upt(I2,K3) ) ) ).

% upt_merge
tff(fact_7754_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B: nat] : map_nat_nat(aTP_Lamm_by(Ofs),upt(A2,B)) = upt(aa_nat_nat(plus_plus_nat(A2),Ofs),aa_nat_nat(plus_plus_nat(B),Ofs)) ).

% map_add_upt'
tff(fact_7755_upt__eq__append__conv,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( upt(I2,J) = append_nat(Xs,Ys) )
      <=> ? [K5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K5)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),J)
            & ( upt(I2,K5) = Xs )
            & ( upt(K5,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_7756_upt__append,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( append_nat(upt(zero_zero_nat,I2),upt(I2,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_7757_butlast__upt,axiom,
    ! [M4: nat,N2: nat] : butlast_nat(upt(M4,N2)) = upt(M4,aa_nat_nat(minus_minus_nat(N2),one_one_nat)) ).

% butlast_upt
tff(fact_7758_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_7759_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: 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(P,I3) )
       => ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U4)) ) ) ) ).

% upt_filter_extend
tff(fact_7760_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,As2: set_nat] :
              ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
               => ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),As2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_hea5848002036693495265it_nat(lim_Product_unit,H)) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_7761_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,As2: set_nat] :
              ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( ( (Y)
                <=> ! [X2: nat] :
                      ( aa_set_nat_o(member_nat(X2),As2)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),aa_hea5848002036693495265it_nat(lim_Product_unit,H)) ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_7762_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,As2: set_nat] :
              ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
               => ~ ! [X4: nat] :
                      ( aa_set_nat_o(member_nat(X4),As2)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_hea5848002036693495265it_nat(lim_Product_unit,H)) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_7763_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,As2: set_nat] :
              ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
               => ( As2 = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_7764_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,As2: set_nat] :
              ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2))
               => ( As2 != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_7765_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,As2: set_nat] :
              ( ( X = aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2) )
             => ( ( (Y)
                <=> ( As2 = bot_bot_set_nat ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),aa_set3979848162462698438et_nat(produc7507926704131184380et_nat(H),As2)) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_7766_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_bg)) ).

% times_int_def
tff(fact_7767_next_Osimps,axiom,
    ! [V: code_natural,W: code_natural] :
      aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V,W)) = $let(
        v: code_natural,
        v:= minus_shift(numera5444537566228673987atural(bit1(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(V,numera5444537566228673987atural(bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(V,numera5444537566228673987atural(bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit0(one)))))))))))))))),
        $let(
          w: code_natural,
          w:= minus_shift(numera5444537566228673987atural(bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(W,numera5444537566228673987atural(bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(W,numera5444537566228673987atural(bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(one)))))))))))))),
          aa_Pro7165575656943818355atural(produc6639722614265839536atural(plus_p4538020629002901425atural(minus_shift(numera5444537566228673987atural(bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),v,plus_p4538020629002901425atural(w,one_one_Code_natural)),one_one_Code_natural)),produc3574140220909816553atural(v,w)) ) ) ).

% next.simps
tff(fact_7768_Random_Orange__def,axiom,
    ! [K3: code_natural] : range(K3) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),K3),aTP_Lamm_ca,one_one_Code_natural),aTP_Lamm_cb(K3)) ).

% Random.range_def
tff(fact_7769_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
tff(fact_7770_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(sup_sup_Product_unit,Uu),Uv) = product_Unity ).

% sup_unit_def
tff(fact_7771_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
tff(fact_7772_Inf__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple2584293577114468500t_unit(Uu) = product_Unity ).

% Inf_unit_def
tff(fact_7773_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Uu) = product_Unity ).

% uminus_unit_def
tff(fact_7774_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
tff(fact_7775_Unity__def,axiom,
    product_Unity = aa_o_Product_unit(product_Abs_unit,$true) ).

% Unity_def
tff(fact_7776_Sup__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple4687483117567863418t_unit(Uu) = product_Unity ).

% Sup_unit_def
tff(fact_7777_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(inf_inf_Product_unit,Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_7778_UNIV__unit,axiom,
    top_to1996260823553986621t_unit = aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,product_Unity),bot_bo3957492148770167129t_unit) ).

% UNIV_unit
tff(fact_7779_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
tff(fact_7780_cone__def,axiom,
    bNF_Cardinal_cone = bNF_Ca7083678892733797993t_unit(aa_set8324764978855220220t_unit(aa_Pro136467997921847252t_unit(insert_Product_unit,product_Unity),bot_bo3957492148770167129t_unit)) ).

% cone_def
tff(fact_7781_times__integer__code_I3_J,axiom,
    ! [M4: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_Pos(M4)),code_Pos(N2)) = code_Pos(times_times_num(M4,N2)) ).

% times_integer_code(3)
tff(fact_7782_times__integer__code_I6_J,axiom,
    ! [M4: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_Neg(M4)),code_Neg(N2)) = code_Pos(times_times_num(M4,N2)) ).

% times_integer_code(6)
tff(fact_7783_times__integer__code_I5_J,axiom,
    ! [M4: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_Neg(M4)),code_Pos(N2)) = code_Neg(times_times_num(M4,N2)) ).

% times_integer_code(5)
tff(fact_7784_times__integer__code_I4_J,axiom,
    ! [M4: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_Pos(M4)),code_Neg(N2)) = code_Neg(times_times_num(M4,N2)) ).

% times_integer_code(4)
tff(fact_7785_times__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] : aa_Code_natural_nat(code_nat_of_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Xa)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_natural_nat(code_nat_of_natural,X)),aa_Code_natural_nat(code_nat_of_natural,Xa)) ).

% times_natural.rep_eq
tff(fact_7786_times__int__code_I3_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,pos(M4)),pos(N2)) = pos(times_times_num(M4,N2)) ).

% times_int_code(3)
tff(fact_7787_times__int__code_I4_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,pos(M4)),neg(N2)) = neg(times_times_num(M4,N2)) ).

% times_int_code(4)
tff(fact_7788_times__int__code_I5_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,neg(M4)),pos(N2)) = neg(times_times_num(M4,N2)) ).

% times_int_code(5)
tff(fact_7789_times__int__code_I6_J,axiom,
    ! [M4: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,neg(M4)),neg(N2)) = pos(times_times_num(M4,N2)) ).

% times_int_code(6)
tff(fact_7790_times__natural_Otransfer,axiom,
    bNF_re1400205106991740995atural(code_pcr_natural,bNF_re3704215830270325841atural(code_pcr_natural,code_pcr_natural),times_times_nat,times_2397367101498566445atural) ).

% times_natural.transfer
tff(fact_7791_times__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,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(times_times_nat,Xa),X)) ).

% times_natural.abs_eq
tff(fact_7792_type__definition__natural,axiom,
    aa_set_nat_o(type_d4410041424927559462al_nat(code_nat_of_natural,code_natural_of_nat),top_top_set_nat) ).

% type_definition_natural
tff(fact_7793_times__natural__def,axiom,
    times_2397367101498566445atural = map_fu7613509917203888380atural(code_nat_of_natural,map_fu1239815594074539274atural(code_nat_of_natural,code_natural_of_nat),times_times_nat) ).

% times_natural_def
tff(fact_7794_times__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),times_times_int,times_3573771949741848930nteger) ).

% times_integer.transfer
tff(fact_7795_times__integer__def,axiom,
    times_3573771949741848930nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),times_times_int) ).

% times_integer_def
tff(fact_7796_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_be,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_1
tff(fact_7797_ATP_Olambda__2,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_ca,Uu) = produc5538323210962509403atural(next,aTP_Lamm_bz(Uu)) ).

% ATP.lambda_2
tff(fact_7798_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bx,Uu) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uu)) ).

% ATP.lambda_3
tff(fact_7799_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_an,Uu) = aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_am,Uu)) ).

% ATP.lambda_4
tff(fact_7800_ATP_Olambda__5,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bw,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_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_5
tff(fact_7801_ATP_Olambda__6,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_bb,Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_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_6
tff(fact_7802_ATP_Olambda__7,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_bd,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_Code_integer_int(code_int_of_integer,Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(plus_plus_int(l2),one_one_int)) ) ).

% ATP.lambda_7
tff(fact_7803_ATP_Olambda__8,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_bc,Uu),Uua) = aa_int7462565939943104719nt_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_8
tff(fact_7804_ATP_Olambda__9,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ar(Uu),Uua) = aa_nat_nat(power_power_nat(aa_nat_nat(binomial(Uu),Uua)),numeral_numeral_nat(bit0(one))) ).

% ATP.lambda_9
tff(fact_7805_ATP_Olambda__10,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_al,Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uua) ) ) ).

% ATP.lambda_10
tff(fact_7806_ATP_Olambda__11,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_ah,Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_11
tff(fact_7807_ATP_Olambda__12,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_as(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(binomial(Uu),Uua)) ).

% ATP.lambda_12
tff(fact_7808_ATP_Olambda__13,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ax(Uu),Uua) = aa_nat_nat(minus_minus_nat(Uua),Uu) ).

% ATP.lambda_13
tff(fact_7809_ATP_Olambda__14,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_az,Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uua),Uu) ).

% ATP.lambda_14
tff(fact_7810_ATP_Olambda__15,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_by(Uu),Uua) = aa_nat_nat(plus_plus_nat(Uua),Uu) ).

% ATP.lambda_15
tff(fact_7811_ATP_Olambda__16,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_am,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_16
tff(fact_7812_ATP_Olambda__17,axiom,
    ! [Uu: set_Product_unit,Uua: product_unit] :
      ( aa_Product_unit_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set8780210009671629607unit_o(member_Product_unit(Uua),Uu) ) ).

% ATP.lambda_17
tff(fact_7813_ATP_Olambda__18,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_ac(Uu),(Uua))
    <=> aa_set_o_o(member_o((Uua)),Uu) ) ).

% ATP.lambda_18
tff(fact_7814_ATP_Olambda__19,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_19
tff(fact_7815_ATP_Olambda__20,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_20
tff(fact_7816_ATP_Olambda__21,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set5091322549519198405at_o_o(member7544390268018397336_nat_o(Uua),Uu) ) ).

% ATP.lambda_21
tff(fact_7817_ATP_Olambda__22,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bh(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_22
tff(fact_7818_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_bg,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_bf(Uu),Uua)) ).

% ATP.lambda_23
tff(fact_7819_ATP_Olambda__24,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_bz(Uu),Uua) = produc6639722614265839536atural(plus_p4538020629002901425atural(Uua,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Uu),numera5444537566228673987atural(bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_24
tff(fact_7820_ATP_Olambda__25,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_cb(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_25
tff(fact_7821_ATP_Olambda__26,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ay(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Uua)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uub),numera6620942414471956472nteger(Uu))),aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Uua)),Uub)) ).

% ATP.lambda_26
tff(fact_7822_ATP_Olambda__27,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_au(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(plus_plus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)),aa_nat_nat(minus_minus_nat(Uub),numeral_numeral_nat(Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),Uub)) ).

% ATP.lambda_27
tff(fact_7823_ATP_Olambda__28,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_av(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua)),one_one_int)),aa_int_int(minus_minus_int(Uub),numeral_numeral_int(Uu))),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua)),Uub)) ).

% ATP.lambda_28
tff(fact_7824_ATP_Olambda__29,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ba(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,aa_Cod6977894654539884864nteger(produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),zero_z3403309356797280102nteger),aa_Cod6977894654539884864nteger(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_29
tff(fact_7825_ATP_Olambda__30,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bv(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bu(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_30
tff(fact_7826_ATP_Olambda__31,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bs(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_31
tff(fact_7827_ATP_Olambda__32,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bq(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_32
tff(fact_7828_ATP_Olambda__33,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bp(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bo(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_33
tff(fact_7829_ATP_Olambda__34,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bn(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bm(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_34
tff(fact_7830_ATP_Olambda__35,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bl(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bk(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_35
tff(fact_7831_ATP_Olambda__36,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_36
tff(fact_7832_ATP_Olambda__37,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_37
tff(fact_7833_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ak(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_38
tff(fact_7834_ATP_Olambda__39,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        | aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_39
tff(fact_7835_ATP_Olambda__40,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_ae(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        & aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_40
tff(fact_7836_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_at(Uu),Uua),Uub) = aa_nat_nat(plus_plus_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_41
tff(fact_7837_ATP_Olambda__42,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(plus_plus_nat(Uub),Uua)) ) ).

% ATP.lambda_42
tff(fact_7838_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_bf(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_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(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_43
tff(fact_7839_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aq(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(binomial(Uub),Uuc))),aa_nat_nat(power_power_nat(Uu),Uuc))),aa_nat_nat(power_power_nat(Uua),aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_44
tff(fact_7840_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ao(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Uu),Uuc)),aa_nat_nat(binomial(Uua),aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_45
tff(fact_7841_ATP_Olambda__46,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bs(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_46
tff(fact_7842_ATP_Olambda__47,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_bq(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_47
tff(fact_7843_ATP_Olambda__48,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bk(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_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_48
tff(fact_7844_ATP_Olambda__49,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bm(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_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_49
tff(fact_7845_ATP_Olambda__50,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bo(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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_50
tff(fact_7846_ATP_Olambda__51,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bu(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_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_51
tff(fact_7847_ATP_Olambda__52,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_52
tff(fact_7848_ATP_Olambda__53,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aw,Uu) = Uu ).

% ATP.lambda_53
tff(fact_7849_ATP_Olambda__54,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ag,Uu)
    <=> $false ) ).

% ATP.lambda_54
tff(fact_7850_ATP_Olambda__55,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_bj,Uu)
    <=> $false ) ).

% ATP.lambda_55

% Helper facts (10)
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__Product____Type__Ounit_T,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( X != Y )
      | aa_Product_unit_o(aa_Pro4895447120693452255unit_o(fequal_Product_unit,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Ounit_T,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ~ aa_Product_unit_o(aa_Pro4895447120693452255unit_o(fequal_Product_unit,X),Y)
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__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_T,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( ( X != Y )
      | aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__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_T,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( ~ aa_fun1675921005893230629at_o_o(aa_fun3380769697021226975at_o_o(fequal1890233700291625924_nat_o,X),Y)
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_fun5526454687433275355o_assn(abs_assn,one_assn_raw)),h)
  <=> ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,h) = bot_bot_set_nat ) ) ).

%------------------------------------------------------------------------------